Chaos, Solitons and Fractals 42 (2009) 1485–1494

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Bifurcations of a singular prey–predator economic model with timedelay and stage structure q

Xue Zhang a,b, Qing-Ling Zhang a,b,*, Chao Liu a,b, Zhong-Yi Xiang c

a Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110004, PR Chinab Key Laboratory of Integrated Automation of Process Industry (Northeastern Univ.), Ministry of Education, Shenyang, Liaoning 110004, PR Chinac Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Accepted 11 March 2009

0960-0779/$ - see front matter � 2009 Published bdoi:10.1016/j.chaos.2009.03.051

q This work was supported by National Science Fouand the Youth Science Foundation of Educational D

* Corresponding author.E-mail addresses: zhangxueer@gmail.com (X. Zha

This paper studies a singular prey–predator economic model with time delay and stagestructure. Compared with other researches on dynamics of prey–predator population, thismodel is described by differential–algebraic equations due to economic factor. For zeroeconomic profit, this model exhibits three bifurcational phenomena: transcritical bifurca-tion, Hopf bifurcation and singular induced bifurcation. For positive economic profit, themodel undergoes a saddle-node bifurcation at critical value of positive economic profit,and the increase of delay destabilizes the positive equilibrium point of the system andbifurcates into small amplitude periodic solution. Finally, by using Matlab software,numerical simulations illustrate the effectiveness of the results.

� 2009 Published by Elsevier Ltd.

1. Introduction

At present, many differential–algebraic equations have been studied in the general power systems [1,2], economic admin-istration [3], robot system [4], mechanical engineering [5] and so on. And a lot of important results have been obtained, suchas local stability, optimal control, singular induced bifurcation, feasibility regions and so on. However, to our best knowledge,the reports of such singular biological systems are few.

In recent years, the growing human needs for more food and more energy have led to increased exploitation of theseresources. The problems related to many fields like fishery, forestry and wildlife. Therefore, mankind is facing the dual prob-lems of resource shortages and environmental degradation.Concerning the conservation for the long-term benefits ofhumanity, there is a wide-range of interest in analysis and modelling of bioeconomic systems. In many earlier studies, ithas been shown that harvesting has a strong impact on population dynamics, ranging from rapid depletion to complete pres-ervation of biological populations. Two main kinds of harvesting were focused on nonzero constant harvesting [6–8] andconstant harvesting effort [9–11]. With constant harvesting, the prey–predator model is found to have interesting dynamicalbehavior including stability, Hopf bifurcation and limit cycle [6,7]. Zhang et al. [8] have investigated a Michaelis–Menten-type predator–prey system with reaction-diffusion and constant harvesting. The conditions for Hopf and Turing bifurcationson the spatial domain were obtained. Das et al. [9] discussed the bioeconomic harvesting of a prey–predator fishery in whichboth species are infected by some toxicant. Gakkhar and Singh [10] have studied the optimal harvesting policy for harvestingpredator and observed that harvesting can control the chaos. Kar and Pahari [11] have investigated the dynamical behaviorof an exploited system consisting of two preys and a predator which is being harvested. The harvesting effort has a stabiliz-

y Elsevier Ltd.

ndation of China (60574011), the Nature Science Foundation of Hubei Province in China (2008CDB075),epartment of Hubei Province in China, the Science Foundation (Q20082906).

ng), qlzhang@mail.neu.edu.cn (Q.-L. Zhang).

1486 X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494

ing effect on equilibrium when it is under a critical level. Most of these discussions are based on differential equations ordifference equations.

In 1954, Gordon [12] studied the effect of harvest effort on ecosystem from an economic perspective and proposed thefollowing economic theory:

Net Economic RevenueðNERÞ ¼ Total RevenueðTRÞ � Total CostðTCÞ:

This provides theoretical evidence for the establishment of differential–algebraic equation. Previous study [13] has shown aclass of singular biological economic model with Holling type II functional form and strength kernel function delay

dxdt ¼ x r1 � a11x� y

aþx

� �;

dydt ¼ y �r2 þ bz

aþz� a22y� �

� Ey;

zðtÞ ¼R t�1 xðsÞ 1

T exp � 1T ðt � sÞ

� �ds;

0 ¼ Eðpy� cÞ �m;

8>>>>>><>>>>>>:

ð1Þ

where x and y represent the prey density and predator density at time t, respectively; z describes the entire past history (be-fore time t) of prey population x in the third equation, where the importance of predator gestation is measured by1T exp 1

T ðt � sÞ� �

and T is the measure of the influence of the past; r1 > 0 and r2 > 0 are the intrinsic growth rate and intrinsicmortality rate of respective species; aii > 0, i ¼ 1;2 represent the strength of respective intraspecific competition; the pred-ator consumes the prey with Holling II functional response xy=ðaþ xÞ and contributes to its growth rate byz=ðaþ zÞ. E rep-resents harvesting effort, p > 0, c > 0 and m P 0 are harvesting reward per unit harvesting effort for unit weight of predator,harvesting cost per unit harvesting effort for predator and the economic profit per unit harvesting effort, respectively.According to Gordon’s economic theory equation mentioned above, NER ¼ m, TR ¼ pEðtÞyðtÞ and TC ¼ cEðtÞ. Hence, the fourthalgebraic equation of the model (1) was established.

