ZHANG ZI-ZHEN,CHu Yu-GuI AND ZHANG XIN

(1.School of Management Science and Engineering,Anhui University of Finance and Economics,Bengbu,Anhui,233030)

(2.School of Mathematics,Jilin University,Changchun,130012)

Communicated by Li Yong

1 Introduction

In recent decades,it has been of great interest to investigate the dynamic interaction between predator species and prey species in both ecology and mathematical ecology(see[1]–[3]).Especially,two species predator-prey systems have been investigated by many scholars at home and abroad(see[4]–[10]),since the pioneering works by Lotka[11]and Volterra[12].However,two species predator-prey systems can describe only a small number of the phenomena that are commonly observed in nature(see[13]).Therefore,it is more realistic to consider a predator-prey system with three or more species to understand complex dynamical behaviors of multiple species predator-prey systems in the real world.Inspired by this idea,Yang and Jia[14]proposed the following three species predator-prey system with reserve area for prey and in the presence of toxicity:

where x(t),y(t)and z(t)denote the biomass densities of the prey species in the unreserve areas,the prey species in the reserve areas and the predator species at time t,respectively.K is the carrying capacity of the prey species in the unreserve areas;r1is the intrinsic growth rate of the prey species in the unreserve areas;r2is the birth rate of the prey species in the reserve areas;v1,v2and ? are the infection rates of the prey species in the unreserve areas,the prey species in the reserve areas and the predator species by an external toxic substance,respectively;q1and q2are the catchability coefficients of the prey species in the unreserved areas and the predator species,respectively;E is the effort applied to harvest the prey species and the predator species in the unreserve areas;σ1(σ2)is the rate at which the prey species in the unreserve(reserve)areas migrate into the reserve(unreserve)areas;d is the death rate of the predator species;a is the capturing rate of the predator species;β is the rate of conversing the prey species in the unreserve areas into the predator species;b is the half saturation rate of the predator species.

Time delays of one type or another have been incorporated into predator-prey systems by many scholars(see[15]–[21]),since delay differential equations exhibit much more complicated dynamics than ordinary ones.Motivated by the work above and considering that the consumption of the prey species by the predator species throughout its past history governs the present birth rate of the predator species,we incorporate the time delay due to the gestation of the predator species into system(1.1)and get the following predator-prey system with time delay:

where τ is the time delay due to the gestation of the predator species.

The rest of this work is organized in this pattern.In the next section,the existence of the Hopf bifurcation is investigated.In Section 3,based on the normal form method and center manifold theory,properties of the Hopf bifurcation are investigated.In Section 4,a numerical example is carried out in order to support the obtained theoretical predictions.The final section gives our conclusion.

2 Local Stability and Hopf Bifurcation

If

then(1.2)has a unique positive equilibrium P?(x?,y?,z?),where

The linearized system of system(1.2)at P?(x?,y?,z?)is given by

with

The characteristic equation of(2.1)is

where

When τ=0,(2.2)becomes

Hence,by Routh-Hurwitz criterion,P?(x?,y?,z?)is asymptotically stable when the condition(H1)holds,that is,

are satis fied.

For τ＞ 0,substituting λ =iω (ω ＞ 0)into(2.2),we obtain{

It follows that

whereLet ω2= ω?.Then

Based on the discussion of the distribution of the roots of(2.6)in[3],we suppose that

Differentiating(2.2)with respect to τ,we have

Further,we have

where

Hence,the transversality condition is satis fied if the condition

holds.Thus,according to the Hopf bifurcation theorem in[22],we have the following.

Theorem 2.1For system(1.2),if the conditions(H1)–(H3)hold,then the infected equilibriumP?(x?,y?,z?)of system(1.2)is asymptotically stable forτ∈ [0,τ0)and system(1.2)undergoes a Hopf bifurcation at the positive equilibriumP?(x?,y?,z?)whenτ= τ0,whereτ0is defined in(2.7).

