Nonautonomous dynamics at work: A Beverton-Holt … · Nonautonomous dynamics at work: A...

Nonautonomous dynamics at work:A Beverton-Holt Ricker model

Christian Pötzsche

Alpen-Adria Universität Klagenfurt, [email protected]

July 19, 2013

Joint work with Thorsten Hüls (Uni Bielefeld)

Motivation

Only wimps treat the most general case— real teachers tackle examples! 1

1from J. Appell: Analysis in Beispielen und Gegenbeispielen, Springer, 2009

Motivation

Beverton-Holt Ricker equation

Franke & Yakubu ’91, Kon ’06, Kang & Smith ’12 investigate{

xn+1 = αxn1+xn+βyn

yn+1 = yneγ−δxn−yn(∆′)

with real parameters α, β, δ > 0 and γ ≥ 0

Exclusion principle

If there exists no fixed point in (0,∞)d , then one species goes extinct

Models in Rd with only rational (resp. exponential) populationdensities

wrong for (∆′)

Motivation

Beverton-Holt model (’57)

xn+1 =αxn

1 + xn(BH ′)

with real parameter α > 0

Basically the time-1-map to

x = λx(1− x)

Ricker model (’54)

yn+1 = yneγ−yn (R′)

with real parameter γ ≥ 0

Motivation

Nonautonomous Beverton-Holt Ricker equation

{xn+1 = anxn

1+xn+bnyn

yn+1 = ynecn−dnxn−yn(∆)

with bounded parameter sequences an, bn, dn > 0 and cn ≥ 0

1 no extinction and coexistence equilibria

2 which linear/nonlinear stability analysis?3 AUTO or CONTENT will not work

Contents

Study the nonautonomous

1 Beverton-Holt equation

2 Ricker equation

3 Beverton-Holt Ricker equation


Goals

1 Nonautonomous dynamics

2 Pullback solutions andattractors

3 Forward dynamics


Autonomous dynamics

Given a fixed f : X → X ,

xn+1 = f (xn)

generates a semigroup

φ(n; x) = f n(x)

for all 0 ≤ n, x ∈ X

Nonautonomous dynamics

For a sequence fn : X → X ,

xn+1 = fn(xn)

generates a process

ϕ(n; n0, x) = fn−1 ◦ . . . ◦ fn0(x)

for all n0 ≤ n, x ∈ X


Consequence 1 (extended state space)

Process of an autonomous equation reads as

ϕ(n; n0, x) = f n−n0(x) for all n0 ≤ n, x ∈ X

For autonomous equations only the elapsed time mattersAutonomous dynamics happens in the state space X

For nonautonomous equations the extended state space Z× Xis appropriate:

X X

φ∗ Z


Consequence 2 (forward and pullback convergence)

Process of an autonomous equation reads as

ϕ(n; n0, x) = f n−n0(x) for all n0 ≤ n, x ∈ X

Long-term behavior n − n0 →∞ can be achieved in two ways:

n→∞ (forward convergence)

n0 →∞ (pullback convergence)


Nonautonomous Beverton-Holt equation

xn+1 =anxn

1 + xn(BH)

with an > 0 bounded

General forward solution

The general forward solution of (BH) (fulfilling x(n0) = x0) reads as

x(n; n0, x0) =Φa(n, n0)x0

1 +∑n−1

j=n0Φa(j, n0)

for all n0 ≤ n, x0 ≥ 0

with the transition operator

Φa(n,m) :=

{an−1 · · · am, m < n,

1, n = m


Pullback limit

For every 0 < x0 one has

ξ∗n := limn0→−∞

x(n; n0, x0) =1∑n−1

j=−∞Φa(j, n)for all n ∈ Z

and (ξ∗n )n∈Z is an entire solution to (BH) (pullback solution)


Absorbing set

A nonautonomous set A ⊆ Z× X with fibers

A(n) := {x ∈ X : (n, x) ∈ A}

is called pullback absorbing, if for every bounded B ⊆ Z× X thereexists a K ∈ N such that

ϕ(n; n − k ,B(n − k)) ⊆ A(n) for all K ≤ k and n ∈ Z

Z

X

B(n � k) A(n)


Example (autonomous equations)

For autonomous equations one has ϕ(n; n − k ,B) = f k (B) and thus

f k (B) ⊆ A for all K ≤ k and n ∈ Z

Pullback attractor

For any absorbing set B ∈ Z× X the pullback attractor

A∗(n) :=⋂

m≥0

⋃

k≥m

ϕ(n; n − k ,B(n − k))

