Research ArticleAnalysis of Stochastic Nicholson-Type Delay System underMarkovian Switching on Patches

Xin Yi and Guirong Liu

School of Mathematical Sciences Shanxi University Taiyuan Shanxi 030006 China

Correspondence should be addressed to Guirong Liu lgr5791sxueducn

Received 28 February 2020 Revised 15 April 2020 Accepted 24 April 2020 Published 23 May 2020

Guest Editor Baltazar Aguirre Hernandez

Copyright copy 2020 Xin Yi and Guirong Liu is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Based on the influence of random environmental perturbations and the patch structure we propose a stochastic Nicholson-typedelay system under Markovian switching on patches Existence of a global positive solution is studied en we show ultimateboundedness and estimation of the sample Lyapunov exponent of the solution Furthermore sufficient conditions for extinctionof species are established which is the main new ingredient of this paper Finally some numerical examples are presented Ourresults improve and generalize previous related results

1 Introduction

In 1980 Gurney et al [1] established Nicholsonrsquos blowfliesequation according to experimental data of Nicholson [2] Inrecent decades there have been a large amount of resultsrelated to the dynamical behaviors for this model and itsmodification see [3ndash13]

In ecosystems the pattern of complex population dy-namics is inevitably subject to some kind of environmentalnoises As a matter of fact the phenomenon of stochasticityplays a critical role in understanding the evolutionary dy-namics and ecological characteristics of species ParticularlyMay [14] has revealed that due to environmental fluctua-tions the parameters in a system should be stochasticEnvironmental noises are classified into two categories thefirst is white noise and the second one is coloured noiseStochastic population models [15ndash20] are more realisticcompared to deterministic population models Wang et al[21] first studied a scalar stochastic Nicholsonrsquos blowfliesdelayed equation

dx(t) minus αx(t) + px(t minus τ)eminus cx(tminus τ)

1113960 1113961dt + σx(t)dB(t) (1)

Notice however that white noise is unable to depict thephenomena that the species may be invaded by the alien

population [22] or suffer sudden catastrophic shocks [23]And in recent years some significant progress has beenmade in the theory of the stochastic population models withregime switching see [24ndash27] and the references therein In[28] Zhu et al considered a stochastic equation withMarkovian switching

dx(t) minus αrtx(t) + prt

x t minus τrt1113872 1113873e

minus crtx tminus τrt( 1113857

1113876 1113877dt + σrtx(t)dB(t)

(2)

where continuous-time Markov chain rt1113864 1113865tge0 is defined on astate space S 1 2 m

On the contrary migration is a ubiquitous phenomenonin the nature Both continuous reaction-diffusion modelsand discrete patchy systems could incorporate and explainthe phenomenology of spatial dispersion [29] in the liter-ature of mathematical ecology Objectively speaking patch-structured models illustrate the spatial heterogeneity ofspecies depending on a lot of factors such as ecologicalsystems in different geographic types (eg nature reservesand other regions) various food-rich patches of habitatsand many other circumstances Besides models in thepatchy environment include disease systems as well such asthe two-compartment model of the cancer cell populationIn order to take the dispersal phenomenon into

HindawiComplexityVolume 2020 Article ID 9078471 13 pageshttpsdoiorg10115520209078471

consideration Berezansky et al [30] introduced the Nich-olson-type delay system on patches as follows

x1prime(t) minus a1x1(t) minus b2x1(t) + b1x2(t) + p1x1(t minus τ)eminus c1x1(tminus τ)

x2prime(t) minus a2x2(t) minus b1x2(t) + b2x1(t) + p2x2(t minus τ)eminus c2x2(tminus τ)

⎧⎨

⎩ (3)

which includes the novel two-compartment models ofleukemia dynamics and the systems of marine protectedareas

In particular considering that the parameters ai ofsystem (3) are affected by the white noise Yi and Liu [31]formulated the stochastic diffusion system which consists oftwo patches

dx1(t) minus a1x1(t) minus b2x1(t) + b1x2(t) + p1x1(t minus τ)eminus c1x1(tminus τ)1113960 1113961dt + σ1x1(t)dB1(t)

dx2(t) minus a2x2(t) minus b1x2(t) + b2x1(t) + p2x2(t minus τ)eminus c2x2(tminus τ)1113960 1113961dt + σ2x2(t)dB2(t)

⎧⎪⎨

⎪⎩(4)

We can further model random shift in different regimesby a continuous-time Markov chain ℓ(t) tge0 defined on astate space M 1 2 N Let ℓ(t) tge0 be right-con-tinuous and Γ (ρij)NtimesN be its generator of ℓ(t) tge0 ie

P ℓ(t + δ) j ∣ ℓ(t) i1113864 1113865 ρijδ + o(δ) if jne i

1 + ρiiδ + o(δ) if j i1113896

(5)

where δ gt 0 ρij ge 0 for ine j and 1113936jisinMρij 0 i j isinMSuppose that ℓ(t) tge0 is irreducible and has the uniquestationary distribution π (π1 π2 πN) Hence weobtain the stochastic Nicholson-type system under Mar-kovian switching on the patch structure as follows

dx1(t) minus a1(ℓ(t))x1(t) minus b2(ℓ(t))x1(t) + b1(ℓ(t))x2(t) + p1(ℓ(t))x1(t minus τ(ℓ(t)))eminus c1(ℓ(t))x1(tminus τ(ℓ(t)))1113960 1113961dt

+σ1(ℓ(t))x1(t)dB1(t)

dx2(t) minus a2(ℓ(t))x2(t) minus b1(ℓ(t))x2(t) + b2(ℓ(t))x1(t) + p2(ℓ(t))x2(t minus τ(ℓ(t)))eminus c2(ℓ(t))x2(tminus τ(ℓ(t)))1113960 1113961dt

+σ2(ℓ(t))x2(t)dB2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(6)

with initial conditions

x(t) φ(t) φ1(t)φ2(t)( 1113857T t isin [minus τ 0] ℓ(0) ℓ0 isinM

(7)

where φh isin C([minus τ 0] [0 +infin)) and φh(0)gt 0 for h 1 2and τ maxiisinM τ(i)

We focus on the meaning of parameters with respect tofish population in marine protected area A1 and fishing areaA2 x1(t) and x2(t) are the number of fish populations in A1and A2 respectively for h 1 2 and i isinM a1(i) and a2(i)

are themortality rate inA1 andA2 respectively let G(xh(t minus

τ(i))) ph(i)xh(t minus τ(i))eminus ch(i)xh(tminus τ(i)) be the fish growthrates p1(i) and p2(i) represent the maximum per adultyearly birth rate in A1 and A2 respectively ch(i)gt 0 1c1(i)

and 1c2(i) are the number at which the reproduction attheir maximum birth rate in A1 and A2 respectively τ(i) isthe maturation time Bh(t) is the standard Brownian motiondefined on the complete probability space (ΩFP) andσh(i)ge 0 for any i isinM and h 1 2 We assume ℓ(t) t⩾0 is

Ft-adapted Nevertheless suppose ℓ(t) t⩾0 and Bh(t) areindependent of each other h 1 2

Especially system (6) can reduce to the model in [32] ifτ(i) equiv τ i isinM By contrast our work differs from andimproves [32] which will be depicted further in detail

In the field of ecology it is important to use mathematicsto study extinction of species see [33 34] and the referencestherein However no work has yet been done on theproblem of extinction for scalar equation (1) not to mentionthe scalar equation withMarkovian switching (2) and system(4) In order to prove the extinction of species the con-ventional method is to construct a proper Lyapunovfunction or functional and then estimate the upper bound ofthe drift term of its It1113954o differential Taking system (6) forexample x1(t) and x2(t) are likely to appear in the de-nominator of the expression of LV and coefficients in frontof them are positive for a general Lyapunov functionV(x1 x2) Unfortunately this leads to some difficulties infinding the upper bound of LV So based on this we give anew method for investigating extinction of species

2 Complexity

Especially system (6) reduces to (1) (2) (4) or the system in[32] when parameters of system (6) assume some specialvaluesat is to say we have derived extinction of the abovesystems at the same time

In this paper system (6) is more general than the modelof [21 28 30ndash32] In addition our results improve andgeneralize the corresponding results in these literaturestudies

e remainder of this paper is built up as follows InSection 2 we show the global existence of almost surelypositive solution e asymptotic estimates for the solutionstochastically ultimate boundedness and boundedness forthe average in time of the θth moment of the solution arethen constructed in Section 3 In Section 4 we discuss thepathwise properties of the solution Sufficient conditions forextinction of species are obtained in Section 5 Numericalinvestigations are then given in Section 6 e last part is aconclusion

2 Preliminary Results

To simplify denote the solution of (6) with initial values (7)

x(t) ≔ x tφ ℓ0( 1113857 (8)

where x(t) (x1(t) x2(t))T Let

β1(i) a1(i) + b2(i)

β2(i) a2(i) + b1(i) i isinM(9)

We denote R+ (0 +infin) R2+ (x1 x2)

T isin1113966

R2 x1 gt 0 x2 gt 0 and R2times2+ (wuv)2times2 isin R2times2 wuv gt1113864

0 u v 1 2 For any Φ M⟶ R let 1113954Φ miniisinMΦ(i)

and Φ maxiisinMΦ(i) Let |middot| denote Euclidean norm in R2Denote the trace norm |A|

trace(ATA)

1113968for matrix A

Lemma 1 Given any initial values (7) system (6) has aunique solution x(t) isin R2

+ for all t isin [minus τinfin) almost surely

Proof We omit the proof since it is analogous to that of [31]by making use of the generalized It1113954o formula (see egeorem 145 in [35]) to 1113936

2h1(xh minus 1 minus logxh)

Remark 1 e delay stochastic Nicholson-type modelunder regime switching on patches (6) is a direct ex-tension of the models in [21 28 30ndash32] From Lemma 1 itis worthy to point out that priori conditions αgt σ22 in[21] are unnecessary erefore Lemma 1 improves andgeneralizes Lemma 22 in [21] In addition this lemmashows that both white noise and telegraph noise will notdestroy a great property that the solution of (3) does notexplode

3 Boundedness

Because of resource constraints asymptotic boundedness isthe core of the research in ecosystems And it is the mainpurpose of the present section For simplicity we use thefollowing notations For any i isinM denote

K1(θ i) ≔ a1(i) minusθ minus 1θ

b1(i) minus b2(i)( 1113857 minus12

(θ minus 1)σ21(i) minus (θ minus 1)

K2(θ i) ≔ a2(i) minusθ minus 1θ

b2(i) minus b1(i)( 1113857 minus12

(θ minus 1)σ22(i) minus (θ minus 1)

Hh(θ i) ≔ θ middot supxhisinR+

minus Kh(θ i)xθh +

ph(i)

ch(i)exθminus 1h1113896 1113897

Hh(θ) ≔ maxiisinMHh(θ i)

A1(θ) ≔2θ2

θ(θ minus 1)1113944iisinM

πi H1(θ i) + H2(θ i)1113858 1113859

A2(θ) ≔1113936

2h1

Hh(θ)

θ(θ minus 1)

A3(θ) ≔ 2θ2 middot1113936

2h1

H(θ)

θ(θ minus 1) h 1 2

(10)

Firstly inspired by the work of Wang and Chen [32] wegive this theorem

Theorem 1 Let θgt 1 such that Kh(θ i)gt 0 h 1 2 i isinMGiven any initial values (7) solution (x1(t) x2(t)) of (6)satisfies

limsupt⟶infin

1t

1113946t

0E xh(s)

11138681113868111386811138681113868111386811138681113868θdsleA1(θ) h 1 2 (11)

andlimsupt⟶infin

E xθ1(t) + x

θ2(t)1113872 1113873leA2(θ) (12)

In particular

limsupt⟶infin

E|x(t)|θ leA3(θ) (13)

at is system (6) is ultimately bounded

Proof Define

V1 x1 x2( 1113857 xθ1 + x

θ2 (14)

e generalized It1113954o formula together with the factph(i)yheminus ch(i)yh le (ph(i)ch(i)e) and the elementary in-equality AεB1minus ε leAε + B(1 minus ε) for any A Bge 0 andε isin [0 1] yields

