STABILITY AND BIFURCATION OF A STAGE-STRUCTURED PREDATOR-PREY MODEL INCORPORATING A CONSTANT PREY REFUGE∗†

Zhen Wei

(School of Electronic and Information Engineering,Fuqing Branch of Fujian Normal University,Fuzhou 350300,E-mail:2903619@qq.com)

Haihui Wu

(Sunshine College,Fuzhou University,Fuzhou 350015)

Changwu Zou

(College of Math.and Computer Science,Fuzhou University,Fuzhou 350116)

STABILITY AND BIFURCATION OF A STAGE-STRUCTURED PREDATOR-PREY MODEL INCORPORATING A CONSTANT PREY REFUGE∗†

Zhen Wei

(School of Electronic and Information Engineering,Fuqing Branch of Fujian Normal University,Fuzhou 350300,E-mail:2903619@qq.com)

Haihui Wu

(Sunshine College,Fuzhou University,Fuzhou 350015)

Changwu Zou

(College of Math.and Computer Science,Fuzhou University,Fuzhou 350116)

A stage-structured predator-prey model incorporating a constant prey refuge is proposed in this paper.The stability analysis of the equilibria is carried out.We also study Hopf bifurcations occurring at the positive equilibrium by choosing a suitable time delay as bifurcating parameter.

delay;Hopf bifurcations;periodic solutions;stability

2000 Mathematics Subject Classification 34K18;92D25

Ann.of Appl.Math.

31:2(2015),212-224

1　Introduction and Motivation

1.1History and Motivation

In the past few years,bifurcation theory of dynamic system has been well developed(see e.g.,[1-3]and references therein).It has been applied to many fields.For example,many authors have studied the stability and bifurcation analysis of biomathematical dynamics(see e.g.,[4-12]and references therein).The classical predator-prey mathematical models have been studied extensively,but questioned by several biologists.They argued that such models should be modified to fit the more realistic environment.For example,stage-structured population growth was introduced in a single species model by[13-15].In the classical predator-prey model,it is assumed that each individual predator admits the same ability of attacking.This assumption is obviously unrealistic for many animals.In the real world, there are many species whose individuals have a life history that take them through two, namely immature and mature stages,where immature predators are raised by their parents, and the rate they attack the prey and the reproductive rate can be ignored.For this reason, a predator-prey model with stage structure for predator was proposed in[16,17].

On the other hand,when predators eat their prey,prey that avoid predation risk can also incur substantial fitness costs through risk-induced changes in survival and reproduction,growth,and morphology.Changes in prey that occur without the predator physically consuming the prey are referred to as non-consumptive effects.One way to reduce the risk of predation is to use a refuge.Recently,the authors[18-22]have incorporated a constant prey refuge m into the predator-prey models.

1.2Model Formulation

Motivated by the above works,in this paper,we consider the following stage-structured predator-prey model incorporating a constant refuge m

where x(t)is the population density of prey species,y1(t)and y2(t)denote the densities of the immature and mature predator species,respectively;r denotes the intrinsic growth rate of prey;a1is the intra-specific competition rate of the prey(due to sharing the nutrients in an closed environment);a2denotes the coefficient in mature predator eating prey;b denotes the rate of conversing mature plant into new predator(immature predator);β and r2denote the death rate of the immature and mature predator,respectively;τ represents a constant time to maturity;b1/b denotes the rate of immature predator becoming into mature predator;m＞0 is the constant prey refuge to reduce the risk of predation.System (1.1)is supplemented with the initial conditions of the form

Note that the first and third equations of(1.1)can be separated from the whole system. Consider the following subsystem of(1.1),and denotefor convenience

In this paper,we mainly discuss the effects of constant refuge on the stage-structured predator-prey model.Stability and bifurcation analysis are carried out in this paper.

2　Model Analysis and Bifurcation of Periodic Solution

Now,we discuss the stability of equilibria of system(1.2).Ifdenotes the equilibrium of system(1.2),it satisfies the algebra equations

Clearly,E0=(0,0),are boundary equilibria.If,it is easy to seethat there is a unique positive equilibrium

To describe the stability of the equilibrium,we introduce the following definition.

Definition 2.1 The equilibriumof system(1.2)is called conditionally stable if it is asymptotically stable for some τjin some intervals,but not necessarily for all delays τj≥0 (1≤j≤m).While theof system(1.2)is called absolutely stable(asymptotically stable independent of the delays)if it is asymptotically stable for all τj≥0(1≤j≤m).