It is well known that many animals need undergo the transition from immaturity to maturity, such as mammalian pop-ulations and some amphibious animals. Stage–structure models have received much attention in recent years [14–17]. Xiaoet al. [14] studied the complicated dynamics occurring in a class of discrete-time models of predator–prey interaction basedon age-structure of predator. Gao and Chen [15,16] studied the dynamics of an exploited single-species discrete model withstage structure in a fish population. Bandyopadhyay and Banerjee [17] introduced a stage-structured predator prey model(stage-structure on prey) with gestation delay and obtained certain thresholds on the rate at which juveniles becomes adultsand the length of delay to preserve stability.

Now, based on the work done by Bandyopadhyay and Banerjee, we develop and study a new singular predator–prey bio-economic model with time delay and stage-structure, which is given by the following differential–algebraic equations

dxidt ¼ r1xm � d1xi � axi � a11x2

i � a12xiy;dxmdt ¼ axi � d2xm;

dydt ¼ a21xiðt � sÞyðt � sÞ � d3y� a22y2 � Ey;

0 ¼ Eðpy� cÞ �m;

8>>>>><>>>>>:

ð2Þ

where xi, xm and y represent densities of immature prey population, mature prey population and predator population at timet, respectively. r1 is the birth rate of the immature prey. a is the rate of transformation of immatures to matures. aii ði ¼ 1;2Þrepresent the strength of the immature prey and predator intraspecific competition, respectively. di ði ¼ 1;2;3Þ are the deathrate of the immature prey, mature prey and predator, respectively. a12 is the rate of predation of predator and the rate ofconversing prey into predator is a21. s is the time required for the gestation of predator. The fourth algebraic equation isthe same as (1). All the parameters mentioned in the system (2) take positive values except that m is nonnegative.

For simplicity, let

f ðX; E;lÞ ¼f1ðX; E;lÞf2ðX; E;lÞf3ðX; E;lÞ

0B@

1CA ¼

r1xm � d1xi � axi � a11x2i � a12xiy

axi � d2xm

a21xiðt � sÞyðt � sÞ � d3y� a22y2 � Ey

0B@

1CA;

gðX; E;lÞ ¼ Eðpy� cÞ �m;

where X ¼ ðxi; xm; yÞT , l is a bifurcation parameter, which will be defined in what follows.In this paper, the effects of time delay and economic profit on the model dynamics are discussed in the region

R4þ ¼ fðxi; xm; y; EÞjxi P 0; xm P 0; y P 0; E P 0g. For zero economic profit, Section 2 discusses the existence and stability of

equilibrium points and gives sufficient conditions of existence for transcritical bifurcation, Hopf bifurcation and singular in-duced bifurcation. For positive economic profit, Section 3 gives sufficient conditions of saddle-node bifurcation and smallamplitude periodic solution around positive equilibrium point. In Section 4, numerical simulations verify the effectivenessof mathematical conclusions. Finally, a brief discussion is given in Section 5 to conclude this paper.

X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494 1487

2. The model with zero economic profit

When the economic profit is zero, the system (2) can be written by

dxidt ¼ r1xm � d1xi � axi � a11x2

i � a12xiy;dxmdt ¼ axi � d2xm;

dydt ¼ a21xiðt � sÞyðt � sÞ � d3y� a22y2 � Ey;

0 ¼ Eðpy� cÞ:

8>>>><>>>>:

ð3Þ

By the analysis of equilibrium points for the system (3), the following result is obtained:

Theorem 1

(i) The system (3) has three boundary equilibrium points P01ð0;0; 0;0Þ, P0

2ðx0i2; x

0m2;0; 0Þ where x0

i2 ¼r1a�d1d2�ad2

a11d2and x0

m2 ¼ ad2

x0i2,

P03ðx0

i3; x0m3; y

03;0Þ where x0

i3 ¼a22ðr1a�d1d2�ad2Þþa12d2d3

d2ða11a22þa12a21Þ, x0

m3 ¼ ad2

x0i3 and y0

3 ¼a21x0

i3�d3

a22. P0

2 exists for r1a� d1d2 � ad2 > 0 and P03

exists for a21ðr1a� d1d2 � ad2Þ > a11d2d3.(ii) The system (3) has a positive equilibrium point P0

4ðx0i4; x

0m4; y

04; E

04Þ, where x0

i4 ¼pðr1a�d1d2�ad2Þ�a12cd2

a11d2p , x0m4 ¼ a

d2x0

i4, y04 ¼ c

p and

E04 ¼ a21x0

i4 � a22y04 � d3. P0

4 exists for a21pðr1a� d1d2 � ad2Þ � a12a21cd2 > a11d2ðpd3 þ a22cÞ.