Complexity 3

LV1 x1 x2 y1 y2 i( 1113857

minus θa1(i)xθ1 minus θb2(i)x

θ1 + θb1(i)x

θminus 11 x2 +

12θ(θ minus 1)σ21(i)x

θ1

minus θa2(i)xθ2 minus θb1(i)x

θ2 + θb2(i)x1x

θminus 12 +

12θ(θ minus 1)σ22(i)x

θ2

+ θ 11139442

h1ph(i)x

θminus 1h yhe

minus ch(i)yh

le θ minus a1(i) +θ minus 1θ

b1(i) minus b2(i)( 1113857 +12

(θ minus 1)σ21(i) +(θ minus 1)1113890 1113891xθ11113888

+ minus a2(i) +θ minus 1θ

b2(i) minus b1(i)( 1113857 +12

(θ minus 1)σ22(i) +(θ minus 1)1113890 1113891xθ2

+ 11139442

h1

ph(i)

ch(i)exθminus 1h

⎞⎠ minus θ(θ minus 1) xθ1 + x

θ21113872 1113873

le 11139442

h1θ minus Kh(θ i)x

θh +

ph(i)

ch(i)exθminus 1h1113890 1113891 minus θ(θ minus 1)V1 x1 x2( 1113857

le 11139442

h1Hh(θ i) minus θ(θ minus 1)V1 x1(t) x2(t)( 1113857

(15)

erefore for tgt 0

dV1 x1(t) x2(t)( 1113857le 11139442

h1Hh(θ ℓ(t)) minus θ(θ minus 1)V1

⎡⎣

middot x1(t) x2(t)( 1113857⎤⎦dt + 1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

(16)

en (16) implies

0leE V1 x1 x2( 1113857( 1113857leφθ1(0) + φθ

2(0) + 1113946t

01113944

2

h1Hh(θ ℓ(s))ds

minus θ(θ minus 1) 1113946t

0EV1 x1(s) x2(s)( 1113857ds

(17)

Noting that the Markov chain ℓ(t) has an invariantdistribution π (πii isinM) and applying the ergodicproperty of the Markov chain it yields

limsupt⟶infin

1t

1113946t

0EV1 x1(s) x2(s)( 1113857dsle limsup

t⟶infin

1θ(θ minus 1)

middot1tV1 φ1(0)φ2(0)( 1113857 +

1t

1113946t

01113944

2

h1Hh(θ ℓ(s))ds⎛⎝ ⎞⎠

le1

θ(θ minus 1)1113944iisinM

πi H1(θ i) + H2(θ i)1113858 1113859 a s

(18)

Furthermore we have

d eθ(θminus 1)t

V1 x1(t) x2(t)( 11138571113960 1113961le eθ(θminus 1)t

1113944

2

h1Hh(θ ℓ(t))dt + e

θ(θminus 1)t1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

le 11139442

h1

Hh(θ)eθ(θminus 1)tdt + e

θ(θminus 1)t1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

(19)

Hence

eθ(θminus 1)t

EV1 x1(t) x2(t)( 1113857leV1 φ1(0)φ2(0)( 1113857

+1113936

2h1

Hh(θ)

θ(θ minus 1)eθ(θminus 1)t

minus1113936

2h1

Hh(θ)

θ(θ minus 1)

(20)

Consequently we infer immediately that (12) holds Onthe contrary according to (12) (18) and the fact that

|x|θ le 2θ2 max x

θ1 x

θ21113966 1113967le 2θ2V1(x) (21)

it follows that (11) and (13) hold e proof is thereforecomplete

Remark 2 Ineorem 1 the parameter θ is greater than 1 inthe result Although ultimate boundedness in the θth mo-ment was derived for θ restricted to the precondition θ gt 1θth moment of system (6) can be obtained when θle 1 byHolderrsquos equality

Remark 3 Without regime switching or without migrationand regime switching eorem 1 improves the corre-sponding results in [21 31] If τ(i) equiv τ system (6) is a directextension of the model in [32] Besides no proof of ultimateboundedness in the pth moment is given in [32] which is

4 Complexity

shown in eorem 1 erefore this theorem extends andimproveseorem 31 in [21]eorem 22 in [28]eorem33 in [31] and eorem 32 in [32]

Theorem 2 Given any initial values (7) solution(x1(t) x2(t)) of (6) satisfies

limsupt⟶infin

E|x(t)|le limsupt⟶infin

E x1(t) + x2(t)1113858 1113859le1113957p1

1113954c1eλ+

1113957p21113954c2eλ

(22)

where λ min 1113954a1 1113954a21113864 1113865 7at is (6) is ultimately bounded inmean

Proof Let V1(t x1 x2) eλt(x1 + x2) en

E x1(t) + x2(t)( 1113857le eminus λt

V1 0φ1(0)φ2(0)( 1113857

+p11113954c1e

+p21113954c2e

1113888 1113889 1113946t

0e

(sminus t)λds

(23)

Finally (22) follows by letting t⟶infin e proof istherefore complete

Remark 4 Compared with eorem 1 this theorem de-scribes the case that θ 1 which does not require anyconditions If τ(i) equiv τ we get (p11113954c1eλ) + (p21113954c2eλ)le (ca)where (ca) is defined in [32] So this theorem improves andextends eorem 31 in [21] and eorem 31 in [32]

Theorem 3 System (6) is stochastically ultimately bounded

Proof By (22) we derive

limsupt⟶infin

E xh(t)1113868111386811138681113868

1113868111386811138681113868lep1

1113954c1eλ+

p21113954c2eλ

h 1 2 (24)

By the Chebyshev inequality it yields for any ε isin (0 1)

limsupt⟶infin

P xh(t)geH1113864 1113865leHminus 1 p1

1113954c1eλ+

p21113954c2eλ

1113888 1113889 ε (25)

where H (1ε)((1113957p11113954c1eλ) + (1113957p21113954c2eλ)) e proof istherefore complete

Remark 5 eorem 3 can be seen as the extension andimprovement of [31 32]

4 Asymptotic Pathwise Estimation

We shall estimate a sample Lyapunov exponent in whatfollows

Lemma 2 If a isin R and b isin R+ then (ax2 + bx1+

x2)leK(a) for x isin R where K(a) (a +a2 + b2

radic2)

By the properties of quadratic functions the proof of thislemma is easy and so is omitted In the process of findingK(a) we know that the precondition is a minus K(a)lt 0 In thiscase we can choose K(a) which satisfies K(a)

(a +a2 + b2

radic2) We have to mention that it has no relation

with the sign of parameter a If alt 0 we get(a +

a2 + b2

radic2)lt minus (b24a) by simple computation So

this lemma is an improvement of Lemma 12 in [28] andLemma 21 in [32]

Theorem 4 Given any initial values (7) solution x(t) of (6)satisfies

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

1tlog|x(t)|le

Q

2 a s (26)

where h 1 2 Q maxiisinM minεisinR+[Q1(i ϵ) + Q2(i ϵ)]1113966 1113967

with

Q1(i ε)

2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 11138592

+ 4 p1(i)c1(i)e( 11138572

1113969

minus 2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 1113859

2

Q2(i ε)

2β2(i) minus σ22(i) minus b1(i) + b2(i)ε( 11138572

+ 4 p2(i)c2(i)e1113857(2

1113969

minus 2β2(i) minus σ22(i) minus b1(i) + b2(i)( 1113857ε( 1113857

2

(27)

Complexity 5

for any positive constant ε Proof e generalized It1113954o formula together with Lemma 2and the CauchyndashSchwarz inequality yields

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ 1113946t

0

minus 2β1(ℓ(s)) minus σ21(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s))( 1113857 isin1113858 1113859x21(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x1(s)

1 + x21(s) + x2

2(s)ds

+ 1113946t

0

minus 2β2(ℓ(s)) minus σ22(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s)) isin( 11138571113858 1113859x22(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x2(s)

1 + x21(s) + x2

2(s)ds

minus 2 11139442

h1σ2h(ℓ(s)) 1113946

t

0

x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

le log 1 + x21(0) + x

22(0)1113872 1113873 + Qt minus 2 1113944

2

h11113946

t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

(28)

where for any h isin 1 2

Mh(t) 21113946t

0

σh(ℓ(s))x2h(s)

1 + x21(s) + x2

2(s)dBh(s) (29)

with the quadratic variation

langMh(t) Mh(t)rang 41113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds (30)

According to the exponential martingale inequality (seeeg [36]) for any integer mgt 0 we have

P sup0letlem

Mh(t) minus 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds

⎧⎨

⎩

⎫⎬

⎭

⎧⎨

⎩

gt 2 logm⎫⎬

⎭ le1

m2 h 1 2

(31)

Since 1113936infinm1 1m2 ltinfin and BorelndashCantellirsquos lemma (see

eg [36]) there existΩ0 isin F with P(Ω0) 1 and an integerm0 m0(ω) such that

Mh(t)le 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds + 2 logm

h 1 2

(32)

for all ω isin Ω0 0le tlem Substituting the above inequalityinto (28) for any ω isin Ω0 mgem0 0le tlem we have

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ Qt + 4 logm(33)

which yields

1tlog 1 + x

21(t) + x

22(t)1113872 1113873le

1m minus 1

log 1 + x21(0) + x

22(0)1113872 11138731113960

+ Qm + 4 logm1113859

(34)

for all ω isin Ω0 0lem minus 1le tlem mgem0 Letting m⟶infinand using the inequality yle (12)(1 + y2) for anyy isin (minus infin +infin) we obtain

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

12(m minus 1)

log 1 + x2(0)1113872 11138731113960

+ Qm + 4 logm1113859 Q

2 a s

(35)e proof is therefore complete

Remark 6 Without migrations we get

Q1(i) (

[2β1(i) minus σ21(i)]2 + 4(p1(i)c1(i)e)21113969

minus [2β1(i) minus

σ21(i)]2) By comparison we find thatQ1(i)leCi where Ci isdefined in [28] In addition without migration and regimeswitching we can get Q ineorem 4 is less than K where K

is defined in [21] Furthermore the condition 2α1 minus σ21 minus

(b1 + b2)ϵgt 0 2α2 minus σ22 minus (b1 + b2)ϵgt 0 in [31] meansthat the parameter ϵ needs to be satisfied(b1 + b2)2α2 minus σ22 lt εlt 2α1 minus σ21(b1 + b2) However weknow that this condition is unnecessary from the abovetheorem Despite all this if we let the parameter ε satisfyε isin (((b1 + b2)2α2 minus σ2) (2α21 minus σ21(b1 + b2))) we computethat Q in eorem 4 is less than Q in [31] erefore theabove work is a promotion of eorem 41 in [21] eorem22 in [28] and eorem 41 in [31]

6 Complexity

5 Extinction

Sufficient conditions for extinction are the subject of thissection Unless otherwise stated we hypothesizeτ(i) equiv τ i isinM in this section We first rewrite (6) asfollows

dx(t) f1(x(t) x(t minus τ) ℓ(t))dt + f2(x(t) ℓ(t))dB(t)

(36)

where the operator f1 R2+ times R2

+ times M⟶ R2+ is defined as

f1(x y i) minus β1(i)x1 + b1(i)x2 + p1(i)y1e

minus c1(i)y1

minus β2(i)x2 + b2(i)x1 + p2(i)y2eminus c2(i)y2

⎛⎝ ⎞⎠

(37)

the operator f2 R2+ times M⟶ R2times2

+ is defined as

f2(x i) σ1(i)x1 0

0 σ2(i)x21113888 1113889 and dB(t)

dB1(t)

dB2(t)1113888 1113889

We first note that

f1(0 0 i) equiv 0

f2(0 i) equiv 0(38)

for i isinM whence (6) admits a trivial solution corre-sponding to φ(0) 0

Before our result we give a lemma

Lemma 3 For system (36) the terms f1(x y i) and f2(x i)

are locally bounded in (x y) while uniformly bounded in i7at is for any mgt 0 there is Km gt 0 satisfying

f1(x y i)1113868111386811138681113868

1113868111386811138681113868 or f2(x i)1113868111386811138681113868

1113868111386811138681113868leKm (39)

for all i isinM x y isin R2+ with |x| or |y|lem

e proof is not particularly difficult so we omit theproof

Theorem 5 Assume that

21113954β1 gt σ21 + b1 +(1 +2

radic)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

21113954β2 gt σ22 +(1 +2

radic)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(40)

en the solution of (36) satisfies limt⟶infinx(t) 0 asfor any initial values (7) at is all populations in system(36) go to extinction with probability one

Proof

Step 1 let

V2 x1 x2( 1113857 xT

112

radic

12

radic 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠x x

21 +

2

radicx1x2 + x

22

(41)

Obviously V2 is positive-definite and radially un-bounded at is

lim|x|⟶infin

V2 x1 x2( 1113857 infin (42)

e generalized It1113954o formula yields

V2 x1(t) x2(t)( 1113857 V2 φ1(0)φ2(0)( 1113857 + 1113946t

0LV2(x(s) x

middot (s minus τ(ℓ(s))) ℓ(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(43)

