# A Predator-Prey Model with Functional Response and Stage Structure for Prey

Authors: Xiao-Ke Sun; Hai-Feng Huo; Xiao-Bing Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A predator-prey system with Holling type II functional response and stage structure for prey is presented. The local and global stability are studied by analyzing the associated characteristic transcendental equation and using comparison theorem. The existence of a Hopf bifurcation at the positive equilibrium is also studied. Some numerical simulations are also given to illustrate our results.


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## Body

## 1. Introduction

Predator-prey system is a very classical and important mathematical model in population dynamics, there are many studies in progress nowadays (see [3–5]). In recent years, many scholars are giving their attention to predator-prey model with stage structure, and many results have been published. In [6, 7], the influences of prey or predator with stage structure to the state of the ecosystem are studied, respectively. A time-delay model of single-species growth incorporating stage structure as a reasonable generation of the logistic model was derived in [8]. This model assumes an average age to maturity which appears as a constant time delay reflecting a delayed birth of immatures and a reduced survival of immatures to their maturity. The model takes the form(1.1)xi.(t)=bxm(t)-d1xi(t)-be-d1τxm(t-τ),x.m(t)=be-d1τxm(t-τ)-d2xm(t)-βxm2(t),
where xi(t) and xm(t) denote the immature and mature population densities, respectively; b represents the birth rate of the immature population; d1 and d2 are the death rate of immature and mature population, respectively; τ denotes the length of time from the birth to maturity. It is shown in [8] that the introduction of stage structure does not affect the permanence of the species, but, under some hypotheses, it may maximize the total carrying capacity of the population.Based on the previous model, several predator-prey models with stage structure are developed and studied (see [9–15]). Recently, Song et al. [16] investigated the following model with stage structure for prey:(1.2)xi.(t)=bxm(t)-d1xi(t)-be-d1τxm(t-τ),xm.(t)=be-d1τxm(t-τ)-βxm2(t)-a1xm(t)y(t)xm(t)+cy(t),y.(t)=a2xm(t)y(t)xm(t)+cy(t)-dy(t).
Their results showed that the positive equilibrium of system (1.2) was globally stable under some conditions.On the other hand, the functional response plays an important role in all prey interactions. A functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. One significant component of the predator-prey relationship is the predator’s functional responses. When the predator spends some time searching for prey and some time, exclusive of searching, processing each captured prey item (i.e., handling time), Holling type II is a useful functional response (see [17–21]). For example, Sun et al. [22] considered the following model with Holling type II functional response and stage structure for predator:(1.3)x.(t)=bx(t)-βx2(t)-a1x(t)y(t)1+cx(t),Y.(t)=a2x(t)y(t)1+cx(t)-d1Y(t)-a2x(t-τ)y(t-τ)1+cx(t-τ)e-d1τ,y.(t)=a2x(t-τ)y(t-τ)1+cx(t-τ)e-d1τ-d2y(t).By analyzing the corresponding characteristic equations, the local stability of equilibria and existence of a Hopf bifurcation at the positive equilibrium are obtained in [22].In this paper, we consider and study the following model with stage structure for prey:(1.4)xi.(t)=bxm(t)-d1xi(t)-be-d1τxm(t-τ),xm.(t)=be-d1τxm(t-τ)-d2xm(t)-βxm2(t)-a1xm(t)y(t)1+cxm(t),y.(t)=a2xm(t)y(t)1+cxm(t)-dy(t).
For continuity of the initial conditions, we further require(1.5)xi(t)>0,xm(t)>0,y(t)>0,t∈[-τ,0],xi(0)=b∫-τ0xm(s)ed1sds.The remainder of the paper is organized as follows: in the next section, results on positivity and boundedness of solutions are presented. In Section3, through using geometric stability switch criteria [1] to analyze the corresponding characteristic equations, we discuss the local stability of equilibria. Further, we study the existence of a Hopf bifurcation at the positive equilibria. The global stability of boundary equilibrium is obtained by using comparison theorem in Section 4. In Section 5, we try to interpret our results by numerical simulation. At last, we give some discussions.