3 Stability of the Bifurcating Periodic Solutions

Let τ= τ0+μ,μ ∈ R.Then μ =0 is the Hopf bifurcation value for system(1.2).Define the space of continuous real-valued functions as C=C([?1,0],R3).Let

The system(1.2)transforms to functional differential equation in C as

where ut=(u1(t),u2(t),u3(t))T∈ C=C([?1,0],R3),

and

where

and

with

By the representation theorem,there exists a 3 × 3 matrix function η(θ,μ),θ ∈ [?1,0]such that

In view of(3.2),we choose

where δ is the Dirac delta function.

For ? ∈ C([?1,0],R3),define

and

Then system(3.1)is equivalent to

where ut(θ)=u(t+ θ)for θ∈ [?1,0].

For φ ∈ C1([0,1],(R3)?),define

and a bilinear inner product

where η(θ)= η(θ,0).Then A(0)and A?are adjoint operators.

Since ±iω0τ0are the eigenvalues of A(0)and A?,respectively.We need to compute eigenvectors of A(0)and A?corresponding to+iω0τ0and ?iω0τ0,respectively.Suppose that ρ(θ)=(1,ρ2,ρ3)Teiω0τ0θis the eigenvector of A(0)belonging to+iω0τ0and ρ?(s)=is the eigenvector of A?(0)belonging to ?iω0τ0.By a direct computation,we can get

From(3.5)we can get

Then we choose

such that 〈ρ?,ρ〉=1.

Next,we can obtain the coefficients by using the method introduced in[22]and a computation process similar as that in[17],[23]and[24]:

with

where E1and E2are given by the following equations,respectively

and

Then we can get the following coefficients which determine the properties of the Hopf bifurcation:

In conclusion,we have the following results.

Theorem 3.1For system(1.2),Ifμ2＞ 0(μ2＜ 0),then the Hopf bifurcation is supercritical(subcritical);Ifβ2＜ 0(β2＞ 0),then the bifurcating periodic solutions are stable(unstable);IfT2＞0(T2＜0),then the bifurcating periodic solutions increase(decrease).

4 Numerical Example

In this section,we present a numerical example of system(1.2)to illustrate our obtained analytical findings.Let r1=1.5,K=4,σ1=2,σ2=2,v1=0.4,a=1,b=0.25,q1=0.1,E=1,r2=1,v2=0.4,β=0.8,d=0.01,?=0.4,q2=0.2.Then we get the following special case of system(1.2):

By a direct computation using Matlab software package,we obtain the unique positive equilibrium P?(0.8026,1.1112,1.6284).Further,we obtain

Let τ=0.83 ∈ [0,τ0).We can see that the positive equilibrium P?(0.8026,1.1112,1.6284)is locally asymptotically stable,which can be shown in Fig.4.1.This reveals that the densities of the three species in system(4.1)will tend to stabilization and this situation does not vary with the time delay τ∈ [0,τ0).However,once the time delay τ passes through the critical value τ0,then the positive equilibrium P?(0.8026,1.1112,1.6284)will lose its stability and a Hopf bifurcation occurs.This property can be illustrated by Fig.4.2.From this we know that the densities of the three species in system(4.1)will oscillate in the vicinity of the three species,respectively.In addition,we also obtain λ′(τ0)=0.4011 ? 0.0367i and C1(0)=?3.4254+1.6580i by some complex computations.Thus,based on(3.6),we haveμ2=8.54＞ 0,β2=?6.8508＜0 and T2=?0.5666＜0.It follows that the Hopf bifurcation at τ0is supercritical and stable,and the periodic solutions decrease.

Fig.4.1 P? is locally asymptotically stable when τ=0.83 ＜ τ0

Fig.4.2 System(4.1)undergoes a Hopf bifurcation when τ=1.43 ＞ τ0

5 Conclusions

In this paper,a delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed by introducing the time delay due to the gestation of the predator species into the model considered in[14].Compared with the work in[14],we mainly investigate the effect of the time delay on the proposed system.

The main results are given in terms of local stability of the positive equilibrium and the local Hopf bifurcation of the system.It is proved that the positive equilibrium is locally asymptotically stable when the time delay is suitable small and a Hopf bifurcation occurs once the value of the time delay is larger than the critical value τ0.Through the numerical simulation,we know that the species in the system can coexist in an oscillatory mode when the certain conditions are satis fied.This is valuable from the viewpoint of biology.

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