By computation we know

LV2(x y i)le minus 2β1(i) minus σ21(i) minus b1(i) minus (1 +2

radic)b2(i) minus p1(i) minus

12

radic p2(i)1113890 1113891x21

minus 2β2(i) minus σ22(i) minus (1 +2

radic)b1(i) minus b2(i) minus

12

radic p1(i) minus p2(i)1113890 1113891x22

+ 1 +12

radic1113888 1113889p1(i)y21 + 1 +

12

radic1113888 1113889p2(i)y22

le minus λ1x21 minus λ2x

22 + 1 +

12

radic1113888 1113889p1y21 + 1 +

12

radic1113888 1113889p2y22

(44)

Complexity 7

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

Especially system (6) reduces to (1) (2) (4) or the system in[32] when parameters of system (6) assume some specialvaluesat is to say we have derived extinction of the abovesystems at the same time

In this paper system (6) is more general than the modelof [21 28 30ndash32] In addition our results improve andgeneralize the corresponding results in these literaturestudies

e remainder of this paper is built up as follows InSection 2 we show the global existence of almost surelypositive solution e asymptotic estimates for the solutionstochastically ultimate boundedness and boundedness forthe average in time of the θth moment of the solution arethen constructed in Section 3 In Section 4 we discuss thepathwise properties of the solution Sufficient conditions forextinction of species are obtained in Section 5 Numericalinvestigations are then given in Section 6 e last part is aconclusion

2 Preliminary Results

To simplify denote the solution of (6) with initial values (7)

x(t) ≔ x tφ ℓ0( 1113857 (8)

where x(t) (x1(t) x2(t))T Let

β1(i) a1(i) + b2(i)

β2(i) a2(i) + b1(i) i isinM(9)

We denote R+ (0 +infin) R2+ (x1 x2)

T isin1113966

R2 x1 gt 0 x2 gt 0 and R2times2+ (wuv)2times2 isin R2times2 wuv gt1113864

0 u v 1 2 For any Φ M⟶ R let 1113954Φ miniisinMΦ(i)

and Φ maxiisinMΦ(i) Let |middot| denote Euclidean norm in R2Denote the trace norm |A|

trace(ATA)

1113968for matrix A

Lemma 1 Given any initial values (7) system (6) has aunique solution x(t) isin R2

+ for all t isin [minus τinfin) almost surely

Proof We omit the proof since it is analogous to that of [31]by making use of the generalized It1113954o formula (see egeorem 145 in [35]) to 1113936

2h1(xh minus 1 minus logxh)

Remark 1 e delay stochastic Nicholson-type modelunder regime switching on patches (6) is a direct ex-tension of the models in [21 28 30ndash32] From Lemma 1 itis worthy to point out that priori conditions αgt σ22 in[21] are unnecessary erefore Lemma 1 improves andgeneralizes Lemma 22 in [21] In addition this lemmashows that both white noise and telegraph noise will notdestroy a great property that the solution of (3) does notexplode

3 Boundedness

Because of resource constraints asymptotic boundedness isthe core of the research in ecosystems And it is the mainpurpose of the present section For simplicity we use thefollowing notations For any i isinM denote

K1(θ i) ≔ a1(i) minusθ minus 1θ

b1(i) minus b2(i)( 1113857 minus12

(θ minus 1)σ21(i) minus (θ minus 1)

K2(θ i) ≔ a2(i) minusθ minus 1θ

b2(i) minus b1(i)( 1113857 minus12

(θ minus 1)σ22(i) minus (θ minus 1)

Hh(θ i) ≔ θ middot supxhisinR+

minus Kh(θ i)xθh +

ph(i)

ch(i)exθminus 1h1113896 1113897

Hh(θ) ≔ maxiisinMHh(θ i)

A1(θ) ≔2θ2

θ(θ minus 1)1113944iisinM

πi H1(θ i) + H2(θ i)1113858 1113859

A2(θ) ≔1113936

2h1

Hh(θ)

θ(θ minus 1)

A3(θ) ≔ 2θ2 middot1113936

2h1

H(θ)

θ(θ minus 1) h 1 2

(10)

Firstly inspired by the work of Wang and Chen [32] wegive this theorem

Theorem 1 Let θgt 1 such that Kh(θ i)gt 0 h 1 2 i isinMGiven any initial values (7) solution (x1(t) x2(t)) of (6)satisfies

limsupt⟶infin

1t

1113946t

0E xh(s)

11138681113868111386811138681113868111386811138681113868θdsleA1(θ) h 1 2 (11)

andlimsupt⟶infin

E xθ1(t) + x

θ2(t)1113872 1113873leA2(θ) (12)

In particular

limsupt⟶infin

E|x(t)|θ leA3(θ) (13)

at is system (6) is ultimately bounded

Proof Define

V1 x1 x2( 1113857 xθ1 + x

θ2 (14)

e generalized It1113954o formula together with the factph(i)yheminus ch(i)yh le (ph(i)ch(i)e) and the elementary in-equality AεB1minus ε leAε + B(1 minus ε) for any A Bge 0 andε isin [0 1] yields

Complexity 3

LV1 x1 x2 y1 y2 i( 1113857

minus θa1(i)xθ1 minus θb2(i)x

θ1 + θb1(i)x

θminus 11 x2 +

12θ(θ minus 1)σ21(i)x

θ1

minus θa2(i)xθ2 minus θb1(i)x

θ2 + θb2(i)x1x

θminus 12 +

12θ(θ minus 1)σ22(i)x

θ2

+ θ 11139442

h1ph(i)x

θminus 1h yhe

minus ch(i)yh

le θ minus a1(i) +θ minus 1θ

b1(i) minus b2(i)( 1113857 +12

(θ minus 1)σ21(i) +(θ minus 1)1113890 1113891xθ11113888

+ minus a2(i) +θ minus 1θ

b2(i) minus b1(i)( 1113857 +12

(θ minus 1)σ22(i) +(θ minus 1)1113890 1113891xθ2

+ 11139442

h1

ph(i)

ch(i)exθminus 1h

⎞⎠ minus θ(θ minus 1) xθ1 + x

θ21113872 1113873

le 11139442

h1θ minus Kh(θ i)x

θh +

ph(i)

ch(i)exθminus 1h1113890 1113891 minus θ(θ minus 1)V1 x1 x2( 1113857

le 11139442

h1Hh(θ i) minus θ(θ minus 1)V1 x1(t) x2(t)( 1113857

(15)

erefore for tgt 0

dV1 x1(t) x2(t)( 1113857le 11139442

h1Hh(θ ℓ(t)) minus θ(θ minus 1)V1

⎡⎣

middot x1(t) x2(t)( 1113857⎤⎦dt + 1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

(16)

en (16) implies

0leE V1 x1 x2( 1113857( 1113857leφθ1(0) + φθ

2(0) + 1113946t

01113944

2

h1Hh(θ ℓ(s))ds

minus θ(θ minus 1) 1113946t

0EV1 x1(s) x2(s)( 1113857ds

(17)

Noting that the Markov chain ℓ(t) has an invariantdistribution π (πii isinM) and applying the ergodicproperty of the Markov chain it yields

limsupt⟶infin

1t

1113946t

0EV1 x1(s) x2(s)( 1113857dsle limsup

t⟶infin

1θ(θ minus 1)

middot1tV1 φ1(0)φ2(0)( 1113857 +

1t

1113946t

01113944

2

h1Hh(θ ℓ(s))ds⎛⎝ ⎞⎠

le1

θ(θ minus 1)1113944iisinM

πi H1(θ i) + H2(θ i)1113858 1113859 a s

(18)

Furthermore we have

d eθ(θminus 1)t

V1 x1(t) x2(t)( 11138571113960 1113961le eθ(θminus 1)t

1113944

2

h1Hh(θ ℓ(t))dt + e

θ(θminus 1)t1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

le 11139442

h1

Hh(θ)eθ(θminus 1)tdt + e

θ(θminus 1)t1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

(19)

Hence

eθ(θminus 1)t

EV1 x1(t) x2(t)( 1113857leV1 φ1(0)φ2(0)( 1113857

+1113936

2h1

Hh(θ)

θ(θ minus 1)eθ(θminus 1)t

minus1113936

2h1

Hh(θ)

θ(θ minus 1)

(20)

Consequently we infer immediately that (12) holds Onthe contrary according to (12) (18) and the fact that

|x|θ le 2θ2 max x

θ1 x

θ21113966 1113967le 2θ2V1(x) (21)

it follows that (11) and (13) hold e proof is thereforecomplete

Remark 2 Ineorem 1 the parameter θ is greater than 1 inthe result Although ultimate boundedness in the θth mo-ment was derived for θ restricted to the precondition θ gt 1θth moment of system (6) can be obtained when θle 1 byHolderrsquos equality

Remark 3 Without regime switching or without migrationand regime switching eorem 1 improves the corre-sponding results in [21 31] If τ(i) equiv τ system (6) is a directextension of the model in [32] Besides no proof of ultimateboundedness in the pth moment is given in [32] which is

4 Complexity

shown in eorem 1 erefore this theorem extends andimproveseorem 31 in [21]eorem 22 in [28]eorem33 in [31] and eorem 32 in [32]

Theorem 2 Given any initial values (7) solution(x1(t) x2(t)) of (6) satisfies

limsupt⟶infin

E|x(t)|le limsupt⟶infin

E x1(t) + x2(t)1113858 1113859le1113957p1

1113954c1eλ+

1113957p21113954c2eλ

(22)

where λ min 1113954a1 1113954a21113864 1113865 7at is (6) is ultimately bounded inmean

Proof Let V1(t x1 x2) eλt(x1 + x2) en

E x1(t) + x2(t)( 1113857le eminus λt

V1 0φ1(0)φ2(0)( 1113857

+p11113954c1e

+p21113954c2e

1113888 1113889 1113946t

0e

(sminus t)λds

(23)

Finally (22) follows by letting t⟶infin e proof istherefore complete

Remark 4 Compared with eorem 1 this theorem de-scribes the case that θ 1 which does not require anyconditions If τ(i) equiv τ we get (p11113954c1eλ) + (p21113954c2eλ)le (ca)where (ca) is defined in [32] So this theorem improves andextends eorem 31 in [21] and eorem 31 in [32]

Theorem 3 System (6) is stochastically ultimately bounded

Proof By (22) we derive

limsupt⟶infin

E xh(t)1113868111386811138681113868

1113868111386811138681113868lep1

1113954c1eλ+

p21113954c2eλ

h 1 2 (24)

By the Chebyshev inequality it yields for any ε isin (0 1)

limsupt⟶infin

P xh(t)geH1113864 1113865leHminus 1 p1

1113954c1eλ+

p21113954c2eλ

1113888 1113889 ε (25)

where H (1ε)((1113957p11113954c1eλ) + (1113957p21113954c2eλ)) e proof istherefore complete

Remark 5 eorem 3 can be seen as the extension andimprovement of [31 32]

4 Asymptotic Pathwise Estimation

We shall estimate a sample Lyapunov exponent in whatfollows

Lemma 2 If a isin R and b isin R+ then (ax2 + bx1+

x2)leK(a) for x isin R where K(a) (a +a2 + b2

radic2)

By the properties of quadratic functions the proof of thislemma is easy and so is omitted In the process of findingK(a) we know that the precondition is a minus K(a)lt 0 In thiscase we can choose K(a) which satisfies K(a)

(a +a2 + b2

radic2) We have to mention that it has no relation

with the sign of parameter a If alt 0 we get(a +

a2 + b2

radic2)lt minus (b24a) by simple computation So

this lemma is an improvement of Lemma 12 in [28] andLemma 21 in [32]

Theorem 4 Given any initial values (7) solution x(t) of (6)satisfies

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

1tlog|x(t)|le

Q

2 a s (26)

where h 1 2 Q maxiisinM minεisinR+[Q1(i ϵ) + Q2(i ϵ)]1113966 1113967

with

Q1(i ε)

2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 11138592

+ 4 p1(i)c1(i)e( 11138572

1113969

minus 2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 1113859

2

Q2(i ε)

2β2(i) minus σ22(i) minus b1(i) + b2(i)ε( 11138572

+ 4 p2(i)c2(i)e1113857(2

1113969

minus 2β2(i) minus σ22(i) minus b1(i) + b2(i)( 1113857ε( 1113857

2

(27)

Complexity 5

for any positive constant ε Proof e generalized It1113954o formula together with Lemma 2and the CauchyndashSchwarz inequality yields

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ 1113946t

0

minus 2β1(ℓ(s)) minus σ21(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s))( 1113857 isin1113858 1113859x21(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x1(s)