(+compactness of fk ) allows the dynamical characterization

A∗ =

{(n, x) ∈ Z× X :

there is a bounded entire solutionof xn+1 = fn(xn) through (n, x)

}


Nonautonomous Beverton-Holt equation

xn+1 =anxn

1 + xn(BH)

has the absorbing set

X := {(n, x) ∈ Z× [0,∞) : 0 ≤ x ≤ an−1}

and the pullback attractor

X ∗a =

{(n, x) ∈ Z× [0,∞) : 0 ≤ x ≤ ξ∗n} ,∏n−1

i=−∞ ai =∞,

Z× {0} , lim supj→−∞

n−1∏

i=j

ai <∞


Constant an

an :≡ α ≥ 1 on Z

yields ξ∗n ≡ α− 1 on Z

æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ

à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à

-30 -20 -10 0 10 20 30

0.5

1.0

1.5

2.0

2.5

3.0

10-periodic an

an := 2 + sin(π

5n)

yields (ξ∗n )n∈Z to be 10-periodic

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Switching down an

an :=

{3, n < 0,

2, n ≥ 0

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Switching up an

an :=

{2, n < 0,

3, n ≥ 0


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Single impulse an

an :=

{2, n 6= 0,

3, n = 0


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Almost periodic an

an := 2 + sin n

yields (ξ∗n )n∈Z to bealmost-periodic

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Forward and pullback convergence

The ”extended” Beverton-Holt equation

xn+1 =anxn

1 + |xn|

has the pullback attractor A∗ = Z× {0} for coefficient sequences

an :≡ 23

on Z

As pullback attractors are uniformly bounded (by definition), they can be characterizedby the bounded entire solutions of the process.

Proposition 2.5. A pullback attractor A ! {Ak : k [ Z} admits the dynamicalcharacterization: for each k [ Z

x [ Ak , there exists a bounded entire solution x with xk ! x:

It is therefore uniquely determined.

Proof. For the implication ")#, pick k [ Z and x [ Ak arbitrarily. Then due to thew-invariance of the pullback attractor A, Proposition 2.1 provides the existence of anentire solution x with xk ! x and xk [ Ak for each k [ Z. Moreover, w is boundedbecause the component sets of the pullback attractor are uniformly bounded.

For the converse implication "(#, if there exists a bounded entire solution x toequation (1), then the set of points Bx :! {xk : k [ Z} is bounded in X. As A pullbackattracts bounded subsets of X, it follows that

0 # dist"xk;Ak# # limn!1

dist w"k; k2 n;Bx#;Ak

! "! 0 for all k [ Z;

so xk [ Ak: A

3. Limitations

The limitations of pullback attraction are discussed in this section through some examples,which show that pullback attractors do not capture the complete dynamics ofnonautonomous systems defined through processes.

First consider the autonomous scalar difference equation

xk$1 !lxk

1$ xkj j "5#

depending on a real parameter l . 0. Its zero solution !x ! 0 exhibits a pitchforkbifurcation at l ! 1, and one obtains the following global behaviour (Figure 1):

. If l # 1, then !x ! 0 is the only fixed point, it is globally asymptotically stable and isthus the global attractor of the autonomous dynamical system generated by thedifference equation (5).

–1

5

2

0

10–5

1

–10 15–2

0

–1

5

2

0

10–5

1

–10 15–2

0

Figure 1. Trajectories of the autonomous difference equation (5) with l ! 0:5 (left) and l ! 1:5(right).

P.E. Kloeden et al.4

Downloaded By: [Rasmussen, Martin][Imperial College] At: 10:40 10 June 2011

an :=

{23 , n < 0,32 , n ≥ 0

. If l . 1, then there exist in addition two nontrivial fixed points x^ :! ^"l2 1#.The trivial solution !x ! 0 is an unstable steady state solution and the symmetricinterval A ! $x2; x%& is the global attractor.

3.1 Piecewise autonomous equation

Consider now the piecewise autonomous equation

xk%1 !lkxk

1% xkj j ; lk :!l; k $ 0;

l21; k , 0

(

"6#

for some l . 1, which corresponds to a switch between the two autonomous problems (5)at k ! 0.