The Jacobian matrix of the system (3) is

A0 ¼ DXf � DEf ðDEgÞ�1DXg ¼�A0

11 r1 �a12xi

a �d2 0a21ye�ks 0 �A0

33

0B@

1CA;

where

A011 ¼ d1 þ aþ 2a11xi þ a12y;

A033 ¼ �a21xie�ks þ d3 þ 2a22yþ E� pEy

py� c:

So we obtain easily that the equilibrium point P01 is a saddle when P0

2 exists.The characteristic polynomial at the equilibrium point P0

2 is

ðk� a21x0i2e�ks þ d3Þ k2 þ d2 � d1 � aþ 2r1a

d2

� �kþ r1a� d1d2 � ad2

� ¼ 0: ð4Þ

Clearly, the eigenvalues of the equation k2 þ d2 � d1 � aþ 2r1ad2

� �kþ r1a� d1d2 � ad2 ¼ 0 always have negative real parts. The

third eigenvalue is given by the solution of the following equation:

k� a21ðr1a� d1d2 � ad2Þa11d2

e�ks þ d3 ¼ 0: ð5Þ

By the analysis of roots for Eq. (5), we obtain that the equilibrium point P02 is a stable focus or node if

a11d2d3 > a21ðr1a� d1d2 � ad2Þ; otherwise P02 is a saddle. The stable equilibrium point P0

2 becomes unstable when the effi-ciency rate of conversing prey into predator a21 increases through a11d2d3

r1a�d1d2�ad2. Therefore, if a21 is regarded as bifurcation

parameter, i.e., l ¼ a21, the following result is obtained:

Theorem 2. If r1a� d1d2 � ad2 > 0, the system (3) undergoes transcritical bifurcation at the equilibrium point P02 when

bifurcation parameter l is increased through a11d2d3r1a�d1d2�ad2

.

Proof. When the bifurcation parameter l ¼ a21 ¼ a11d2d3r1a�d1d2�ad2

, the characteristic polynomial (4) at the equilibrium point P02

has a simple zero eigenvalue with left eigenvector uT ¼ ð0 0 1Þ and right eigenvector vT ¼ ða12d2 a12a � a11d2ÞT . And

uTðDXDlfRÞvT jP02

¼ 0 0 1ð Þ0 0 0

0 0 0

0 0 x0i2

0BB@

1CCA

a12d2

a12a

�a11d2

0B@

1CA ¼ d1d2 þ ad2 � r1a;

uT D2XfRðvT ;vT ;vTÞjP0

2¼ uT

P3i¼1

eiuTT DXðDXfiÞTvT

� �P0

2

¼ �2a11d22ða11a22 þ a12a21Þ;

where ei ði ¼ 1;2;3Þ are unit vectors.According to the literature [18], the system (3) undergoes transcritical bifurcation at the equilibrium point P0

2. h

1488 X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494

Remark 1. Transcritical bifurcation implies that the equilibrium point P02 remains stable if the rate of conversing prey into

predator a21 is lower than the critical point a11d2d3r1a�d1d2�ad2

, otherwise the stability of P02 is lost and the equilibrium point P0

3 exists.

The characteristic polynomial at the equilibrium point P03 x0

i3; x0m3; y

03;0

� �is given by

k3 þ p1k2 þ p2kþ p3 þ p4k

2 þ p5kþ p6

� �e�ks ¼ 0; ð6Þ

where

p1 ¼ a11x0i3 þ

r1ad2þ d2 þ a21x0

i3 þ a22y03;

p2 ¼ a21x0i3 þ a22y0

3

� �a11x0

i3 þr1ad2þ d2

� �þ d2a11x0

i3;

p3 ¼ a21x0i3 þ a22y0

3

� �d2a11x0

i3;

p4 ¼ �a21x0i3;

p5 ¼ �a21x0i3 a11x0

i3 þr1ad2þ d2 � a12y0

3

� �;

p6 ¼ �a21d2x0i3 a11x0

i3 � a12y03

� �:

In fact, Bandyopadhyay and Banerjee [17] have investigated the dynamics of the system (3) without harvesting. Hence,assuming that a purely imaginary solution of the form k ¼ ix0 exists for Eq. (6), where x0 is the root of the followingequation:

x6 þ p21 � 2p2 � p2

4

� �x4 þ p2

2 � p25 � 2p1p3 þ 2p4p6

� �x2 þ p2

3 � p26 ¼ 0; ð7Þ

it is easy to obtain that if the following condition:

a21ð3a11a22 � a12a21Þðr1a� d1d2 � ad2Þ þ ð3a12a21 � a11a22Þa11d2d3 < 0 ð8Þ

holds, the system (3) undergoes Hopf bifurcation at the equilibrium point P03 when

s0k ¼

1x0

arccosp5x2

0 x20 � p2

� �þ p1x2

0 � p3

� �p6 � p4x2

0

� �p2

5x20 þ p6 � p4x2

0

� �2 þ 2kpx0

; k ¼ 0;1;2; . . .

It is noted that det DEg ¼ 0 at the equilibrium point P04. Regarding m as the bifurcation parameter, i.e., l ¼ m, the following

singular induced bifurcation is discussed, which is particular in differential–algebraic equations.

Theorem 3. If a21pðr1a� d1d2 � ad2Þ � a12a21cd2 > a11d2ðpd3 þ a22cÞ, the system (3) undergoes singular induced bifurcation atthe equilibrium point P0

4 when the bifurcation parameter l increases through 0. Moreover, the stability of the equilibrium point P04

changes, i.e., from stable to unstable.

Proof. Let D ¼ DEg ¼ py� c. At the equilibrium point P04, D has a simple zero eigenvalue and

DXf DEf

DXg DEg

P0

4

¼

�d1 � a� 2a11xi � a12y r1 �a12xi 0a �d2 0 0

a21y 0 �a22y �y

0 0 pE py� c

P0

4

¼ pa11d2x0i4y0

4E04;

traceðDEfadjðDEgÞDXgÞjP04¼ trace

00�y

0B@

1CA 0 0 pEð Þ

0B@

1CA

P04

¼ �cE04;

DXf DEf Dlf

DXg DEg Dlg

DXD DED DlD

P04

¼

� r1ad2� a11xi r1 �a12xi 0 0

a �d2 0 0 0a21y 0 �a22y �y 0

0 0 pE py� c �10 0 p 0 0

P0

4

¼ pa11d2x0i4y0

4:

On the other hand, we get

c1 ¼ �traceðDEfadjðDEgÞDXgÞjP04¼ cE0

4 > 0;

c2 ¼ DlD� ðDXD DEDÞDXf DEfDXg DEg

� ��1 Dlf

Dlg

� � !P0

4

¼ 1E0

4

> 0:

X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494 1489

According to [19], the system (3) undergoes singular induced bifurcation (the description and explanation of singular in-duced bifurcation theorem can be seen in Appendix) at the equilibrium point P0

4 when the bifurcation parameterl ¼ m ¼ 0. If l increases through 0, one eigenvalue of the system (3) moves from C� (the open complex left half plane)to Cþ (the open complex right half plane) along the real axis by diverging through1, which causes impulsive phenomenonof singular system, i.e., rapid expansion of the population from the view of biological explanation. By a simple calculation, theothers remain bounded and stay at C� away from the origin. Therefore, the stability of the equilibrium point P0

4 changes, i.e.,from stable to unstable. h

Remark 2. Local bifurcation analysis of singular system often results in three different bifurcation types: saddle-node bifur-cation, Hopf bifurcation and singularity induced bifurcation (SIB). The SIB is a new type of bifurcations and does not occur inordinary differential equation system. For the differential–algebraic equations (DAEs) with a one-dimensional parameter lmentioned in Appendix, SIB occurs when an equilibrium point crosses the following singular surface:

S :¼ fðx; y;lÞjgðx; y;lÞ ¼ 0; det½Dygðx; y;lÞ� ¼ 0g:

At the SIB, the equilibrium point undergoes stability exchanges and one eigenvalue of the system Jacobian matrixðDxf � Dyf ðDygÞ�1DxgÞ becomes unbounded. One important implication of the occurrence of SIB is that it causes impulsivephenomenon, which might yield catastrophic consequences, for example, hard fault [20] of electrical circuits and rapidexpansion of population for biological system. More details of SIB can be found in [19].

3. The model with positive economic profit

In this section, we investigate how economic profit and time delay influence the stability of the system (2).