1 + x21(s) + x2

2(s)ds

+ 1113946t

0

minus 2β2(ℓ(s)) minus σ22(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s)) isin( 11138571113858 1113859x22(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x2(s)

1 + x21(s) + x2

2(s)ds

minus 2 11139442

h1σ2h(ℓ(s)) 1113946

t

0

x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

le log 1 + x21(0) + x

22(0)1113872 1113873 + Qt minus 2 1113944

2

h11113946

t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

(28)

where for any h isin 1 2

Mh(t) 21113946t

0

σh(ℓ(s))x2h(s)

1 + x21(s) + x2

2(s)dBh(s) (29)

with the quadratic variation

langMh(t) Mh(t)rang 41113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds (30)

According to the exponential martingale inequality (seeeg [36]) for any integer mgt 0 we have

P sup0letlem

Mh(t) minus 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds

⎧⎨

⎩

⎫⎬

⎭

⎧⎨

⎩

gt 2 logm⎫⎬

⎭ le1

m2 h 1 2

(31)

Since 1113936infinm1 1m2 ltinfin and BorelndashCantellirsquos lemma (see

eg [36]) there existΩ0 isin F with P(Ω0) 1 and an integerm0 m0(ω) such that

Mh(t)le 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds + 2 logm

h 1 2

(32)

for all ω isin Ω0 0le tlem Substituting the above inequalityinto (28) for any ω isin Ω0 mgem0 0le tlem we have

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ Qt + 4 logm(33)

which yields

1tlog 1 + x

21(t) + x

22(t)1113872 1113873le

1m minus 1

log 1 + x21(0) + x

22(0)1113872 11138731113960

+ Qm + 4 logm1113859

(34)

for all ω isin Ω0 0lem minus 1le tlem mgem0 Letting m⟶infinand using the inequality yle (12)(1 + y2) for anyy isin (minus infin +infin) we obtain

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

12(m minus 1)

log 1 + x2(0)1113872 11138731113960

+ Qm + 4 logm1113859 Q

2 a s

(35)e proof is therefore complete

Remark 6 Without migrations we get

Q1(i) (

[2β1(i) minus σ21(i)]2 + 4(p1(i)c1(i)e)21113969

minus [2β1(i) minus

σ21(i)]2) By comparison we find thatQ1(i)leCi where Ci isdefined in [28] In addition without migration and regimeswitching we can get Q ineorem 4 is less than K where K

is defined in [21] Furthermore the condition 2α1 minus σ21 minus

(b1 + b2)ϵgt 0 2α2 minus σ22 minus (b1 + b2)ϵgt 0 in [31] meansthat the parameter ϵ needs to be satisfied(b1 + b2)2α2 minus σ22 lt εlt 2α1 minus σ21(b1 + b2) However weknow that this condition is unnecessary from the abovetheorem Despite all this if we let the parameter ε satisfyε isin (((b1 + b2)2α2 minus σ2) (2α21 minus σ21(b1 + b2))) we computethat Q in eorem 4 is less than Q in [31] erefore theabove work is a promotion of eorem 41 in [21] eorem22 in [28] and eorem 41 in [31]

6 Complexity

5 Extinction

Sufficient conditions for extinction are the subject of thissection Unless otherwise stated we hypothesizeτ(i) equiv τ i isinM in this section We first rewrite (6) asfollows

dx(t) f1(x(t) x(t minus τ) ℓ(t))dt + f2(x(t) ℓ(t))dB(t)

(36)

where the operator f1 R2+ times R2

+ times M⟶ R2+ is defined as

f1(x y i) minus β1(i)x1 + b1(i)x2 + p1(i)y1e

minus c1(i)y1

minus β2(i)x2 + b2(i)x1 + p2(i)y2eminus c2(i)y2

⎛⎝ ⎞⎠

(37)

the operator f2 R2+ times M⟶ R2times2

+ is defined as

f2(x i) σ1(i)x1 0

0 σ2(i)x21113888 1113889 and dB(t)

dB1(t)

dB2(t)1113888 1113889

We first note that

f1(0 0 i) equiv 0

f2(0 i) equiv 0(38)

for i isinM whence (6) admits a trivial solution corre-sponding to φ(0) 0

Before our result we give a lemma

Lemma 3 For system (36) the terms f1(x y i) and f2(x i)

are locally bounded in (x y) while uniformly bounded in i7at is for any mgt 0 there is Km gt 0 satisfying

f1(x y i)1113868111386811138681113868

1113868111386811138681113868 or f2(x i)1113868111386811138681113868

1113868111386811138681113868leKm (39)

for all i isinM x y isin R2+ with |x| or |y|lem

e proof is not particularly difficult so we omit theproof

Theorem 5 Assume that

21113954β1 gt σ21 + b1 +(1 +2

radic)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

21113954β2 gt σ22 +(1 +2

radic)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(40)

en the solution of (36) satisfies limt⟶infinx(t) 0 asfor any initial values (7) at is all populations in system(36) go to extinction with probability one

Proof

Step 1 let

V2 x1 x2( 1113857 xT

112

radic

12

radic 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠x x

21 +

2

radicx1x2 + x

22

(41)

Obviously V2 is positive-definite and radially un-bounded at is

lim|x|⟶infin

V2 x1 x2( 1113857 infin (42)

e generalized It1113954o formula yields

V2 x1(t) x2(t)( 1113857 V2 φ1(0)φ2(0)( 1113857 + 1113946t

0LV2(x(s) x

middot (s minus τ(ℓ(s))) ℓ(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(43)

By computation we know

LV2(x y i)le minus 2β1(i) minus σ21(i) minus b1(i) minus (1 +2

radic)b2(i) minus p1(i) minus

12

radic p2(i)1113890 1113891x21

minus 2β2(i) minus σ22(i) minus (1 +2

radic)b1(i) minus b2(i) minus

12

radic p1(i) minus p2(i)1113890 1113891x22

+ 1 +12

radic1113888 1113889p1(i)y21 + 1 +

12

radic1113888 1113889p2(i)y22

le minus λ1x21 minus λ2x

22 + 1 +

12

radic1113888 1113889p1y21 + 1 +

12

radic1113888 1113889p2y22

(44)

Complexity 7

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

LV1 x1 x2 y1 y2 i( 1113857

minus θa1(i)xθ1 minus θb2(i)x

θ1 + θb1(i)x

θminus 11 x2 +

12θ(θ minus 1)σ21(i)x

θ1

minus θa2(i)xθ2 minus θb1(i)x

θ2 + θb2(i)x1x

θminus 12 +

12θ(θ minus 1)σ22(i)x

θ2

+ θ 11139442

h1ph(i)x

θminus 1h yhe

minus ch(i)yh

le θ minus a1(i) +θ minus 1θ

b1(i) minus b2(i)( 1113857 +12

(θ minus 1)σ21(i) +(θ minus 1)1113890 1113891xθ11113888

+ minus a2(i) +θ minus 1θ

b2(i) minus b1(i)( 1113857 +12

(θ minus 1)σ22(i) +(θ minus 1)1113890 1113891xθ2

+ 11139442

h1

ph(i)

ch(i)exθminus 1h

⎞⎠ minus θ(θ minus 1) xθ1 + x

θ21113872 1113873

le 11139442

h1θ minus Kh(θ i)x

θh +

ph(i)

ch(i)exθminus 1h1113890 1113891 minus θ(θ minus 1)V1 x1 x2( 1113857

le 11139442

h1Hh(θ i) minus θ(θ minus 1)V1 x1(t) x2(t)( 1113857

(15)

erefore for tgt 0

dV1 x1(t) x2(t)( 1113857le 11139442

h1Hh(θ ℓ(t)) minus θ(θ minus 1)V1

⎡⎣

middot x1(t) x2(t)( 1113857⎤⎦dt + 1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

(16)

en (16) implies

0leE V1 x1 x2( 1113857( 1113857leφθ1(0) + φθ

2(0) + 1113946t

01113944

2

h1Hh(θ ℓ(s))ds

minus θ(θ minus 1) 1113946t

0EV1 x1(s) x2(s)( 1113857ds

(17)

Noting that the Markov chain ℓ(t) has an invariantdistribution π (πii isinM) and applying the ergodicproperty of the Markov chain it yields

limsupt⟶infin

1t

1113946t

0EV1 x1(s) x2(s)( 1113857dsle limsup

t⟶infin

1θ(θ minus 1)

middot1tV1 φ1(0)φ2(0)( 1113857 +

1t

1113946t

01113944

2

h1Hh(θ ℓ(s))ds⎛⎝ ⎞⎠

le1

θ(θ minus 1)1113944iisinM

πi H1(θ i) + H2(θ i)1113858 1113859 a s

(18)

Furthermore we have

d eθ(θminus 1)t

V1 x1(t) x2(t)( 11138571113960 1113961le eθ(θminus 1)t

1113944

2

h1Hh(θ ℓ(t))dt + e

θ(θminus 1)t1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

le 11139442

h1

Hh(θ)eθ(θminus 1)tdt + e

θ(θminus 1)t1113944

2

h1θσh(ℓ(t))x

θh(t)dBh(t)

(19)

Hence

eθ(θminus 1)t

EV1 x1(t) x2(t)( 1113857leV1 φ1(0)φ2(0)( 1113857

+1113936

2h1

Hh(θ)

θ(θ minus 1)eθ(θminus 1)t

minus1113936

2h1

Hh(θ)

θ(θ minus 1)

(20)

Consequently we infer immediately that (12) holds Onthe contrary according to (12) (18) and the fact that

|x|θ le 2θ2 max x

θ1 x

θ21113966 1113967le 2θ2V1(x) (21)

it follows that (11) and (13) hold e proof is thereforecomplete

Remark 2 Ineorem 1 the parameter θ is greater than 1 inthe result Although ultimate boundedness in the θth mo-ment was derived for θ restricted to the precondition θ gt 1θth moment of system (6) can be obtained when θle 1 byHolderrsquos equality

Remark 3 Without regime switching or without migrationand regime switching eorem 1 improves the corre-sponding results in [21 31] If τ(i) equiv τ system (6) is a directextension of the model in [32] Besides no proof of ultimateboundedness in the pth moment is given in [32] which is

4 Complexity

shown in eorem 1 erefore this theorem extends andimproveseorem 31 in [21]eorem 22 in [28]eorem33 in [31] and eorem 32 in [32]

Theorem 2 Given any initial values (7) solution(x1(t) x2(t)) of (6) satisfies

limsupt⟶infin

E|x(t)|le limsupt⟶infin

E x1(t) + x2(t)1113858 1113859le1113957p1

1113954c1eλ+

1113957p21113954c2eλ

(22)

where λ min 1113954a1 1113954a21113864 1113865 7at is (6) is ultimately bounded inmean

Proof Let V1(t x1 x2) eλt(x1 + x2) en

E x1(t) + x2(t)( 1113857le eminus λt

V1 0φ1(0)φ2(0)( 1113857

+p11113954c1e

+p21113954c2e

1113888 1113889 1113946t

0e

(sminus t)λds

(23)

Finally (22) follows by letting t⟶infin e proof istherefore complete

Remark 4 Compared with eorem 1 this theorem de-scribes the case that θ 1 which does not require anyconditions If τ(i) equiv τ we get (p11113954c1eλ) + (p21113954c2eλ)le (ca)where (ca) is defined in [32] So this theorem improves andextends eorem 31 in [21] and eorem 31 in [32]

Theorem 3 System (6) is stochastically ultimately bounded

Proof By (22) we derive

limsupt⟶infin

E xh(t)1113868111386811138681113868

1113868111386811138681113868lep1

1113954c1eλ+

p21113954c2eλ

h 1 2 (24)

By the Chebyshev inequality it yields for any ε isin (0 1)

limsupt⟶infin

P xh(t)geH1113864 1113865leHminus 1 p1

1113954c1eλ+

p21113954c2eλ

1113888 1113889 ε (25)

where H (1ε)((1113957p11113954c1eλ) + (1113957p21113954c2eλ)) e proof istherefore complete

Remark 5 eorem 3 can be seen as the extension andimprovement of [31 32]

4 Asymptotic Pathwise Estimation

We shall estimate a sample Lyapunov exponent in whatfollows

Lemma 2 If a isin R and b isin R+ then (ax2 + bx1+

x2)leK(a) for x isin R where K(a) (a +a2 + b2

radic2)

By the properties of quadratic functions the proof of thislemma is easy and so is omitted In the process of findingK(a) we know that the precondition is a minus K(a)lt 0 In thiscase we can choose K(a) which satisfies K(a)