## 2. Positive and Bounded

In this section, we discuss the positivity and boundedness of the solutions of system (1.4). Positivity means that the species is persistent and boundedness implies a natural restriction, which shows our model (1.4) is reasonable.Theorem 2.1.
The solutions of system (1.4) with initial conditions (1.5) are positive for all t≥0.Proof.
It is easy to see thatxm(t) and y(t) are positive for all t≥0.
We only showxi(t)>0.
Since(2.1)xm(t)>0,
hence
(2.2)xi.(t)>-d1xi(t)-be-d1τxm(t-τ).
Considering the equation
(2.3)u.(t)=-d1u(t)-be-d1τxm(t-τ),u(0)=xi(0),
we have xi(t)>u(t) on t∈(0,τ], and from variation-of-constants formula, we get
(2.4)u(t)=e-d1t[xi(0)-∫0tbed1(s-τ)xm(s-τ)ds],
hence,
(2.5)u(τ)=e-d1τ[xi(0)-∫0τbed1(s-τ)xm(s-τ)ds]=e-d1τ[xi(0)-∫-τ0bed1θxm(θ)dθ]=0.
Thus, xi(t)>0 on t∈[0,τ]. By induction, we can show that xi(t)>0 for all t≥0. The proof is complete.Theorem 2.2.
Every solution of system (1.4) with initial conditions (1.5) are bounded for all t≥0, and all of these solutions are ultimately bounded.Proof.
LetV(t)=a2xi(t)+a2xm(t)+a1y(t), calculating the derivative of V(t) with respect to t along the positive solution of system (1.4), we have
(2.6)V.(t)=a2x.i(t)+a2x.m(t)+a1y.(t)=a2[bxm(t)-d1xi(t)-d2xm(t)-βxm2(t)-a1xm(t)y(t)1+cxm(t)]+a1[-dy(t)+a2xm(t)y(t)1+cxm(t)]=(a2b-a2d2)xm(t)-a2d1xi(t)-a1dy(t)-a2βxm2(t).
For a small positive constant s≤min{d1,d},
(2.7)V.(t)+sV(t)=(a2b1-a2d2+a2s)xm(t)-(a2d1-a2s)xi(t)-(a1d-a1s)y(t)-a2βxm2(t)<a2(b-d2+s)x2(t)-a2βx22(t).
Hence there exists a positive constant K, such that
(2.8)V.(t)+sV(t)≤K,
thus we get
(2.9)V(t)≤(V(0)-Ks)e-st+Ks.
Therefore, V(t) is ultimately bounded, that is, each solution z(t)=(xi(t),xm(t),y(t)) of system (1.4) is ultimately bounded. The proof is complete.


## 3. Equilibria Analysis

Note that, in the system (1.4), the equation for the variable xi has a form(3.1)xi.(t)=bxm(t)-d1xi(t)-be-d1τxm(t-τ):=-d1x1(t)+f(xm(t),xm(t-τ)).
If xm(t) is bounded and xm(t)→x* as t→∞, then xi(t)→f(x*,x*)/d1 as t→∞; that is, the asymptotic behavior of xi(t) is dependent on that of xm(t). Therefore, in following sections we just need to study the asymptotic behavior of the following system:(3.2)x.(t)=be-d1τx(t-τ)-d2x(t)-βx2(t)-a1x(t)y(t)1+cx(t),y.(t)=a2x(t)y(t)1+cx(t)-dy(t).By a straightforward calculation, we can get two nonnegative equilibria:E0=(0,0) and E1=(x1,0)=((be-d1τ-d2)/β,0), E1 exists if and only if(H1):
be-d1τ>d2 holds.Further, if(H2):
a2-cd>0 and (be-d1τ-d2) (a2-cd)-βd>0 are satisfied, then system (3.2) has a unique positive equilibrium E=(x*,y*)=(d/(a2-cd),(a2[(be-d1τ-d2)(a2-cd)-βd])/a1(a2-cd)2).Remark 3.1.
(H1) is equivalent to τ∈[0,τ1), where τ1=(1/d1)ln(b/d2). From (H2), we have τ∈[0,τ2)and τ2=(1/d1)ln(b(a2-cd)/(βd+d2(a2-cd))); hence, if E is a positive equilibrium, there must be
(3.3)(b-d2)(a2-cd)>βd,(3.4)τ1>τ2.
In some of the subsequent sections, we always assume that (3.3) is valid; otherwise, there is no positive equilibrium of system (3.2).