3.1. The system (2) without time delay

We consider the system (2) in the absence of time delay, which can be written as follows:

dxidt ¼ r1xm � d1xi � axi � a11x2

i � a12xiy;dxmdt ¼ axi � d2xm;

dydt ¼ a21xiy� d3y� a22y2 � Ey;

0 ¼ Eðpy� cÞ �m:

8>>>>><>>>>>:

ð9Þ

If the system (9) has positive equilibrium point, coordinate of y is the root of the following equation:

s1y2 þ s2yþ s3 ¼ 0; ð10Þ

where

s1 ¼ pða11a22 þ a12a21Þ;

s2 ¼ �pa21r1ad2� d1 � a

� �þ a11d3p� cða11a22 þ a12a21Þ;

s3 ¼ ca21r1ad2� d1 � a

� �� a11d3c þ a11m:

By the analysis of the roots in Eq. (10), we can obtain that if there exists

msn ¼½a11d2d3p� a21pðr1a� d1d2 � ad2Þ þ cd2ða11a22 þ a12a21Þ�2

4pa11d22ða11a22 þ a12a21Þ

> 0;

(1) for m > msn, there are no positive equilibrium point,m m m m m� �

m 1 r1a m� �

m a m

(2) for m ¼ msn, only one positive equilibrium point P� xi� ; xm�; y� ; E� , where xi� ¼ a11 d2� d1 � a� a12y� , xm� ¼ d2

xi� ,

Em� ¼ msn

pym� �c and ym

� ¼a21pðr1a�d1d2�ad2Þ�a11pd2d3þcd2ða11a22þa12a21Þ

2pd2ða11a22þa12a21Þ,

(3) for m < msn, two positive equilibrium points Pmþ xm

iþ; xmmþ; y

mþ ; E

mþ

� �and Pm

�ðxmi�; x

mm�; y

m� ; E

m�Þ.

Therefore, if m is regarded as bifurcation parameter again, i.e., l ¼ m, the following result is obtained:

Theorem 4. If cp <

a21pðr1a�d1d2�ad2Þ�a11pd2d3þcd2ða11a22þa12a21Þ2pd2ða11a22þa12a21Þ < 1

a12

r1ad2� d1 � a

� �, the system (9) undergoes saddle-node bifurcation

at m ¼ msn.

1490 X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494

Proof. The characteristic polynomial of the system (9) at the equilibrium point Pm� is given by

k3 þ l1k2 þ l2kþ l3 ¼ 0; ð11Þ

where

l1 ¼ a11xmi� þ

r1ad2þ d2 þ a22ym

� �pEm� ym�

pym� � c

;

l2 ¼ a22ym� �

pEm� ym�

pym� � c

� �a11xm

i� þr1ad2þ d2

� �þ d2a11xm

i� þ a12a21xmi�y

m� ;

l3 ¼ a11d2xmi� a22ym

� �pEm� ym�

pym� � c

� �þ a12a21d2xm

i�ym� :

Clearly, it has a geometrically simple zero eigenvalue with left eigenvector usn ¼ d2 r1a11d2xm

i�a21ym

�

� �and right eigenvector

v sn ¼ d2 a � a11d2a12

� �T. And

usnDlfRjPm�¼ usnðDlf � DEf ðDEgÞ�1DlgÞjPm

�¼ � a11d2xm

i�a21ðpym

� � cÞ ;

usnD2XfRðvsn;v sn;v snÞjPm

�¼ usn

P3i¼1

eivTsnDXðDXfiÞTvsn

� �Pm�¼ � 2a2

11d32xm

i�a2

12a21ym�ða11a22 þ a12a21Þ;

where ei ði ¼ 1;2;3Þ are unit vectors.According to [19], the system (9) undergoes saddle-node bifurcation at the equilibrium point Pm

� when m ¼ msn. h

3.2. The analysis of the system (2)

In this section, we assume that economic profit satisfies 0 < m < msn. From the discussion of Section 3.1, there exist twopositive equilibrium points Pm

þ and Pm� .

The characteristic polynomial of the system (2) has the following form:

k3 þ q1k2 þ q2kþ q3 þ q4k

2 þ q5kþ q6

� �e�ks ¼ 0; ð12Þ

where

q1 ¼ a11xi þr1ad2þ d2 þ a21xi þ a22y� pEy

py� c;

q2 ¼ a21xi þ a22y� pEypy� c

� �a11xi þ

r1ad2þ d2

� �þ d2a11xi;

q3 ¼ a21xi þ a22y� pEypy� c

� �d2a11xi;

q4 ¼ �a21xi;

q5 ¼ �a21xi a11xi þr1ad2þ d2 � a12y

� �;

q6 ¼ �a21d2xiða11xi � a12yÞ:

If ix is a root of the system (12), separating the real and imaginary parts, we have

x3 � q2x� q5xcosxsþ q6 � q4x2� �

sinxs ¼ 0;q1x2 � q3 � q6 � q4x2

� �cosxs� q5xsinxs ¼ 0;

ð13Þ

which leads to

x6 þ q21 � 2q2 � q2

4

� �x4 þ q2

2 � q25 � 2q1q3 þ 2q4q6

� �x2 þ q2

3 � q26 ¼ 0: ð14Þ

Clearly, the coefficients qi ði ¼ 1;2; . . . ;6Þ of Eq. (12) are those of Eq. (6) at the equilibrium point P03. Since f ðX; E;lÞ and

gðX; E;lÞ are continuous and P03 is stable for the system (3) without time delay, there exists a m1 such that when

m 2 ð0;m1Þ, the system (9) has a locally asymptotically stable equilibrium point. Without any loss of generality, assume thatPmþ xm

iþ; xmmþ; y

mþ ; E

mþ

� �is locally asymptotically stable equilibrium point of the system (9). From Eq. (14), it is seen that q2

3 � q26

corresponds to p23 � p2

6 of Eq. (7). The condition (8) implies p23 � p2

6 < 0 (see [17]), hence there exists a m2 so that q23 � q2

6 < 0for m 2 ð0;m2Þ, i.e., Eq. (14) has one positive solution xm when m 2 ð0;m2Þ. The critical values of the delay corresponding toxm is given by

smk ¼

1xm

arccosq5x2

m x2m � q2

� �þ q1x2

m � q3

� �q6 � q4x2

m

� �q2

5x2m þ q6 � q4x2

m

� �2 þ 2kpxm

; k ¼ 0;1;2; . . . ð15Þ

X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494 1491

Differentiating (12) with respect to the delay s, we have

Redkds

� ��1

smk

¼2x6

m þ q21 � q2

4 � 2q2

� �x4

m þ q26 � q2

3

x2m ðq3 � q1x2

mÞ2 þx2

m q2 �x2m

� �2h i :

From Section 2, we can obtain that p21 � p2

4 � 2p2 ¼ a11x0i3 þ

r1ad2

� �2þ d2

2 þ 2r1aþ a22y03 a22y0

3 þ 2a21x0i3

� �> 0 at the equilibrium

point P03. So, there exists a m3 so that q2

1 � q24 � 2q2 > 0 for m 2 ð0;m3Þ. When the economic profit m 2 ð0;minfm1;

m2;m3;msngÞ, we have

dðRekÞds

� �sm

k

> 0:

Then the following result is obtained:

Theorem 5. If the economic profit m 2 ð0;minfm1;m2;m3;msngÞ, the system (2) undergoes Hopf bifurcation at the equilibriumpoint Pm

þ when s ¼ smk , k ¼ 0;1;2; . . . Furthermore, for s > sm

0 and s� sm0

� 1, an attracting invariant closed curve bifurcatesfrom the equilibrium point Pm

þ .

4. Numerical simulation

To illustrate the results obtained, let us consider the following particular cases.Case I. For the parameter values r1 ¼ 3; d1 ¼ 1:3; a ¼ 1:7; a11 ¼ 0:2; a12 ¼ 1:5; d2 ¼ 1; a21 ¼ 1; d3 ¼ 0:7; a22 ¼ 0:8;

p ¼ 3; c ¼ 1; m ¼ 0, the system (3) has a positive equilibrium point P04ð8;13:6;0:333;7:033Þ. In the neighbor of m ¼ 0,

two values of economic profit are chosen: when m ¼ �0:007, the eigenvalues of the system (3) at the equilibrium pointP0

4 are �0.214, �7.029, �7.487; when m ¼ 0:01, the eigenvalues are �0.214, �7.484, 3520.7. Obviously, two eigenvalues re-main almost constant and another moves from C� to Cþ along the real axis by diverging through 1.

Case II. When r1 ¼ 8; d1 ¼ 1:2; a ¼ 2:6; a11 ¼ 0:2; a12 ¼ 1:5; d2 ¼ 0:9; a21 ¼ 1:5; d3 ¼ 0:7; a22 ¼ 0:8; p ¼ 2; c ¼ 0:5;

m ¼ msn ¼ 826:352, there is a unique positive equilibrium point Pm� ð50:763;146:650;6:106;70:561Þ for the system (9). If

the economic profit m < msn, the system (9) has two equilibrium points. Taking m ¼ 400 for example, we obtain two positiveequilibrium points Pm

�ð82:309;237:781;1:899;121:244Þ and Pmþð19:218;55:518;10:312;19:877Þ. The eigenvalues at the equi-

librium point Pm� and Pm

þ are �0:333;�38:141;136:074 and �7:139;�4:297þ i5:646;�4:297� i5:646, respectively. This im-plies that the system (9) occurs saddle-node bifurcation at m ¼ 826:352.