(a +a2 + b2

radic2) We have to mention that it has no relation

with the sign of parameter a If alt 0 we get(a +

a2 + b2

radic2)lt minus (b24a) by simple computation So

this lemma is an improvement of Lemma 12 in [28] andLemma 21 in [32]

Theorem 4 Given any initial values (7) solution x(t) of (6)satisfies

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

1tlog|x(t)|le

Q

2 a s (26)

where h 1 2 Q maxiisinM minεisinR+[Q1(i ϵ) + Q2(i ϵ)]1113966 1113967

with

Q1(i ε)

2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 11138592

+ 4 p1(i)c1(i)e( 11138572

1113969

minus 2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 1113859

2

Q2(i ε)

2β2(i) minus σ22(i) minus b1(i) + b2(i)ε( 11138572

+ 4 p2(i)c2(i)e1113857(2

1113969

minus 2β2(i) minus σ22(i) minus b1(i) + b2(i)( 1113857ε( 1113857

2

(27)

Complexity 5

for any positive constant ε Proof e generalized It1113954o formula together with Lemma 2and the CauchyndashSchwarz inequality yields

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ 1113946t

0

minus 2β1(ℓ(s)) minus σ21(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s))( 1113857 isin1113858 1113859x21(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x1(s)

1 + x21(s) + x2

2(s)ds

+ 1113946t

0

minus 2β2(ℓ(s)) minus σ22(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s)) isin( 11138571113858 1113859x22(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x2(s)

1 + x21(s) + x2

2(s)ds

minus 2 11139442

h1σ2h(ℓ(s)) 1113946

t

0

x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

le log 1 + x21(0) + x

22(0)1113872 1113873 + Qt minus 2 1113944

2

h11113946

t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

(28)

where for any h isin 1 2

Mh(t) 21113946t

0

σh(ℓ(s))x2h(s)

1 + x21(s) + x2

2(s)dBh(s) (29)

with the quadratic variation

langMh(t) Mh(t)rang 41113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds (30)

According to the exponential martingale inequality (seeeg [36]) for any integer mgt 0 we have

P sup0letlem

Mh(t) minus 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds

⎧⎨

⎩

⎫⎬

⎭

⎧⎨

⎩

gt 2 logm⎫⎬

⎭ le1

m2 h 1 2

(31)

Since 1113936infinm1 1m2 ltinfin and BorelndashCantellirsquos lemma (see

eg [36]) there existΩ0 isin F with P(Ω0) 1 and an integerm0 m0(ω) such that

Mh(t)le 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds + 2 logm

h 1 2

(32)

for all ω isin Ω0 0le tlem Substituting the above inequalityinto (28) for any ω isin Ω0 mgem0 0le tlem we have

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ Qt + 4 logm(33)

which yields

1tlog 1 + x

21(t) + x

22(t)1113872 1113873le

1m minus 1

log 1 + x21(0) + x

22(0)1113872 11138731113960

+ Qm + 4 logm1113859

(34)

for all ω isin Ω0 0lem minus 1le tlem mgem0 Letting m⟶infinand using the inequality yle (12)(1 + y2) for anyy isin (minus infin +infin) we obtain

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

12(m minus 1)

log 1 + x2(0)1113872 11138731113960

+ Qm + 4 logm1113859 Q

2 a s

(35)e proof is therefore complete

Remark 6 Without migrations we get

Q1(i) (

[2β1(i) minus σ21(i)]2 + 4(p1(i)c1(i)e)21113969

minus [2β1(i) minus

σ21(i)]2) By comparison we find thatQ1(i)leCi where Ci isdefined in [28] In addition without migration and regimeswitching we can get Q ineorem 4 is less than K where K

is defined in [21] Furthermore the condition 2α1 minus σ21 minus

(b1 + b2)ϵgt 0 2α2 minus σ22 minus (b1 + b2)ϵgt 0 in [31] meansthat the parameter ϵ needs to be satisfied(b1 + b2)2α2 minus σ22 lt εlt 2α1 minus σ21(b1 + b2) However weknow that this condition is unnecessary from the abovetheorem Despite all this if we let the parameter ε satisfyε isin (((b1 + b2)2α2 minus σ2) (2α21 minus σ21(b1 + b2))) we computethat Q in eorem 4 is less than Q in [31] erefore theabove work is a promotion of eorem 41 in [21] eorem22 in [28] and eorem 41 in [31]

6 Complexity

5 Extinction

Sufficient conditions for extinction are the subject of thissection Unless otherwise stated we hypothesizeτ(i) equiv τ i isinM in this section We first rewrite (6) asfollows

dx(t) f1(x(t) x(t minus τ) ℓ(t))dt + f2(x(t) ℓ(t))dB(t)

(36)

where the operator f1 R2+ times R2

+ times M⟶ R2+ is defined as

f1(x y i) minus β1(i)x1 + b1(i)x2 + p1(i)y1e

minus c1(i)y1

minus β2(i)x2 + b2(i)x1 + p2(i)y2eminus c2(i)y2

⎛⎝ ⎞⎠

(37)

the operator f2 R2+ times M⟶ R2times2

+ is defined as

f2(x i) σ1(i)x1 0

0 σ2(i)x21113888 1113889 and dB(t)

dB1(t)

dB2(t)1113888 1113889

We first note that

f1(0 0 i) equiv 0

f2(0 i) equiv 0(38)

for i isinM whence (6) admits a trivial solution corre-sponding to φ(0) 0

Before our result we give a lemma

Lemma 3 For system (36) the terms f1(x y i) and f2(x i)

are locally bounded in (x y) while uniformly bounded in i7at is for any mgt 0 there is Km gt 0 satisfying

f1(x y i)1113868111386811138681113868

1113868111386811138681113868 or f2(x i)1113868111386811138681113868

1113868111386811138681113868leKm (39)

for all i isinM x y isin R2+ with |x| or |y|lem

e proof is not particularly difficult so we omit theproof

Theorem 5 Assume that

21113954β1 gt σ21 + b1 +(1 +2

radic)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

21113954β2 gt σ22 +(1 +2

radic)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(40)

en the solution of (36) satisfies limt⟶infinx(t) 0 asfor any initial values (7) at is all populations in system(36) go to extinction with probability one

Proof

Step 1 let

V2 x1 x2( 1113857 xT

112

radic

12

radic 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠x x

21 +

2

radicx1x2 + x

22

(41)

Obviously V2 is positive-definite and radially un-bounded at is

lim|x|⟶infin

V2 x1 x2( 1113857 infin (42)

e generalized It1113954o formula yields

V2 x1(t) x2(t)( 1113857 V2 φ1(0)φ2(0)( 1113857 + 1113946t

0LV2(x(s) x

middot (s minus τ(ℓ(s))) ℓ(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(43)

By computation we know

LV2(x y i)le minus 2β1(i) minus σ21(i) minus b1(i) minus (1 +2

radic)b2(i) minus p1(i) minus

12

radic p2(i)1113890 1113891x21

minus 2β2(i) minus σ22(i) minus (1 +2

radic)b1(i) minus b2(i) minus

12

radic p1(i) minus p2(i)1113890 1113891x22

+ 1 +12

radic1113888 1113889p1(i)y21 + 1 +

12

radic1113888 1113889p2(i)y22

le minus λ1x21 minus λ2x

22 + 1 +

12

radic1113888 1113889p1y21 + 1 +

12

radic1113888 1113889p2y22

(44)

Complexity 7

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

shown in eorem 1 erefore this theorem extends andimproveseorem 31 in [21]eorem 22 in [28]eorem33 in [31] and eorem 32 in [32]

Theorem 2 Given any initial values (7) solution(x1(t) x2(t)) of (6) satisfies

limsupt⟶infin

E|x(t)|le limsupt⟶infin

E x1(t) + x2(t)1113858 1113859le1113957p1

1113954c1eλ+

1113957p21113954c2eλ

(22)

where λ min 1113954a1 1113954a21113864 1113865 7at is (6) is ultimately bounded inmean

Proof Let V1(t x1 x2) eλt(x1 + x2) en

E x1(t) + x2(t)( 1113857le eminus λt

V1 0φ1(0)φ2(0)( 1113857

+p11113954c1e

+p21113954c2e

1113888 1113889 1113946t

0e

(sminus t)λds

(23)

Finally (22) follows by letting t⟶infin e proof istherefore complete

Remark 4 Compared with eorem 1 this theorem de-scribes the case that θ 1 which does not require anyconditions If τ(i) equiv τ we get (p11113954c1eλ) + (p21113954c2eλ)le (ca)where (ca) is defined in [32] So this theorem improves andextends eorem 31 in [21] and eorem 31 in [32]

Theorem 3 System (6) is stochastically ultimately bounded

Proof By (22) we derive

limsupt⟶infin

E xh(t)1113868111386811138681113868

1113868111386811138681113868lep1

1113954c1eλ+

p21113954c2eλ

h 1 2 (24)

By the Chebyshev inequality it yields for any ε isin (0 1)

limsupt⟶infin

P xh(t)geH1113864 1113865leHminus 1 p1

1113954c1eλ+

p21113954c2eλ

1113888 1113889 ε (25)

where H (1ε)((1113957p11113954c1eλ) + (1113957p21113954c2eλ)) e proof istherefore complete

Remark 5 eorem 3 can be seen as the extension andimprovement of [31 32]

4 Asymptotic Pathwise Estimation

We shall estimate a sample Lyapunov exponent in whatfollows

Lemma 2 If a isin R and b isin R+ then (ax2 + bx1+

x2)leK(a) for x isin R where K(a) (a +a2 + b2

radic2)

By the properties of quadratic functions the proof of thislemma is easy and so is omitted In the process of findingK(a) we know that the precondition is a minus K(a)lt 0 In thiscase we can choose K(a) which satisfies K(a)

(a +a2 + b2

radic2) We have to mention that it has no relation

with the sign of parameter a If alt 0 we get(a +

a2 + b2

radic2)lt minus (b24a) by simple computation So

this lemma is an improvement of Lemma 12 in [28] andLemma 21 in [32]

Theorem 4 Given any initial values (7) solution x(t) of (6)satisfies

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

1tlog|x(t)|le

Q

2 a s (26)

where h 1 2 Q maxiisinM minεisinR+[Q1(i ϵ) + Q2(i ϵ)]1113966 1113967

with

Q1(i ε)

2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 11138592

+ 4 p1(i)c1(i)e( 11138572

1113969

minus 2β1(i) minus σ21(i) minus b1(i) + b2(i)( 1113857ϵ1113858 1113859

2

Q2(i ε)

2β2(i) minus σ22(i) minus b1(i) + b2(i)ε( 11138572

+ 4 p2(i)c2(i)e1113857(2

1113969

minus 2β2(i) minus σ22(i) minus b1(i) + b2(i)( 1113857ε( 1113857

2

(27)

Complexity 5

for any positive constant ε Proof e generalized It1113954o formula together with Lemma 2and the CauchyndashSchwarz inequality yields

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ 1113946t

0

minus 2β1(ℓ(s)) minus σ21(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s))( 1113857 isin1113858 1113859x21(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x1(s)

1 + x21(s) + x2

2(s)ds

+ 1113946t

0

minus 2β2(ℓ(s)) minus σ22(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s)) isin( 11138571113858 1113859x22(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x2(s)

1 + x21(s) + x2

2(s)ds

minus 2 11139442

h1σ2h(ℓ(s)) 1113946

t

0

x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

le log 1 + x21(0) + x

22(0)1113872 1113873 + Qt minus 2 1113944

2

h11113946

t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

(28)

where for any h isin 1 2

Mh(t) 21113946t

0

σh(ℓ(s))x2h(s)

1 + x21(s) + x2

2(s)dBh(s) (29)

with the quadratic variation

langMh(t) Mh(t)rang 41113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds (30)

According to the exponential martingale inequality (seeeg [36]) for any integer mgt 0 we have

P sup0letlem

Mh(t) minus 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds

⎧⎨

⎩

⎫⎬

⎭

⎧⎨

⎩

gt 2 logm⎫⎬

⎭ le1

m2 h 1 2

(31)

Since 1113936infinm1 1m2 ltinfin and BorelndashCantellirsquos lemma (see

eg [36]) there existΩ0 isin F with P(Ω0) 1 and an integerm0 m0(ω) such that

Mh(t)le 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds + 2 logm

h 1 2

(32)

for all ω isin Ω0 0le tlem Substituting the above inequalityinto (28) for any ω isin Ω0 mgem0 0le tlem we have

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ Qt + 4 logm(33)

which yields

1tlog 1 + x

21(t) + x

22(t)1113872 1113873le

1m minus 1

log 1 + x21(0) + x

22(0)1113872 11138731113960

+ Qm + 4 logm1113859

(34)

for all ω isin Ω0 0lem minus 1le tlem mgem0 Letting m⟶infinand using the inequality yle (12)(1 + y2) for anyy isin (minus infin +infin) we obtain