To determine the local stability of the equilibria E0,E1and E∗,we need linearize(1.2) aboutthen(1.2) can be transformed into

The linear part of system(2.2)is

The corresponding characteristic equation about the unknown λ is

that is,

Theorem 2.1 The equilibrium point E0=(0,0)is unstable.

that is

Obviously,λ1=r is a positive root.Therefore,E0is unstable.

that is

Obviously,λ=-r is a negative root.If,that is,,we set

It is easy to see that

and

Consequently,there exists a λ0＞0 such that F(λ)=0.That is,F(λ)=0 has a positive root.Then the equilibrium pointis unstable.The first assertion of Theorem 2.2 is proved immediately.

(ii)Now we consider the caseNote that λ1=-r is a negative root in this case.Let

To show the asymptotic stability of E1,it suffices to prove that all the roots of F1(λ)=0 have negative real parts.In fact,when τ=0,F1(λ)=0 reduces toThat is,is the unique negative root.So by Rouché’s theorem(see Dieudonné[25],Theorem 9.17.4),we only need to prove that F1(λ)=0 does not have any purely imaginary roots.To this end,by way of contradiction,assume that iω is a purely imaginary root of F1(λ)=0.Rewrite F1(λ)=0 in terms of its real and imaginary part as

which implies

This is a contradiction.Therefore,if,E1is asymptotically stable.

in(2.4),we have where

When τ=0,(2.5)reduces to

and

It is not difficult to see that all the roots of(2.6)have negative real parts in this case.

To determine the local stability of the positive equilibrium E∗more precisely,we let λ=α+iω,α,ω∈R,and rewrite(2.5)in terms of its real and imaginary arts as

When α=0,(2.7)reduces to

It follows by taking the sum of squares that

Denote

The two roots of(2.9)can be expressed as follows:

Thus,

(i)if b0＞0 or Δ0＜0,then neither ofandis positive,that is,(2.9)does not have positive roots.Therefore,characteristic equation(2.5)does not have purely imaginary roots.Since q-h＞0 ensures that all roots of(2.6)have negative real parts,by Rouché’s theorem(see Dieudonné[25],Theorem 9.17.4),it follows that the roots of(2.6)does not have any purely imaginary roots;

(ii)if b0＜0 and Δ0=0,then(2.9)has a positive root

(iii)if b0＜0 and Δ0＞0,then(2.9)has two positive rootsand

In both cases of(ii)and(iii),the characteristic equation(2.5)has purely imaginary roots when τ takes certain values.This critical valuesof τ can be determined from system (2.8)given by

From the above analysis,we have the following lemma.

(1)If either b0＞0 or Δ0＜0,all the roots of(2.5)have negative real parts for all τ≥0.

(2)If b0＜0 and Δ0=0,then(2.5)has a pair of purely imaginary roots±iω+.

(3)If b0＜0 and Δ0＞0,then(2.5)has two pair of purely imaginary roots±iω+(±iωrespectively).

Then we claim that the real part of some root of(2.5)is positive whenand τ＜τj-.To prove this,denote

where the root of(2.5)satisfies

By simple computation,it is not difficult to verify that the following transversality conditions hold:

It follows that τ±jare bifurcation values.Thus,we have the following theorem about the distribution of the characteristic roots of(2.5).

Theorem 2.3 Let τj±be defined by(2.11).In addition to,assume that:

(i)If b0＜0 and Δ0=0,then when τ∈[0,τ+0)all roots of(2.5)have negative real parts;when τ=τ0+(2.5)has a pair of imaginary roots±ω+;when τ＞τ0+(2.5)has at least one root with positive real part.

(ii)If b0＜0 and Δ0＞0,then there is a positive integer k such that there are switches from stability to instability,that is,when

all roots of(2.5)have negative real parts,and when

(2.5)has at least one root with positive real part.

Theorem 2.5 Let ω+and τj+be defined by(2.10)and(2.11),respectively.In addition to,if b0＜0 and Δ0=0,then the equilibrium E∗of system(1.2)is conditionally

stable.More precisely,

(i)if τ∈[0,τ0+),then E∗is asymptotically stable;

(ii)if τ＞τ0+,then E∗is unstable;

(iii)if τ=τj+(j=0,1,2,···),then system(1.2)undergoes Hopf bifurcations at E∗.

Theorem 2.6Let ω±and τj±be defined by(2.10)and(2.11),respectively.In addition to,if b0＜0 and Δ0＞ 0,then there is a positive integer k such that the equilibrium E∗switches k times from stability to instability,that is,when τ∈ [0,τ0+),(τ0-,τ1+),···,,the positive equilibrium E∗of(1.2)is asymptotically stable; whe n τ∈[τ0+,τ0-),[τ1+,τ1-),···,and τ＞τk+,the positive equilibrium E∗of (1.2)is unstable.