Case III. Consider that the economic profit m ¼ 400 and the other parameters are the same as case II. Eq. (14) has a simplepositive root xm ¼ 1:119. Substituting it into Eq. (15), we have sm

0 ¼ 0:6305. Hence, Pmþ remains stable for s < sm

0 , which canbe seen in Fig. 1. When s passes through the critical value sm

0 , the equilibrium point Pmþ loses its stability and Hopf bifurcation

occurs. The bifurcating period solution from Pmþ at sm

0 is stable, which is depicted in Fig. 2.When m ¼ 350 and other parameters remain invariable, prey, predator and harvesting all converge to their equilibrium

values (see Fig. 3). This indicates that the economic profit has an effect of stabilizing the equilibrium point of the singularprey–predator economic system.

5. Discussion

So far, considerable attention has been focused on differential or difference systems in biology. However, to our bestknowledge, the reports of differential–algebraic biological systems are few. Considering the economic theory of fishery re-source proposed by Gordon in 1954, this paper mainly discusses bifurcational phenomena of a singular prey–predator eco-nomic model with time delay and stage structure.

For zero economic profit, transcritical bifurcation, singular induced bifurcation and Hopf bifurcation are obtained due toregarding the rate of conversing prey into predator, economic profit and time delay as different bifurcation parameters,respectively. Transcritical bifurcation implies that the equilibrium point P0

2 remains stable if the rate of conversing prey intopredator a21 is less than the critical point a11d2d3=ðr1a� d1d2 � ad2Þ, otherwise the stability is lost and the equilibrium pointP0

3 exists. In fact, stable P02 means that prey population will stay at a positive value and predator population is to be extinct,

which links with low rate of conversing prey into predator. Singular induced bifurcation implies that sufficient small positiveeconomic profit leads to impulsive phenomenon, i.e., rapid expansion of biological population, which may induce ecosystemunbalance and even biological disaster. Hence, necessary measures can be adopted to regulate the harvesting and economicprofit so that biological population stay at steady state and impulsive phenomenon can be eliminated.

Furthermore, this paper studies the effects of positive economic profit and gestation delay on dynamical behavior of thesingular prey–predator economic system. The analysis shows that time delay can alter the stability of biological populationand cause population oscillations. Although immature preys are available in abundance, the overflow of predator populationwill not happen due to gestation delay. In addition, numerical simulation shows that adjusting economic profit may restrain

0 20 40 60 80 10018.2

18.4

18.6

18.8

19

19.2

19.4

19.6

19.8

20

20.2

t

x i

0 20 40 60 80 10054

54.5

55

55.5

56

56.5

57

t

x m

0 10 20 30 40 50 60 70 80 90 100

9.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

t

y

0 10 20 30 40 50 60 70 80 90 10018

18.5

19

19.5

20

20.5

21

21.5

t

E

a b

c d

Fig. 1. Stable dynamics of the system (2) at the equilibrium point Pmþ with s ¼ 0:3.

1492 X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494

unstable or oscillational phenomenon. That means that time delay and economic profit are both responsible for the dynam-ical behavior of biological population.

Generally speaking, gestation delay can be regarded as an inherence of biological population. Thus, it is easier to adjusteconomic profit than gestation delay in order to eliminate bifurcation phenomenon and keep biological population stay atsteady state, such as adjusting revenue, drawing out favorable policy to encourage or improve fishery and so on.

Acknowledgements

The authors gratefully thank the anonymous authors for their valuable suggestions.

17 18 19 20 21 22 23

5055

60658

9

10

11

12

13

xixm

y

Fig. 2. Bifurcation period solutions from the equilibrium point Pmþ occur at sm

0 .

17.217.3

17.417.5

17.617.7

49.550

50.551

51.510.5

10.52

10.54

10.56

10.58

10.6

xixm

y

17.2 17.3 17.4 17.5 17.6 17.7

49.550

50.551

51.5

16.85

16.9

16.95

17

17.05

17.1

xixm

E

a

b

Fig. 3. Behavior of the populations for m ¼ 350, remaining other parameters invariable.