### 3.1.E0=(0,0)

First, we analyze the stability of equilibriumE0, and the associated characteristic equation has the form(3.5)(λ+d2-be-λτ-d1τ)(λ+d)=0.Theorem 3.2.
(1) If d2>b, then the equilibrium E0 of system (3.2) is uniformly asymptotically stable.
(2) If d2<b, then the equilibrium E0 of system (3.2) is(i)
unstable for allτ∈[0,τ1) (see Remark 3.1),(ii)
stable, but not asymptotically stable whenτ=τ1,(iii)
asymptotically stable for allτ∈(τ1,+∞).Proof.
λ=-d has always negative real part. Denote G(λ,τ)=λ+d2-be-d1τe-λτ.
(1) Let τ=0 in G(λ,τ), then we have that
(3.6)G(λ,0)=λ+d2-b=0,
if d2>b, then
(3.7)λ=b-d2<0,
that is, the trivial solution E0 of system (3.2) is stable when there is no delay τ. If the increasing of τ leads to instability of system (3.2), then there is an ω>0 such that G(ωi,τ)=0 for some τ>0. Note that
(3.8)d2=be-d1τcos⁡⁡ωτ,ω=-be-d1τsin⁡⁡ωτ.
Hence,
(3.9)ω2=(be-d1τ)2-d22.
But, from hypothesis d2>b, we know that (3.9) has no real root. Hence, the equilibrium E0 of system (3.2) is uniformly asymptotic stable for any delay τ>0.
(2) If d2<b, then the trivial solution E0 of system (3.2) is unstable when there is no delay τ.
(i) Whenτ∈[0,τ1), we know that (3.9) has a positive root ω(τ). For τ∈[0,τ1), let θ(τ)∈(0,2π) be defined by
(3.10)cos⁡⁡θ(τ)=d2be-d1τ,sin⁡⁡θ(τ)=-ω(τ)be-d1τ,
where, jointly with (3.9), we define the following maps:
(3.11)Sn(τ)=τ-θ(τ)+2nπω(τ),n∈N.
IfSn(τ)=0 has no roots in [0,τ1) when n=0, then we know that the equilibrium E0 of system (3.2) is unstable through using geometric stability switch criteria [1]. Otherwise, let τ*=min{τ:τ∈[0,τ1),S0(τ)=0}, then
(3.12)sign⁡{dReλdτ|λ=ω(τ*)i}=sign⁡{dSn(τ)dτ|τ=τ*}>0(becauseofS0(0)<0),
which straightforwards implies (i).
(ii) Whenτ=τ1, we have G(λ,τ1)=λ+d2-d2e-λτ1, and λ=0 is always a simple eigenvalue. Suppose that G(λ,τ1)=0 has a root λ=u+iv, where u>0, then
(3.13)(u+d2)2+v2=d22e-2uτ1≤d22,
a contradiction. Hence, (ii) is true.
(iii) Whenτ∈(τ1,+∞), (3.9) has no positive root, which means that the stability of E0 does not change as τ increases. Let τ10=τ1+ε, where ε is a positive constant; we consider the following characteristic equation with delay-independent coefficients:
(3.14)λ+d2-be-d1τ10e-λτ=0,d2>be-d1τ10.
By Theorem3.4.1 [2, page 66] we know that (3.14) has only negative real-part eigenvalues; thus, (iii) is valid. The proof is complete.Remark 3.3.
From Theorem3.2 we know that when the birth rate of prey species is less than death rate of itself, then the prey species tend to be extinct; therefore, the predator species reduces to zero since there is no enough foods, which accords with the basic evolution law of biology. We also see that time delay can stabilize an unstable system and keep this state to τ large enough. This phenomenon cannot appear in the case that the coefficients of a corresponding characteristic equation are independent of the delay.


### 3.2.E1=(x1,0)

Now, we analyze the stability ofE1. The characteristic equation of system (3.2) at E1 takes the form(3.15)(λ-d2+2be-d1τ-be-λτ-d1τ)(λ+d-a2x11+cx1)=0.DenoteH1(λ,τ)=λ-d2+2be-d1τ-be-λτ-d1τ and H2(λ)=λ+d-a2(x1)/(1+cx1).Obviously, the root of equationH2(λ)=0 is positive when τ∈[0,τ2) and negative when τ∈(τ2,τ1); hence, we only need to consider H1(λ,τ)=0. By the same method with the proof of Theorem 3.2, we know that H1(λ,τ)=0 has only negative real-part characteristic roots when τ∈[0,τ1).Summarizing the discussion above, we get the following conclusion.Theorem 3.4.
If (H1) is satisfied, then the equilibrium E1 of system (3.2) is
(1) unstable when τ∈[0,τ2) (see Remark 3.1),
(2) stable, but not asymptotically stable when τ=τ2,
(3) asymptotically stable when τ∈(τ2,τ1).