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

12(m minus 1)

log 1 + x2(0)1113872 11138731113960

+ Qm + 4 logm1113859 Q

2 a s

(35)e proof is therefore complete

Remark 6 Without migrations we get

Q1(i) (

[2β1(i) minus σ21(i)]2 + 4(p1(i)c1(i)e)21113969

minus [2β1(i) minus

σ21(i)]2) By comparison we find thatQ1(i)leCi where Ci isdefined in [28] In addition without migration and regimeswitching we can get Q ineorem 4 is less than K where K

is defined in [21] Furthermore the condition 2α1 minus σ21 minus

(b1 + b2)ϵgt 0 2α2 minus σ22 minus (b1 + b2)ϵgt 0 in [31] meansthat the parameter ϵ needs to be satisfied(b1 + b2)2α2 minus σ22 lt εlt 2α1 minus σ21(b1 + b2) However weknow that this condition is unnecessary from the abovetheorem Despite all this if we let the parameter ε satisfyε isin (((b1 + b2)2α2 minus σ2) (2α21 minus σ21(b1 + b2))) we computethat Q in eorem 4 is less than Q in [31] erefore theabove work is a promotion of eorem 41 in [21] eorem22 in [28] and eorem 41 in [31]

6 Complexity

5 Extinction

Sufficient conditions for extinction are the subject of thissection Unless otherwise stated we hypothesizeτ(i) equiv τ i isinM in this section We first rewrite (6) asfollows

dx(t) f1(x(t) x(t minus τ) ℓ(t))dt + f2(x(t) ℓ(t))dB(t)

(36)

where the operator f1 R2+ times R2

+ times M⟶ R2+ is defined as

f1(x y i) minus β1(i)x1 + b1(i)x2 + p1(i)y1e

minus c1(i)y1

minus β2(i)x2 + b2(i)x1 + p2(i)y2eminus c2(i)y2

⎛⎝ ⎞⎠

(37)

the operator f2 R2+ times M⟶ R2times2

+ is defined as

f2(x i) σ1(i)x1 0

0 σ2(i)x21113888 1113889 and dB(t)

dB1(t)

dB2(t)1113888 1113889

We first note that

f1(0 0 i) equiv 0

f2(0 i) equiv 0(38)

for i isinM whence (6) admits a trivial solution corre-sponding to φ(0) 0

Before our result we give a lemma

Lemma 3 For system (36) the terms f1(x y i) and f2(x i)

are locally bounded in (x y) while uniformly bounded in i7at is for any mgt 0 there is Km gt 0 satisfying

f1(x y i)1113868111386811138681113868

1113868111386811138681113868 or f2(x i)1113868111386811138681113868

1113868111386811138681113868leKm (39)

for all i isinM x y isin R2+ with |x| or |y|lem

e proof is not particularly difficult so we omit theproof

Theorem 5 Assume that

21113954β1 gt σ21 + b1 +(1 +2

radic)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

21113954β2 gt σ22 +(1 +2

radic)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(40)

en the solution of (36) satisfies limt⟶infinx(t) 0 asfor any initial values (7) at is all populations in system(36) go to extinction with probability one

Proof

Step 1 let

V2 x1 x2( 1113857 xT

112

radic

12

radic 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠x x

21 +

2

radicx1x2 + x

22

(41)

Obviously V2 is positive-definite and radially un-bounded at is

lim|x|⟶infin

V2 x1 x2( 1113857 infin (42)

e generalized It1113954o formula yields

V2 x1(t) x2(t)( 1113857 V2 φ1(0)φ2(0)( 1113857 + 1113946t

0LV2(x(s) x

middot (s minus τ(ℓ(s))) ℓ(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(43)

By computation we know

LV2(x y i)le minus 2β1(i) minus σ21(i) minus b1(i) minus (1 +2

radic)b2(i) minus p1(i) minus

12

radic p2(i)1113890 1113891x21

minus 2β2(i) minus σ22(i) minus (1 +2

radic)b1(i) minus b2(i) minus

12

radic p1(i) minus p2(i)1113890 1113891x22

+ 1 +12

radic1113888 1113889p1(i)y21 + 1 +

12

radic1113888 1113889p2(i)y22

le minus λ1x21 minus λ2x

22 + 1 +

12

radic1113888 1113889p1y21 + 1 +

12

radic1113888 1113889p2y22

(44)

Complexity 7

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

for any positive constant ε Proof e generalized It1113954o formula together with Lemma 2and the CauchyndashSchwarz inequality yields

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ 1113946t

0

minus 2β1(ℓ(s)) minus σ21(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s))( 1113857 isin1113858 1113859x21(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x1(s)

1 + x21(s) + x2

2(s)ds

+ 1113946t

0

minus 2β2(ℓ(s)) minus σ22(ℓ(s)) minus b1(ℓ(s)) + b2(ℓ(s)) isin( 11138571113858 1113859x22(s) + 2p1(ℓ(s))c1(ℓ(s))e( 1113857x2(s)

1 + x21(s) + x2

2(s)ds

minus 2 11139442

h1σ2h(ℓ(s)) 1113946

t

0

x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

le log 1 + x21(0) + x

22(0)1113872 1113873 + Qt minus 2 1113944

2

h11113946

t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572 ds + M1(t) + M2(t)

(28)

where for any h isin 1 2

Mh(t) 21113946t

0

σh(ℓ(s))x2h(s)

1 + x21(s) + x2

2(s)dBh(s) (29)

with the quadratic variation

langMh(t) Mh(t)rang 41113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds (30)

According to the exponential martingale inequality (seeeg [36]) for any integer mgt 0 we have

P sup0letlem

Mh(t) minus 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds

⎧⎨

⎩

⎫⎬

⎭

⎧⎨

⎩

gt 2 logm⎫⎬

⎭ le1

m2 h 1 2

(31)

Since 1113936infinm1 1m2 ltinfin and BorelndashCantellirsquos lemma (see

eg [36]) there existΩ0 isin F with P(Ω0) 1 and an integerm0 m0(ω) such that

Mh(t)le 21113946t

0

σ2h(ℓ(s))x4h(s)

1 + x21(s) + x2

2(s)( 11138572ds + 2 logm

h 1 2

(32)

for all ω isin Ω0 0le tlem Substituting the above inequalityinto (28) for any ω isin Ω0 mgem0 0le tlem we have

log 1 + x21(t) + x

22(t)1113872 1113873le log 1 + x

21(0) + x

22(0)1113872 1113873

+ Qt + 4 logm(33)

which yields

1tlog 1 + x

21(t) + x

22(t)1113872 1113873le

1m minus 1

log 1 + x21(0) + x

22(0)1113872 11138731113960

+ Qm + 4 logm1113859

(34)

for all ω isin Ω0 0lem minus 1le tlem mgem0 Letting m⟶infinand using the inequality yle (12)(1 + y2) for anyy isin (minus infin +infin) we obtain

limsupt⟶infin

1tlog xh(t)le limsup

t⟶infin

12(m minus 1)

log 1 + x2(0)1113872 11138731113960

+ Qm + 4 logm1113859 Q

2 a s

(35)e proof is therefore complete

Remark 6 Without migrations we get

Q1(i) (

[2β1(i) minus σ21(i)]2 + 4(p1(i)c1(i)e)21113969

minus [2β1(i) minus

σ21(i)]2) By comparison we find thatQ1(i)leCi where Ci isdefined in [28] In addition without migration and regimeswitching we can get Q ineorem 4 is less than K where K

is defined in [21] Furthermore the condition 2α1 minus σ21 minus

(b1 + b2)ϵgt 0 2α2 minus σ22 minus (b1 + b2)ϵgt 0 in [31] meansthat the parameter ϵ needs to be satisfied(b1 + b2)2α2 minus σ22 lt εlt 2α1 minus σ21(b1 + b2) However weknow that this condition is unnecessary from the abovetheorem Despite all this if we let the parameter ε satisfyε isin (((b1 + b2)2α2 minus σ2) (2α21 minus σ21(b1 + b2))) we computethat Q in eorem 4 is less than Q in [31] erefore theabove work is a promotion of eorem 41 in [21] eorem22 in [28] and eorem 41 in [31]

6 Complexity

5 Extinction

Sufficient conditions for extinction are the subject of thissection Unless otherwise stated we hypothesizeτ(i) equiv τ i isinM in this section We first rewrite (6) asfollows

dx(t) f1(x(t) x(t minus τ) ℓ(t))dt + f2(x(t) ℓ(t))dB(t)

(36)

where the operator f1 R2+ times R2

+ times M⟶ R2+ is defined as

f1(x y i) minus β1(i)x1 + b1(i)x2 + p1(i)y1e

minus c1(i)y1

minus β2(i)x2 + b2(i)x1 + p2(i)y2eminus c2(i)y2

⎛⎝ ⎞⎠

(37)

the operator f2 R2+ times M⟶ R2times2

+ is defined as

f2(x i) σ1(i)x1 0

0 σ2(i)x21113888 1113889 and dB(t)

dB1(t)

dB2(t)1113888 1113889

We first note that

f1(0 0 i) equiv 0

f2(0 i) equiv 0(38)

for i isinM whence (6) admits a trivial solution corre-sponding to φ(0) 0

Before our result we give a lemma

Lemma 3 For system (36) the terms f1(x y i) and f2(x i)

are locally bounded in (x y) while uniformly bounded in i7at is for any mgt 0 there is Km gt 0 satisfying

f1(x y i)1113868111386811138681113868

1113868111386811138681113868 or f2(x i)1113868111386811138681113868

1113868111386811138681113868leKm (39)

for all i isinM x y isin R2+ with |x| or |y|lem

e proof is not particularly difficult so we omit theproof

Theorem 5 Assume that

21113954β1 gt σ21 + b1 +(1 +2

radic)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

21113954β2 gt σ22 +(1 +2

radic)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(40)

en the solution of (36) satisfies limt⟶infinx(t) 0 asfor any initial values (7) at is all populations in system(36) go to extinction with probability one

Proof

Step 1 let

V2 x1 x2( 1113857 xT

112

radic

12

radic 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠x x

21 +

2

radicx1x2 + x

22

(41)

Obviously V2 is positive-definite and radially un-bounded at is

lim|x|⟶infin

V2 x1 x2( 1113857 infin (42)

e generalized It1113954o formula yields

V2 x1(t) x2(t)( 1113857 V2 φ1(0)φ2(0)( 1113857 + 1113946t

0LV2(x(s) x

middot (s minus τ(ℓ(s))) ℓ(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(43)

By computation we know

LV2(x y i)le minus 2β1(i) minus σ21(i) minus b1(i) minus (1 +2

radic)b2(i) minus p1(i) minus

12

radic p2(i)1113890 1113891x21

minus 2β2(i) minus σ22(i) minus (1 +2

radic)b1(i) minus b2(i) minus

12

radic p1(i) minus p2(i)1113890 1113891x22

+ 1 +12

radic1113888 1113889p1(i)y21 + 1 +

12

radic1113888 1113889p2(i)y22

le minus λ1x21 minus λ2x

22 + 1 +

12

radic1113888 1113889p1y21 + 1 +

12

radic1113888 1113889p2y22

(44)

Complexity 7

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

5 Extinction

Sufficient conditions for extinction are the subject of thissection Unless otherwise stated we hypothesizeτ(i) equiv τ i isinM in this section We first rewrite (6) asfollows

dx(t) f1(x(t) x(t minus τ) ℓ(t))dt + f2(x(t) ℓ(t))dB(t)

(36)

where the operator f1 R2+ times R2

+ times M⟶ R2+ is defined as

f1(x y i) minus β1(i)x1 + b1(i)x2 + p1(i)y1e

minus c1(i)y1

minus β2(i)x2 + b2(i)x1 + p2(i)y2eminus c2(i)y2

⎛⎝ ⎞⎠

(37)

the operator f2 R2+ times M⟶ R2times2

+ is defined as

f2(x i) σ1(i)x1 0

0 σ2(i)x21113888 1113889 and dB(t)

dB1(t)

dB2(t)1113888 1113889

We first note that

f1(0 0 i) equiv 0

f2(0 i) equiv 0(38)

for i isinM whence (6) admits a trivial solution corre-sponding to φ(0) 0

Before our result we give a lemma

Lemma 3 For system (36) the terms f1(x y i) and f2(x i)

are locally bounded in (x y) while uniformly bounded in i7at is for any mgt 0 there is Km gt 0 satisfying

f1(x y i)1113868111386811138681113868

1113868111386811138681113868 or f2(x i)1113868111386811138681113868

1113868111386811138681113868leKm (39)

for all i isinM x y isin R2+ with |x| or |y|lem

e proof is not particularly difficult so we omit theproof

Theorem 5 Assume that

21113954β1 gt σ21 + b1 +(1 +2

radic)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

21113954β2 gt σ22 +(1 +2

radic)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(40)

en the solution of (36) satisfies limt⟶infinx(t) 0 asfor any initial values (7) at is all populations in system(36) go to extinction with probability one