Due to Proposition 2.5, the pullback attractor A of the resulting nonautonomous systemhas component sets Ak ; {0} for all k [ Z corresponding to the zero entire solution. Notethat the trivial fixed point !x ! 0 is ‘asymptotically stable’ for k , 0 and then ‘unstable’ fork $ 0. Moreover, the interval $x2; x%& is like a global attractor for the whole equation on Z,but it is not really one because it is not invariant or minimal for k , 0:

The nonautonomous difference equation (6) is asymptotic autonomous in bothdirections, but the pullback attractor does not reflect the full limiting dynamics (Figure 2(left)), in particular in the forward time direction.

If lk do not switch from one constant to another, but increase monotonically, e.g. suchas lk ! 1% "0:9k="1% jkj##, then the dynamics is similar, although the limiting dynamicsis not so obvious from the equations (Figure 2 (left)).

3.2 Fully nonautonomous equation

Now let {lk}k[Z be a monotonically increasing sequence with limk!^1lk ! !l^1 for!l . 1. The nonautonomous problem

xk%1 ! f k"xk# :!lkxk

1% xkj j "7#

is asymptotically autonomous in both directions with the limiting systems given in theprevious subsection.

Its pullback attractor A has component sets Ak ; {0} for all k [ Z corresponding tothe zero entire solution, which is the only bounded entire solution. As above, the trivial

–1

5

2

0

10–5

1

–10 15–2

–1

2

0

1

–20 5 10–5–10 150

Figure 2. Trajectories of the piecewise autonomous difference equation (6) with l ! 1:5 (left) andthe asymptotically autonomous difference equation (7) with lk ! 1% "0:9k="1% jkj## (right).

Journal of Difference Equations and Applications 5

Downloaded By: [Rasmussen, Martin][Imperial College] At: 10:40 10 June 2011


Transcritical bifurcation

xn+1 =λanxn

1 + xn(BHλ)

with an > 0 being bounded away from 0

an ≡ 1 an ≡ 1.9 + 0.2 sin n

1 Solution bifurcation (Langa, Robinson & Suárez ’06)

2 Attractor bifurcation (Rasmussen ’06)

2 Ricker equation

Goals

1 Pullback solutions

2 Nonautonomous flipbifurcation

3 Forward behavior

2 Ricker equation

Nonautonomous Ricker equation

yn+1 = ynecn−yn (R)

with cn ≥ 0 bounded

Theorem

The pullback attractor of the Ricker equation (R) is

Y∗c =

{{(n, y) : 0 ≤ y ≤ η∗n} , supn∈Z cn ≤ 1,{

(n, y) : 0 ≤ y ≤ ecn−1

e

}, 1 ≤ infn∈Z cn

with the pullback solution η∗ given by

η∗n := limk→−∞

y(n; k , 1) for all n ∈ Z.

2 Ricker equation

Nontrival equilibrium

Although the nonautonomous Ricker equation (R) has no nontrivialequilibrium anymore, the pullback limit

η∗n := limk→−∞

y(n; k , 1) for all n ∈ Z

exists under the assumption∏n−1

j=−∞ ecj−2 = 0

Example

In case cn ≡ γ one hasη∗n ≡ γ on Z

2 Ricker equation

Robustness of a flip bifurcation

yn+1 = ynecn(γ)−yn

autonomous case

cn(γ) = γ

pullback and forwardconvergence to

γ for γ < 2

a 2-periodic solution forγ > 2

nonautonomous case

cn(γ) = γ + 0.02 sin n

pullback convergence to

η∗(γ) for γ < 2

(y(n; n − k , 1))k∈Npossesses two convergentsubsequences for γ > 2

2 Ricker equation

Robustness of a flip bifurcation

yn+1 = ynecn(γ)−yn

cn(γ) = γ cn(γ) = γ + 0.02 sin n

QUALITATIVE ANALYSIS OF A NONAUTONOMOUS BEVERTON-HOLT-RICKER MODEL 19

and suppose the parameter sequence is given as

cn(!) := 2 + !"n for all n ! I,with a bounded sequence ("n)n!N and a real parameter !. Even in this perturbed au-tonomous situation the problem arises that the established nonautonomous bifurcation re-sults (cf. [22, 30]) require a whole family (parametrized by !) of bounded entire solutions#(!)I along which the bifurcation occurs at ! = 0 and #(0) = ".