### 3.3.E=(x*,y*)

Finally, we analyze the stability of positive equilibriumE of system (3.2). The linearizing equation of system (3.2) at E is(3.16)x.(t)=be-d1τx(t-τ)-[d2+a1y*(1+cx*)2+2βx*]x(t)-a1x*1+cx*y(t),y.(t)=a2y*(1+cx*)2x(t)+(a2x*1+cx*-d2)y(t),
hence, the characteristic equation takes the form(3.17)D(λ,τ)=λ2+p(τ)λ+q(τ)+r(τ)λe-λτ=0,
where(3.18)p(τ)=d2+a1y*(1+cx*)2+2βx*,q(τ)=a1dy*(1+cx*)2,r(τ)=-d2-βx*-a1y*1+cx*.Note that these are dependent on time delayτ since y* includes τ. In order to apply the geometric criterion of Beretta and Kuang [1], we rewrite D(λ,τ)=0 into(3.19)D(λ,τ)=P(λ,τ)+Q(λ,τ)e-λτ=0,
where(3.20)P(λ,τ)=λ2+p(τ)λ+q(τ),Q(λ,τ)=r(τ)λ.For convenience, letf(τ)=βa2-c((be-d1τ-d2)(a2-cd)-βd),τ∈[0,τ2), then f(τ) is strictly monotone increasing function in τ. If f(0)>0, hence, f(τ)>0 for all τ∈[0,τ2). Otherwise, that is, if f(0)<0, since f(τ2)=βa2>0, there is a unique τ3∈(0,τ2) such that f(τ3)=0.Lemma 3.5.
(1) If (H2) and (H3): βa2-c[(b-d2)(a2-cd)-βd]>0 hold, then q(τ)>0 and p(τ)+r(τ)>0 for all τ∈[0,τ2).
(2) If (H2) and (H4): βa2-c[(b-d2)(a2-cd)-βd]<0 hold, then q(τ)>0 for all τ∈[0,τ2), p(τ)+r(τ)<0 when τ∈[0,τ3) and p(τ)+r(τ)>0 when τ∈(τ3,τ2).Proof.
Obviously,q(τ)>0. From (3.18), we have
(3.21)p(τ)+r(τ)=βx*-ca1y*x*(1+cx*)2=x*(1+cx*)2[β(1+cx*)2-ca1y*]=x*(1+cx*)(a2-cd)[βa2-c((be-d1τ-d2)(a2-cd)-βd)].(1)
Iff(0)>0, then f(τ)>0 for all τ∈[0,τ2); hence, p(τ)+r(τ)>0 for all τ∈[0,τ2).(2)
Iff(0)<0, then f(τ)<0 when τ∈[0,τ3) and f(τ)>0 when τ∈(τ3,τ2); hence, (2) is true. The proof is complete.Step 1.
Whenτ=0, (3.17) becomes
(3.22)D(λ,0)=λ2+(p(0)+r(0))λ+q(0)=0,
by Routh-Hurwitz Criterion we know that D(λ,0)=0 has only negative real-part eigenvalue if (H3) holds and has positive real-part eigenvalue if (H4) holds. Hence, we obtain the following theorem.Theorem 3.6.
(1) If (H2) and (H3) are satisfied, then the equilibrium E of system (3.2) is asymptotically stable when τ=0.
(2) If (H2) and (H4) are satisfied, then the equilibrium E of system (3.2) is unstable when τ=0.Step 2.
SupposeD(ωi,τ)=0 for some τ∈[0,τ2) and ω>0, we get
(3.23)sin⁡⁡ωτ=(ω2-q)ωr,cos⁡⁡ωτ=-pr.
Hence,(3.24)F(ω,τ)=|P(ωi,τ)|2-|Q(ωi,τ)|2=ω4+A(τ)ω2+B(τ)=0,
where
(3.25)A(τ)=p2-2q-r2,B(τ)=q2.
Now, we take hypotheses (H3) or (H4) into account, respectively.Case 1 (βa2-c[(b-d2)(a2-cd)-βd]>0).
In this case, since(3.26)▵(τ)=A(τ)2-4B(τ)=(p2-2q-r2)2-4q2=(p2-4q-r2)(p-r)(p+r),B(τ)=q2(τ)>0,p-r>0,p+r>0.
Thus, if A(τ)≥0, then (3.24) has no positive real root, and if A(τ)<0, then ▵(τ)<0 and (3.24) has no real root, that is, the stability of equilibrium E cannot change for all τ∈[0,τ2). Summarizing the discussion above, we obtain the following theorem.Theorem 3.7.