Proof

Step 1 let

V2 x1 x2( 1113857 xT

112

radic

12

radic 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠x x

21 +

2

radicx1x2 + x

22

(41)

Obviously V2 is positive-definite and radially un-bounded at is

lim|x|⟶infin

V2 x1 x2( 1113857 infin (42)

e generalized It1113954o formula yields

V2 x1(t) x2(t)( 1113857 V2 φ1(0)φ2(0)( 1113857 + 1113946t

0LV2(x(s) x

middot (s minus τ(ℓ(s))) ℓ(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(43)

By computation we know

LV2(x y i)le minus 2β1(i) minus σ21(i) minus b1(i) minus (1 +2

radic)b2(i) minus p1(i) minus

12

radic p2(i)1113890 1113891x21

minus 2β2(i) minus σ22(i) minus (1 +2

radic)b1(i) minus b2(i) minus

12

radic p1(i) minus p2(i)1113890 1113891x22

+ 1 +12

radic1113888 1113889p1(i)y21 + 1 +

12

radic1113888 1113889p2(i)y22

le minus λ1x21 minus λ2x

22 + 1 +

12

radic1113888 1113889p1y21 + 1 +

12

radic1113888 1113889p2y22

(44)

Complexity 7

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

where λ1 21113954β1 minus σ21 minus b1 minus (1 +2

radic)b2 minus p1 minus (1

2

radic)

p2 and λ2 21113954β2 minus σ22 minus (1 +2

radic)b1 minus b2 minus (1

2

radic)

p1 minus p2It is straightforward to see from (40) that λh gt 0 h

1 2 For simplicity we let

F1(x) min λ1 λ21113864 1113865|x|2

F2(x) 1 +12

radic1113888 1113889max p1 p21113864 1113865|x|2

(45)

By condition (40) again we obtain that

F(x) ≔ F1(x) minus F2(x)

min λ1 λ21113864 1113865 minus 1 +12

radic1113888 1113889max p1 p21113864 11138651113890 1113891|x|2 gt 0 xne 0

(46)

Applying (44) and (46) we derive

1113946t

0LV2(x(s) x(s minus τ(ℓ(s))) ℓ(s))ds

le 11139460

minus τF2(x(s))ds minus 1113946

t

0F(x(s))ds

(47)

Substituting the preceding equality into (43) it yields

V2 x1(t) x2(t)( 1113857leV2 φ1(0)φ2(0)( 1113857 + 11139460

minus τF2(x(s))ds

minus 1113946t

0F(x(s))ds

+ 1113946t

0

z

zxV2 x1(s) x2(s)( 11138571113890 1113891f2(x(s) ℓ

middot (s))dB(s)

(48)

en the nonnegative semimartingale convergencetheorem (see eg [37]) implies

limsupt⟶infin

V2 x1(t) x2(t)( 1113857ltinfin a s (49)

Moreover we obtain from (48) that

E 1113946t

0F(x(s))ds leV2 φ1(0)φ2(0)( 1113857 + 1113946

0

minus τF2(x(s))ds

(50)

en letting t⟶infin together with the Fubini the-orem we have

E 1113946infin

0F(x(t))dt ltinfin (51)

Let Ak ω|Y(ω) 1113938infin0 F(x(sω))dsgt 2k1113966 1113967 where

k 1 2 Obviously Ak1113864 1113865Ak+1 Combining Che-

byshevrsquos inequality and (51) we see that1113936infink1 P(Ak)ltinfin By BorelndashCantellirsquos lemma one can

show that P(limk⟶infinAk) P ω|Y(ω) infin 0 thatis

1113946infin

0F(x(t))dt ltinfin a s (52)

Step 2 from (52) we observe

lim inft⟶infin

F(x(t)) 0 a s (53)

One now needs to consider

limt⟶infin

F(x(t)) 0 a s (54)

If the above conclusion would not hold thenP limsupt⟶infinF(x(t))gt 01113864 1113865gt 0 So there isε isin (0 (13)) satisfying

P Ω1( 1113857ge 3ε (55)

where

Ω1 limsupt⟶infin

F(x(t))gt 2ε1113896 1113897 (56)

Noting that Lyapunov function V2(x(t)) and the so-lution x(t) of (6) are all continuous together with (49)it yields

supminus τletltinfin

V2(x(t))ltinfin as (57)

Define

](r) inf|x|gek

V2(x) for kgt 0 (58)

8 Complexity

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

Clearly

supminus τletltinfin

](|x(t)|)le supminus τletltinfin

V2(x(t))ltinfin a s (59)

In addition by (42) we get

limk⟶infin

](k) infin (60)

So

supminus τletltinfin

|x(t)|ltinfin a s (61)

Recalling (7) we know that the initial values satisfyφh isin C([minus τ 0] [0 +infin)) for h 1 2 We thereforecould find an integer mgt 0 depending on ε sufficientlylarge for |φ(s)|ltm for s isin [minus τ 0] almost surely while

P Ω2( 1113857ge 1 minus ε (62)

where Ω2 supminus τletltinfin|x(t)| ltm1113864 1113865 By (55) and (62)one implies

P Ω1 capΩ2( 1113857geP Ω1( 1113857 minus P Ωc2( 1113857ge 2ε (63)

where Ωc2 is the complement of Ω2 Let

ρ1 inf tge 0 F(x(t))ge 2ε

ρ2j inf tge ρ2jminus 1 F(x(t))le ε1113966 1113967 j 1 2

ρ2j+1 inf tge ρ2j F(x(t)) ge 2ε1113966 1113967 j 1 2

σm inf tge 0 |x(t)|gem

(64)

From (53) and the definitions of Ω1 and Ω2 we have

Ω1 capΩ21113864 1113865 sub σm infin1113864 1113865cap capinfin

j1ρj ltinfin1113966 11139671113888 11138891113896 1113897 (65)

Hence we define ζ(t) x(tand σm)1113864 1113865 for tgt minus τ andits differential is

dζ(t) 1113957f1(t)dt + 1113957f2(t)dB(t) (66)

where

1113957f1(t) f1(x(t) x(t minus τ) ℓ(t))I 0σm[ )(t)

1113957f2(t) f2(x(t) ℓ(t))I 0σm[ )(t)(67)

Here IA is the indicator function of A RecallingLemma 3 we know

1113957f1(t)1113868111386811138681113868

1113868111386811138681113868 or 1113957f2(t)1113868111386811138681113868

1113868111386811138681113868leKm a s (68)

for any tge minus τ ℓ(t) isinM and |x(t)|or |x(t minus τ)|lemMoreover the definition of yields |ζ(t)|lem tge 0 Wealso note that for all ω isin Ω1 capΩ2 and jge 1

F ζ ρ2jminus 11113872 11138731113872 1113873 minus F ζ ρ2j1113872 11138731113872 1113873 ε

F(ζ(t))ge ε t isin ρ2jminus 1 ρ2j1113960 1113961(69)

In the close ball Sm x isin R2 |x|lem1113864 1113865 F(middot) is uni-formly continuous erefore there exists ξ ξ(ε)gt 0small sufficiently such that

|F(ζ) minus F(ζ)|lt ε ζ ζ isin Sm with |ζ minus ζ|lt ξ (70)

For ω isin Ω1 capΩ2 we emphasize that if |ζ(ρ2jminus 1 + t) minus

ζ(ρ2jminus 1)|lt ξ for some Tgt 0 and t isin [0 T] thenρ2j minus ρ2jminus 1 geT Furthermore let the numberT T(ε ξ m)gt 0 be small enough such that

2K2mT(T + 4)le εξ2 (71)

By (63) and (65) we can obtain that

P ρ2j ltinfin1113872 1113873ge 2ε (72)

In particular if ρ2j ltinfin then |ζ(ρ2j)|ltm Hence thedefinition of ζ(t) implies ρ2j lt σm So

ζ(tω) x(tω) (73)

for all 0le tle ρ2j ω isin ρ2j ltinfin1113966 1113967 en from theHolder inequality and the BurkholderndashDavisndashGundyinequality it follows that

Complexity 9

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 1113873

11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

leE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

dζ(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds + 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎡⎣ ⎤⎦

le 2E I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)ds

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

+ 1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)dB(s)

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

le 2TE I ρ2jminus 1ltinfin1113864 1113865sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f1(s)1113868111386811138681113868

11138681113868111386811138682ds1113890 1113891 + 8E I ρ2jminus 1ltinfin1113864 1113865

sup0letleT

1113946ρ2jminus 1+t

ρ2jminus 1

1113957f2(s)1113868111386811138681113868

11138681113868111386811138682dB(s)1113890 1113891

le 2K2mT(T + 4)

(74)

which together with (71) and the Chebyshev inequalityimply easily that

P ρ2jminus 1 ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868ge ξ1113896 11138971113888 1113889le ε

(75)

Obviously we observe that ρ2jminus 1 ltinfin if ρ2j ltinfin Henceby (72) and (75) we can derive that

P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 1113873 minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868lt ξ1113896 11138971113888 1113889

P ρ2j ltinfin1113872 1113873 minus P ρ2j ltinfin1113966 1113967cap sup0letleT

ζ ρ2jminus 1 + t1113872 11138731113868111386811138681113868111386811138961113888

minus ζ ρ2jminus 11113872 111387311138681113868111386811138681113868ge ξ11138831113875

ge ε(76)

is together with (70) we can get that

P ρ2j ltinfin1113966 1113967cap ρ2j minus ρ2jminus 1 geT1113966 11139671113872 1113873ge ε (77)

Recalling (51) (69) and (77) for all ω isin Ω1 capΩ2 andjge 1 we compute

infingtE 1113946infin

0F(x(t))dt

ge 1113944infin

j1E I ρ2jltinfin1113864 1113865 1113946

ρ2j

ρ2jminus 1

F(ζ(t))dt1113890 1113891

ge ε1113944infin

j1E I ρ2jltinfin1113864 1113865

ρ2j minus ρ2jminus 11113872 11138731113876 1113877

ge εT 1113944infin

j1ε infin

(78)

which is a contradiction Consequently we infer (54)

Step 3 by (46) and (54) we now derive limt⟶infinx(t)

0 a s e proof is therefore complete

Corollary 1 Assume that are nonnegative constantsτrtequiv τ ge 0 and

2minrtisinS

αrt1113966 1113967gt max

rtisinSσrt

1113966 11139672

+ 3 +3

2

radic

21113888 1113889max

rtisinSprt

1113966 1113967 (79)

en solution x(t) of (2) obeys

limt⟶infin

x(t) 0 a s (80)

for any initial value x(t) ϕ(t) t isin [minus τ 0]

ϕ(0)gt 0ϕ isin C([minus τ 0] [0 +infin)) at is all populations inequation (2) go to extinction with probability one

Corollary 2 Assume that ah bh ph ch σh τ are nonneg-ative constants h 1 2 and

2a1 gt σ21 + b1 +(

2

radicminus 1)b2 + 2 +

12

radic1113888 1113889p1 +(1 +2

radic)p2

2a2 gt σ22 +(

2

radicminus 1)b1 + b2 +(1 +

2

radic)p1 + 2 +

12

radic1113888 1113889p2

(81)

en solution x(t) of (4) obeys

limt⟶infin

x(t) 0 a s (82)

for any initial value xh(t) ϕh(t) t isin [minus τ 0] ϕh

(0)gt 0ϕh isin C([minus τ 0] [0 +infin)) at is all populations inmodel (4) go to extinction with probability one

Remark 7 is theorem reveals that the solutions of (6) willall tend to the origin asymptotically with probability onewhen the intensities of noises and the parameters satisfycondition (40) However [21 28 31 32] do not study ex-tinction of populations Besides this method can be ex-tended to research extinction in the above literature studies

10 Complexity

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

Corollaries 1 and 2 give the conditions of extinction of (2)and (4) respectively erefore our work is the extension of[21 28 31 32]

6 Numerical Simulations

Based on [38] we show numerical simulations in the presentsection

Here we consider model (6) with the same initial dataφ1(0) 1 φ2(0) 05 and the same ℓ(t) tge0 onM 1 2 3 with