Numerically, we can approximate such bounded solutions #(!) by solving

yn+1 = ynecn(!)"yn , n = n", . . . , n+ " 1

• with periodic boundary conditions yn! = yn+

• or by computing a least squares solution #y[n!,n+]#2 = min.Assuming hyperbolicity (i.e. an exponential dichotomy of the variational equation) it turnsout that approximation errors decay exponentially fast towards the midpoint of the finiteinterval, see [15] in case of boundary value problems.

For "n ! ["1, 1] chosen randomly, we compute bounded trajectories in Fig. 10 for! = 0 (left) and ! = 0.02 (right).

nn

ynyn

""

periodic orbits

FIGURE 10. Bounded trajectories of the Ricker model for ! = 0 (left)and ! = 0.02 (right).

Thus, we observe that the autonomous flip bifurcation in (R’) at " = 2 turns into a”nonautonomous flip bifurcation” in (R!) for ! > 0.

Remark 4.1 (nonautonomous flip bifurcation). Once a reference solution #(!) is knownanalytically, a nonautonomous flip bifurcation can be approached analytically as follows:Inspired from the autonomous situation, where a period doubling is established as pitchforkbifurcation in the second iterate of (R’), one introduces the equation of perturbed motion

yn+1 = (yn + #(!)n)e2+!"n"yn"#(!)n =: gn(yn,!)

and applies the nonautonomous bifurcation criteria from [22, 30] to the difference equation

yn+1 = Gn(yn,!), Gn(y,!) := gn+1(gn(y,!),!).

Due to the tedious computations we skip the details here.

Before addressing forward convergence of solutions, let us illustrate that the trivial so-lution to (R) can be both pullback attracting and unstable:

γ < 2 Pullbackconvergence to γ

γ > 2 Pullbackconvergence to 2-periodicsolution

cn(γ) = γ + 0.02 sin n

2 Ricker equation

Theorem (forward convergence)

If the sequence c satisfies

n−1∏

j=−∞ecj−2 =

∞∏

j=n0

ecj−2 = 0 for some n, n0 ∈ Z,

then one has the limit relation

limn→∞

(y(n; n0, y0)− η∗n) = 0 for all n0 ∈ Z, y0 > 0

Proof: Path stability (Krause ’09)


Goals

1 Pullback attractor

2 Dichotomy spectrum3 Nonautonomous center

manifolds

4 Bifurcations in 2d


Nonautonomous Beverton-Holt Ricker equation

{xn+1 = anxn

1+xn+bnyn

yn+1 = ynecn−dnxn−yn(∆)

with bounded parameter sequences an, bn, dn > 0 and cn ≥ 0

Pullback attractor A∗Due to the existence of the absorbing set

A :=

{(n, x , y) ∈ Z× R2

+ : 0 ≤ x ≤ an−1, 0 ≤ y ≤ ecn−1

e

}

the pullback attractor A∗ of (∆) is invariant, compact, connected,pullback attracts all bounded subsets of Z× R2

+ and satisfies

A∗ ⊆ A


Properties of A∗

1 X ∗a × {0} ⊆ A∗ and {0} × Y∗c ⊆ A∗2 Ricker dominates

n−1∏

j=−∞aj = 0 ⇒ A∗ = {0} × Y∗c

3 If cn ≤ 1, then (∆) is order-preserving w.r.t. the south-west cone

C := R+ × (−∞, 0]

and thus

A∗ ⊆{

(n, x , y) ∈ Z× R2+ : 0 ≤ x ≤ ξ∗n , 0 ≤ y ≤ η∗n

}


Forward dynamics of (∆)

If there exist N ∈ Z, µ ∈ (0, 1) such that

asnecn max

{(sbn)se

1bn−s,(

sdn

)sedn−s

}≤ 1− µ for all n ≥ N

holds for some s ≥ supn≥N max{

1bn, dn

}, then

limn→∞

ϕ(n; n0, x0, y0) = 0 for all n ∈ I, x0 > 0, y0 ≥ 0

Proof: V (x , y) := yxs is a Lyapunov function for (∆) (Franke & Yakubu

’91); nonautonomous invariance principle (LaSalle ’76)


How about the dynamics inside of the pullback attractor A∗?