3　Direction and Stability of Hopf Bifurcation

In the previous section,some sufficient conditions are obtained to guarantee that system (1.2)undergoes Hopf bifurcation at the positive equilibrium E∗when τ=τj±(j=0,1,2,···). In this section,based on the normal form and the center manifold theory developed by Hassard,Kazarinoff and Wan[26],we shall derive the explicit formulaes determining the direction,stability,and period of these periodic solutions bifurcating from equilibrium E∗at these critical values of τ.Without loss of generality,denote any one of these critical values τ=τj±(j=0,1,2,···)by,at which(2.5)has a pair of purely imaginary roots iω and system(1.1)undergoes a Hopf bifurcation from E∗.

Let u1(t)=x(τt)-x∗,u2(t)=y(τt)-y∗andThenµ=0 is the Hopf bifurcation value of system(1.2)and system(1.2)can be rewritten as

Thus,we can work in the fixed phase space C=C([-1,0],R2),which does not depend on the delay τ.In the space C=C([-1,0],R2),system(3.1)is transformed into a FDE as

where u=(u1,u2)T,ut(θ)=u(t+θ)∈C,Lµand f are given respectively by

where φ=(φ1,φ2)∈C.

By the Riesz representation theorem,there exists a matrix whose components are bounded variation functions η(µ,θ)in θ∈[-1,0]such that

where the bounded variation functions η(µ,θ)can be chosen as

where δ is the Dirac function.For φ∈C1([-1,0],R2),define

Then system(3.2)is equivalent to

where u=(u1,u2)T,ut(θ)=u(t+θ),θ∈[-1,0].

For ψ∈C1([0,1],(R2)∗),define

and a bilinear inner product

where η(θ)=η(0,θ).Then A(0)and A∗are adjoint operators.In addition,from Section 2 we know thatare eigenvalues of A(0).Thus,they are also eigenvalues of A∗.We first need to compute the eigenvector of A(0)and A∗corresponding to,respectively.

To this end,suppose that q(θ)=(1,α0)Teiω˜τθis the eigenvector of A(0)corresponding to,thenIt follows from the definition of A(0),(3.3)and(3.5)that

Thus,we can choose

Similarly,letting q∗(s)=D(β0,1)T=D(β0,1)eiω˜τsbe the eigenvector of A∗corresponding,we can compute

In order to assure〈q∗(s),q(θ)〉=1,we need to determine the value of D.From(3.7),we have

Thus,we can choose a D

It is also easy to verify thatNow we compute the coordinates to describe the center manifold C0atµ=0.Let utbe a solution to(3.6)whenµ=0.Define

On the center manifold C0,we have

where

We rewrite the equation as

where

It follows from(3.8)and(3.9)that

which together with(3.4)gives

Comparing the coefficients with(3.10),we have

Since there are W20(θ)and W11(θ)in g21,we still need to compute them.

From(3.6)and(3.8),we have

Substituting the corresponding series into(3.14)and comparing the coefficients,we obtain

From(3.14),it is easy to see that for θ∈[-1,0),

Comparing the coefficients with(3.15)gives that

It follows from(3.16),(3.18)and the definition of A that

Note that q(θ)=(1,α0)Teiω˜τθ.Hence

Similarly,it follows from(3.16)and(3.19)that

In what follows,we shall seek appropriate E1and E2.From the definition of A and (3.16),we obtain where η(θ)=η(0,θ).By(3.14),we have

Note that the fact

and

Substituting(3.20)and(3.24)into(3.22)gives

which leads to

Solving(3.26),we have

where

Similarly,substituting(3.21)and(3.25)into(3.23)gives

Solving(3.27),we have

where

Thus,we can determine W20(θ)and W11(θ)from(3.20)and(3.21).Furthermore,g21in (3.13)can be expressed by the parameters and delay.Therefore,we can compute the following values:

which determine the quantities of bifurcating periodic solutions at the critical value,that is,µ2determines the directions of the Hopf bifurcation:Ifµ2＞0(µ2＜0),then the Hopf bifurcation is supercritical(subcritical)and the bifurcating periodic solutions exist for τ＞β2determines the stability of the bifurcating periodic solutions:the bifurcating periodic solutions in the center manifold are stable(unstable)if β2＜0(β2＞0);and T2determines the period of the bifurcating periodic solutions:the period increase(decrease)if T2＞0(T2＜0).Further,it follows from(2.13)and(3.28)that the following result holds.

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(edited by Liangwei Huang)

∗This project was supported by the school scientific research project under grant KY2008022,the science and technology Program Foundation of Fujian Provincial Education Department(JB12254) and the Natural Science Foundation of Fujian Province under grant No.2013J01010.

†Manuscript March 17,2015