Γ

minus 10 4 6

2 minus 3 1

3 5 minus 8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (83)

en we know the Markov chain ℓ(t)tge0 is irreducibleand has a unique stationary distributionπ (01845 06019 02136)

In Figure 1 we give a simulation of the sample path ofℓ(t)tge0 with ℓ(0) 3

In Figure 2 we can choose a1 [1 2 3]b1 [012 023 022] p1 [4 5 6] c1 [04 05 06]

σ1 [12 11 2] a2 [3 1 2] b2 [023 016 012] p2

[7 4 3] c2 [03 04 05] σ2 [045 15 025] τ

[1 2 3] It is easy to see that 1 a22 lt (σ2222) 1125 whenξ(t) 2 ere is a good agreement between Lemma 1 andFigure 2 By eorem 2 we know ( p1 1113954c1eλ)+

( p2 1113954c2eλ) 141020 Furthermore we get Q max95553 63244 38081 by calculation erefore condi-tions of eorem 4 have been checked Solim supt⟶infin(1t)lnxh(t)le 47777 a s

In Figure 3 we can choose a1 [21 22 2] b1

[011 013 012] p1 [32 3 35] c1 [083 085 086]

σ1 [062 061 063] a2 [2 21 22] b2 [011 013

012] p2 [31 33 32] c2 [081 08 083] σ2 [071

075 071] Let θ 2 en conditions of eorem 1 couldbe checked By calculation we get lim supt⟶infin(1t)1113938

t

0 E|xh(s)|θdsle 319797 lim supt⟶infinE(xθ1(t) + xθ

2(t))le193208 lim supt⟶infinE|x(t)|θ le 386416 Figure 3 clearlysupports this result

In Figure 4 we can choose a1 [17 18 19] b1

[014 035 015] p1 [455 6 543] c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175] b2 [05 013

022] p2 [323 467 6] c2 [03 04 08] σ2 [049

1 065] τ [3 3 3] By calculation we get τ(1) τ(2)

τ(3) 3 35 21113954β1 gt σ21 + b1 + (1 +2

radic) b2 + (2 + (1

2

radic)

)p1+ (1 +2

radic)p2 asymp 325350 and 3328 21113954β2 gt σ22 + (1+

2radic

)b1 + b2 + (1 +2

radic)p1 + (2 + (1

2

radic))p2 asymp 330729

erefore conditions of eorem 5 have been checked

us from eorem 5 all species become extinct Figure 4clearly supports this result

ξ (t)

0

1

2

3

4

5

6

200 400 600 800 10000Time t

ξ (0) = 3

Figure 1 Sample path of ℓ(t) tge0 with ℓ(0) 3

Popu

latio

n

0

05

1

15

2

25

3

35

100 200 300 400 5000Time t

x1 (t)x2 (t)

Figure 2 Numerical solutions of (6) with a1 [1 2 3] b1 [012

023 022] p1 [4 5 6] c1 [04 05 06] σ1 [12 11 2]a2 [3 1 2] b2 [023 016 012] p2 [7 4 3] c2 [03 04

05] σ2 [045 15 025] and τ [1 2 3]

Complexity 11

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

7 Conclusions

By the conclusion of Lemma 1 it is worthy to point out thatthe Brownian noise and colored noise will not destroy a greatproperty that the solution of (6) may not explode Especiallysystem (6) reduces to (1)ndash(4) or the model in [32] whenparameters of system (6) take some special values FromLemma 1 the condition αgt (σ22) in [21] is too strict and

unnecessary In eorem 1 we comprehensively analyzeultimate boundedness in the θth moment and boundednessfor the average in time of the θth moment of solution whichis the improvement of eorem 31 in [21] eorem 22 in[28] eorem 33 in [31] and eorem 32 in [32] Ineorem 4 we find an upper bound Q2 of the sampleLyapunov exponent When parameters of system (6) takesome special values we compute that the upper bound Q2 isless than the corresponding upper bound in [21 28] Fur-thermore we find that the condition 2α1 minus σ21 minus (b1+

b2)ϵgt 0 2α2 minus σ22 minus b1 + b2ϵgt 0 in [31] is not necessaryDespite all this if we let parameter ϵ satisfy the aboveconditions we compute that Q2 is less than the upperbound in [31] One point should be stressed is that themethod for extinction ineorem 5 can be used successfullyfor the models in [21 28 31 32] And then Corollaries 1 and2 give the conditions of extinction of (2) and (4) respec-tively From Remarks 1ndash7 our work is a generalization andpromotion of the corresponding work in [21 28 30ndash32] Tosome extent our proposed approaches are both more robustand more efficient than the existing methods

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed equally and significantly towriting this paper All authors read and approved the finalmanuscript

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (no 11971279)

References

[1] W S C Gurney S P Blythe and R M Nisbet ldquoNicholsonrsquosblowflies revisitedrdquo Nature vol 287 no 5777 pp 17ndash211980

[2] A Nicholson ldquoAn outline of the dynamics of animal pop-ulationsrdquoAustralian Journal of Zoology vol 2 no 1 pp 9ndash651954

[3] W Chen and B Liu ldquoPositive almost periodic solution for aclass of Nicholsonrsquos blowflies model with multiple time-varying delaysrdquo Journal of Computational and AppliedMathematics vol 235 no 8 pp 2090ndash2097 2011

[4] M Yang ldquoExponential convergence for a class of Nicholsonrsquosblowflies model withmultiple time-varying delaysrdquoNonlinearAnalysis Real World Applications vol 12 no 4 pp 2245ndash2251 2011

[5] L Berezansky E Braverman and L Idels ldquoNicholsonrsquosblowflies differential equations revisited main results andopen problemsrdquo Applied Mathematical Modelling vol 34no 6 pp 1405ndash1417 2010

Popu

latio

n

0

02

04

06

08

1

12

14

16

18

20 40 60 80 1000Time t

x1 (t)x2 (t)

Figure 3 Numerical solutions of (6) with a1 [21 22 2]b1 [011 013 012] p1 [32 3 35] c1 [083 085 086]σ1 [062 061 063] a2 [2 21 22] b2 [011 013 012]p2 [31 33 32] c2 [081 08 083] σ2 [071 075 071] τ

[1 2 3]

Popu

latio

n

0

02

04

06

08

1

12

5 10 150Time t

x1 (t)x2 (t)

Figure 4 Numerical solutions of (6) e parameters of model (5)are a1 [17 18 19] b1 [014 035 015] p1 [455 6 543]

c1 [08 05 06] σ1 [05 045 012] a2 [165 169 175]b2 [05 013 022] p2 [323 467 6] c2 [03 04 08] σ2

[049 1 065] and τ [3 3 3]

12 Complexity

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13

[6] T Faria and G Rost ldquoPersistence permanence and globalstability for an n-dimensional Nicholson systemrdquo Journal ofDynamics and Differential Equations vol 26 no 3pp 723ndash744 2014

[7] C Huang X Yang and J Cao ldquoStability analysis of Nich-olsonrsquos blowflies equation with two different delaysrdquo Math-ematics and Computers in Simulation vol 171 pp 201ndash2062020

[8] X Long and S Gong ldquoNew results on stability of Nicholsonrsquosblowflies equation with multiple pairs of time-varying delaysrdquoApplied Mathematics Letters vol 100 Article ID 1060272020

[9] J Zhang and C Huang ldquoDynamics analysis on a class ofdelayed neural networks involving inertial termsrdquo Advancesin Difference Equations vol 2020 p 120 2020

[10] C Qian and Y Hu ldquoNovel stability criteria on nonlineardensity-dependent mortality Nicholsonrsquos blowflies systems inasymptotically almost periodic environmentsrdquo Journal ofInequalities and Applications vol 2020 p 13 2020

[11] W Wang F Liu and W Chen ldquoExponential stability ofpseudo almost periodic delayed Nicholson-type system withpatch structurerdquo Mathematical Methods in the Applied Sci-ences vol 42 no 2 pp 592ndash604 2019

[12] C Huang X Long L Huang and S Fu ldquoStability of almostperiodic Nicholsonrsquos blowflies model with involoving patchstructure and mortality termsrdquo Canadian MathematicalBulletin vol 63 no 2 pp 405ndash422 2019

[13] C Huang H Zhang and L Huang ldquoAlmost periodicityanalysis for a delayed Nicholsonrsquos blowflies model withnonlinear density-dependent mortality termrdquo Communica-tions on Pure amp Applied Analysis vol 18 no 6 pp 3337ndash33492019

[14] R M May Stability and Complexity in Model EcosystemsPrinceton University Press Princeton NJ USA 1973

[15] R Liu and G Liu ldquoAsymptotic behavior of a stochastic two-species competition model under the effect of diseaserdquoComplexity vol 2018 Article ID 3127404 15 pages 2018

[16] W Wang C Shi and W Chen ldquoStochastic Nicholson-typedelay differential systemrdquo International Journal of Control2019 In press

[17] W Wang and W Chen ldquoStochastic delay differential neo-classical growth modelrdquo Advances in Difference Equationsvol 2019 p 355 2019

[18] X Mao ldquoStationary distribution of stochastic populationsystemsrdquo Systems amp Control Letters vol 60 no 6pp 398ndash405 2011

[19] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexityvol 2017 Article ID 1950907 15 pages 2017

[20] F Bian W Zhao Y Song and R Yue ldquoDynamical analysis ofa class of prey-predator model with Beddington-DeAngelisfunctional response stochastic perturbation and impulsivetoxicant inputrdquo Complexity vol 2017 Article ID 374219718 pages 2017

[21] W Wang L Wang and W Chen ldquoStochastic Nicholsonrsquosblowflies delayed differential equationsrdquoAppliedMathematicsLetters vol 87 pp 20ndash26 2019

[22] J M Drake ldquoAllee effects and the risk of biological invasionrdquoRisk Analysis vol 24 no 4 pp 795ndash802 2004

[23] M Assaf A Kamenev and B Meerson ldquoPopulation ex-tinction risk in the aftermath of a catastrophic eventrdquo PhysicalReview E vol 79 no 1 Article ID 011127 2009

[24] X Mao ldquoStability of stochastic differential equations withMarkovian switchingrdquo Stochastic Processes and 7eir Appli-cations vol 79 no 1 pp 45ndash67 1999

[25] Q Luo and X Mao ldquoStochastic population dynamics underregime switching IIrdquoMathematical Analysis and Applicationsvol 355 no 2 pp 577ndash593 2007

[26] T K Siu ldquoBond pricing under a Markovian regime-switchingjump-augmented Vasicek model via stochastic flowsrdquoAppliedMathematics and Computation vol 216 no 11 pp 3184ndash3190 2010

[27] A Settati and A Lahrouz ldquoStationary distribution of sto-chastic population systems under regime switchingrdquo AppliedMathematics and Computation vol 244 pp 235ndash243 2014

[28] Y Zhu K Wang Y Ren and Y Zhuang ldquoStochasticNicholsonrsquos blowflies delay differential equation with regimeswitchingrdquo Applied Mathematics Letters vol 94 pp 187ndash1952019

[29] L J S Allen ldquoPersistence extinction and critical patchnumber for island populationsrdquo Journal of MathematicalBiology vol 24 no 6 pp 617ndash625 1987

[30] L Berezansky L Idels and L Troib ldquoGlobal dynamics ofNicholson-type delay systems with applicationsrdquo NonlinearAnalysis Real World Applications vol 12 no 1 pp 436ndash4452011

[31] X Yi and G Liu ldquoAnalysis of stochastic Nicholson-type delaysystem with patch structurerdquo Applied Mathematics Lettersvol 96 pp 223ndash229 2019

[32] W Wang and W Chen ldquoStochastic Nicholson-type delaysystem with regime switchingrdquo Systems amp Control Lettersvol 136 p 104603 2020

[33] A Lahrouz and A Settati ldquoNecessary and sufficient conditionfor extinction and persistence of SIRS system with randomperturbationrdquo Applied Mathematics and Computationvol 233 pp 10ndash19 2014

[34] A Settati S Hamdoune A Imlahi and A Akharif ldquoEx-tinction and persistence of a stochastic Gilpin-Ayala modelunder regime switching on patchesrdquo Applied MathematicsLetters vol 90 pp 110ndash117 2019

[35] X Mao Ren and C Yuan Stochastic Differential Equationswith Markovian Switching Imperical College Press LondonUK 2006

[36] X Mao Stochsatic Differential Equations and ApplicationsHorwood Chichester UK 2007

[37] R S Liptser and A N Shiryayev 7eory of MartingalesKluwei Academic Publisherm Dordrecht Netherlands 1989

[38] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

Complexity 13