For an entire solution (ξn, ηn) to (∆) consider the variational equation

(xn+1

yn+1

)= F ′n(ξn, ηn)

(xn

yn

)(V )

with the coefficient matrices

F ′n(ξ, η) =

(an(1+bnη)

(1+ξ+bnη)2 − anbnξ(1+ξ+bnη)2

−dnηecn−dnξ−η ecn−dnξ−η(1− η)

)

σ(F ′n(ξn, ηn)) of (V ) is of no use!


Ambient spectrum Σ(ξ, η)

Σ(ξ, η) yields asymptotic stability (or instability)

gaps in Σ(ξ, η) allow to construct invariant manifolds

nice perturbation theory

computable


Spectral theory for linear difference equations I

A linear difference equation

xn+1 = Anxn, An ∈ Rd×d

is said to have an exponential dichotomy (ED for short), if Z× Rd

allows a hyperbolic splitting

X

Z

N (Pt )

R(Pt )


Spectral theory for linear difference equations II

The dichotomy spectrum reads as

Σ(A) :={γ > 0 : xn+1 = 1

γAnxn does not have an ED}

and is of the form (Sacker & Sell ’78, Aulbach & Siegmund ’01)

Σ(A) =k⋃

j=1

[aj , bj ] for some k ≤ d

Ra1 b1 a2 b2 a3 = b3 a4 b40


Continuity properties

1 Only upper-semicontinuous in (An)n∈Z w.r.t. the `∞-topology

2 Set of discontinuity points is meagre

Stability properties

Applied to the variational equation (V ) of an entire solution (ξ, η) it is

1 Σ(ξ, η) ⊆ (0, 1) yields uniform asymptotic stability (i.e.exponential stability)

2 Each gap in Σ(ξ, η) gives rise to a (generalized) saddle-pointstructure around (ξ, η)


Example (autonomous equations)

xn+1 = Axn has the dichotomy spectrum Σ(A) = |σ(A)|

Example (scalar equations)

xn+1 = anxn has the dichotomy spectrum

Σ(a) =[β(a), β(a)

]

with the Bohl exponents

β(a) := limn→∞

infk∈Z

n

√√√√k+n−1∏

j=k

|aj |, β(a) := limn→∞

supk∈Z

n

√√√√k+n−1∏

j=k

|aj |


Example (triangular equations)

The difference equation

xn+1 = Anxn, An :=

(an cn

0 bn

)

has a dichotomy spectrum Σ(A) satisfying

[β(a), β(a)

]∆[β(b), β(b)

]⊆ Σ(A) ⊆

[β(a), β(a)

]∪[β(b), β(b)

]


Computation of Σ(A) (Dieci & van Vleck ’05. . . )

Replace Z by a sufficiently large {κ, . . . , κ} and set Qκ := Id1 Compute the QR-decomposition

Aκ = Qκ+1Rκ

with Qκ+1 orthogonal and Rκ upper-triangular

2 For k = κ, . . . , κ− 1 compute the QR-decompositions

Ak+1Qk+1 = Qk+2Rk+1

3 Determine the Bohl exponents for the diagonal elements of theupper triangular difference equation

xn+1 = Rnxn, Rn = Q−1n+1AnQn


Beverton-Holt equilibrium

For the Beverton-Holt equilibrium

(ξ∗n , 0

)=(

1∑n−1j=−∞ Φa(j,n)

, 0)

one obtains the variational equation

(xn+1

yn+1

)=

(an

(1+ξ∗n )2 − anbnξ(1+ξ∗n )2

0 ecn−dnξ∗n

)(xn

yn

)

0 0.5 1 1.5 2 2.5 30

1

2

3

delt

a

an :=

{2, n ≥ 0,32 , n < 0

, bn :=

{2, n ≥ 0,

1, n < 0

cn :=

{1, n ≥ 0,12 , n < 0

, dn := δ


Beverton-Holt equilibrium

For the Beverton-Holt equilibrium(ξ∗n , 0

)=(

1∑n−1j=−∞ Φa(j,n)

, 0)

with

the dichotomy spectrum Σ(ξ∗, 0) = σ1 ∪ σ2 one has:

hyperbolic situation

R

R

R0 1

σ1

σ1

σ1

σ2

σ2

σ2

(h1)

(h2)

(h3)

(h1) sink

(h2) saddle-point and

(h3) source

nonhyperbolic situation

R

R

σ1

σ1

σ2

σ2

0 1

(n1)

(n2)

maxσ1 < minσn ≤ 1


Theorem (reduction principle)