X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494 1493

Appendix

Singular system is also called differential–algebraic system, descriptor system, degenerate system, constrained system,etc. The general form of singular system can be described by differential equations and algebraic equations. Consider the fol-lowing DAEs with a one-dimensional parameter l:

_x ¼ f ðx; y;lÞ; f : Rnþmþ1 ! Rn;

0 ¼ gðx; y;lÞ; g : Rnþmþ1 ! Rm;

where x 2 Rn, y 2 Rm.Singularity Induced Bifurcation Theorem. Suppose that the following conditions are satisfied at ð0;0;l0Þ:

(SI1) f ð0;0;l0Þ ¼ 0, gð0;0;l0Þ ¼ 0, trace½DyfadjðDygÞDxg� is nonzero and Dyg has a simple zero eigenvalue.(SI2) Dxf Dyf

� �

Dxg Dyg

is nonsingular.(SI3) D f D f D f

0 1

x y l

Dxg Dyg Dlg

DxD DyD DlD

B@ CA

is nonsingular.

Then there exists a smooth curve of equilibria in Rnþmþ1 which passes through ð0;0;l0Þ and is transversal to the singular surface atð0;0;l0Þ. When l increases through l0, one eigenvalue of the system, (i.e., an eigenvalue of

J ¼ Dxf � Dyf ðDygÞ�1Dxg

1494 X. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1485–1494

evaluated along the equilibrium locus), moves from C� to Cþ if bc > 0 (respectively, from Cþ to C� if b

c < 0) along the real axis bydiverging through 1. The other ðn� 1Þ eigenvalues remain bounded and stay away from the origin. The constants b and c canbe computed by evaluating

b ¼ �trace½DyfadjðDygÞDxg�;

c ¼ DlD� DxD DyDð ÞDxf Dyf

Dxg Dyg

� ��1 Dlf

Dlg

� �

at ð0;0;l0Þ.

References

[1] Ayasun S, Nwankpa CO, Kwatny HG. Computation of singular and singularity induced bifurcation points of differential–algebraic power system model.IEEE Trans Circ Syst – I: Fundam Theor Appl 2004;51(8):1525–38.

[2] Riaza R. Singularity-induced bifurcations in lumped circuits. IEEE Trans Circ Syst – I: Fundam Theor Appl 2005;52(7):1442–50.[3] Luenberger DJ. Nonsingular descriptor system. J Econ Dynam Contr 1979;1:219–42.[4] Krishnan H, McClamroch NH. Tracking in nonlinear differential-algebra control systems with applications to constrained robot systems. Automatica

1994;30:1885–97.[5] Bloch AM, Reyhanoglu M, McClamroch NH. Control and stabilization of nonholonomic dynamic systems. IEEE Trans Automat Contr 1992;37:1746–57.[6] Ling L, Wang WM. Dynamics of a Ivlev-type predator–prey system with constant rate harvesting. Chaos, Solitons & Fractals 2009;41(4):2139–53.[7] Liu ZH, Yuan R. Stability and bifurcation in a harvested one-predator–two-prey model with delays. Chaos, Solitons & Fractals 2006;27(5):1395–407.[8] Zhang L, Wang WH, Xue YK. Spatiotemporal complexity of a predator–prey system with constant harvest rate. Chaos, Solitons & Fractals

2009;41(1):38–46.[9] Das T, Mukherjee RN, Chaudhuri KS. Harvesting of a prey–predator fishery in the presence of toxicity. Appl Math Model 2008. doi:10.1016/

j.apm.2008.06.008.[10] Gakkhar S, Singh B. The dynamics of a food web consisting of two preys and a harvesting predator. Chaos, Solitons & Fractals 2007;34(4):1346–56.[11] Kar TK, Pahari UK. Non-selective harvesting in prey–predator models with delay. Commun Nonlinear Sci Numer Simulat 2006;11(4):499–509.[12] Gordon HS. Economic theory of a common property resource: the fishery. J Polit Econ 1954;63:116–24.[13] Zhang X, Zhang QL, Zhang Y. Bifurcations of a class of singular biological economic models. Chaos, Solitons & Fractals 2009;40(3):1309–18.[14] Xiao YN, Cheng DZ, Tang SY. Dynamic complexities in predator–prey ecosystem models with age-structure for predator. Chaos, Solitons & Fractals

2002;14(9):1403–11.[15] Gao SJ, Chen LS, Sun LH. Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos, Solitons & Fractals 2005;25(5):1209–19.[16] Gao SJ, Chen LS. The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses. Chaos, Solitons &

Fractals 2005;24(4):1013–23.[17] Bandyopadhyay M, Banerjee S. A stage-structured prey–predator model with discrete time delay. Appl Math Comput 2006;182(2):1385–98.[18] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer; 1983.[19] Venkatasubramanian V, Schättler H, Zaborszky J. Local bifurcation and feasibility regions in differential–algebraic systems. IEEE Trans Automat Contr

1995;40(12):1992–2013.[20] Li F, Woo PY. Fault detection for linear analog IC – the method of short-circuit admittance parameters. IEEE Trans Circ Syst – I: Fundam Theor Appl

2002;49(1):105–8.