Under the assumptions

β( aZ

(1+ξZ)2

)< β

(ecZ−dZξZ

)≤ 1, β

( aZ(1+ξZ)2

)< β

(ecZ−dZξZ

)2,

the stability properties of the Beverton-Holt solution (ξ∗, 0) to theplanar system (∆) correspond to the stability of the trivial solution tothe scalar equation (the reduced equation)

yn+1 = ecn−dnξ∗n(

yn−(1+dnt∗n )y2n +

(1 + dnt∗n )2 − dnωn

2y3

n +O(y4n ))

Proof: Nonautonomous center manifold reduction (Rasmussen & P.,’05)


Bifurcation

For parameter-dependent equations{

xn+1 = anxn1+xn+bnyn

yn+1 = ynecn−δxn−yn(∆)

the Beverton-Holt equilibrium (ξ∗, 0) becomes unstable in form of atranscritical bifurcation into an asymptotically stable entire boundedsolution (coexistence equilibrium) as δ > 0 increases


Computation of the coexistence equilibrium (Hüls ’09)

Solve the boundary value problem{

xn+1 = anxn1+xn+bnyn

yn+1 = ynecn−dnxn−ynfor all n− ≤ n < n+ − 1

w.r.t. periodic boundary conditions (xn− , yn−) = (xn+ , yn+)

QUALITATIVE ANALYSIS OF A NONAUTONOMOUS BEVERTON-HOLT-RICKER MODEL 31

7.1. Beverton-Holt entire solution. We investigate the behavior near the entire solution(!!

Z, 0) to the planar difference equation

(!!)

!xn+1 = anxn

1+xn+bnyn,

yn+1 = ynecn"!xn"yn

being a special case of (!) with constant coupling dn ! ".Let us consider " as a bifurcation parameter to understand the loss of stability in the

Beverton-Holt solution (!!Z, 0) as the coupling strength " changes. The associate dichotomy

spectrum reads as

"(!!Z, 0) =

"##

aZ(1+"!

Z )2

$,##

aZ(1+"!

Z )2

$%""##ecZ"!"!

Z$,##ecZ"!"!

Z$%

,

and we suppose that

##

aZ(1+"!

Z )2

$< #

#ecZ"!!"!

Z$, #

#ecZ"!!"!

Z$

= 1(7.1)

holds for some critical value "! > 0. A reduction to the center fiber bundle as in Sub-sect. 6.2 yields that the bifurcation in (!!) is determined by the scalar difference equation

yn+1 = ecn"!!"!n&(1 # (" # "!)!!

n)yn # (1 + "!!!n)y2

n

'+O(("#"!)2yn, ("#"!)y2

n, y3n)

uniformly in n $ Z. This indicates a transcritical bifurcation as " decreases through "!.We verify the spectral assumption (7.1) numerically with the algorithm from Section B.1.

Let rZ $ [#0.05, 0.05]Z be a uniformly distributed random sequence. For n $ Z we choosethe parameters an = rn + 1.5, bn = rn + 0.1, cn = rn + 1, " = 2.5 and obtain from ournumerical experiments:

##

aZ(1+"!

Z )2

$= 0.66771 < #

#ecZ"!"!

Z$

= 0.77618, ##ecZ"!!"!

Z$

= 0.78163.

For "! = 2, (7.1) is approximately satisfied:

##

aZ(1+"!

Z )2

$= 0.66771 < #

#ecZ"!!"!

Z$

= 0.99781, ##ecZ"!!"!

Z$

= 1.00250.

Note that neither the bounded Beverton-Holt trajectory !!Z, nor the Ricker trajectory depend

on ". For " < 2 a bounded coexistence trajectory exists, that bifurcates from the Beverton-Holt solutution, see Fig. 16 and Fig. 17 for the dichotomy spectrum.

0

10

20

30

00.2

0.40.6

0

0.5

1

x

y

n

" = 0

" = 2

Beverton-Holtsolution

FIGURE 16. Coexistence trajectories (cyan) for various values of " $ [0, 2].

Conclusions

Conclusions

1 bounded entire solutions replace equilibria (fixed points)

2 spectral intervals replace eigenvalue real parts

but, qualitative analysis of nonautonomous models requires numericalmethods for

1 continuation algorithms for entire solutions

2 the dichotomy spectrum and Bohl exponents

Þakka Þér

Nonautonomous dynamics at work: A Beverton-Holt … · Nonautonomous dynamics at work: A...

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