Assume that (H2) and (H3) are satisfied, then the equilibrium E of system (3.2) is asymptotically stable for all τ∈[0,τ2).Case 2 (βa2-c[(b-d2)(a2-cd)-βd]<0).
In this case, whenτ∈(τ3,τ2), by the similar analysis we also obtain that the stability of equilibrium E cannot change for all τ∈(τ3,τ2). Let τ0=τ3+ε, where ε is a positive constant; we consider the following characteristic equation with delay-independent coefficients:
(3.27)D(λ,τ)=λ2+p(τ0)λ+q(τ0)+r(τ0)λe-λτ=0,
where p(τ0)+r(τ0)>0 and q(τ0)>0. By Theorem 3.4.1 [2, page 66] we know that (3.27) has only negative real-part eigenvalues, then the equilibrium E is asymptotically stable for all τ∈(τ3,τ2).
Whenτ∈[0,τ3), since p(τ)+r(τ)<0, hence, A(τ)<0,B(τ)>0, and ▵(τ)>0; we easily know that (3.24) has two positive real roots ω+(τ) and ω-(τ), where
(3.28)ω±(τ)=-A(τ)±A2(τ)-4B(τ)2.
Forτ∈[0,τ3), let θ(τ)∈(0,2π) be defined by
(3.29)sin⁡θ±(τ)=(ω±2-q)rω±,cos⁡θ±(τ)=-pr,
where, jointly with (3.23), we define the following maps:
(3.30)Sn±(τ)=τ-θ±(τ)+2nπω±(τ),n∈N.
Obviously,sinθ+(τ)=(ω+2-q)/rω+<0, sinθ-(τ)=(ω-2-q)/(rω-)>0, and cosθ±(τ)=-p/r>0, which imply
(3.31)θ+(τ)=2π-arccos-pr,θ-(τ)=arccos-pr.
Furthermore, we note thatS0-(0)<0 and limτ→τ3S0-(τ)=τ3>0 (because of limτ→τ3θ-(τ)=0); thus, equation S0-(τ)=0 has at least one root in [0,τ3); let τ*=min{τ:τ∈[0,τ3),S0-(τ)=0}, then
(3.32)δ(τ*):=sign⁡{dReλdτ|λ=ω-(τ*)i}=-sign⁡{dS0-(τ)dτ|τ=τ*}<0,
that is, the pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right. Summarizing the discussion above, we obtain the following result.Theorem 3.8.
Assume that (H2) and (H4) are satisfied; if equation S0+(0)=0 has no root in [0,τ*), then the equilibrium E of system (3.2) is unstable for all τ∈[0,τ*) and becomes asymptotically stable for τ staying in some right neighborhood of τ*; hence, system (3.2) undergoes Hopf bifurcation when τ=τ*. The equilibrium E of system (3.2) is asymptotically stable for τ∈(τ3,τ2).


## 3.1.E0=(0,0)

First, we analyze the stability of equilibriumE0, and the associated characteristic equation has the form(3.5)(λ+d2-be-λτ-d1τ)(λ+d)=0.Theorem 3.2.
(1) If d2>b, then the equilibrium E0 of system (3.2) is uniformly asymptotically stable.
(2) If d2<b, then the equilibrium E0 of system (3.2) is(i)
unstable for allτ∈[0,τ1) (see Remark 3.1),(ii)
stable, but not asymptotically stable whenτ=τ1,(iii)
asymptotically stable for allτ∈(τ1,+∞).Proof.
λ=-d has always negative real part. Denote G(λ,τ)=λ+d2-be-d1τe-λτ.
(1) Let τ=0 in G(λ,τ), then we have that
(3.6)G(λ,0)=λ+d2-b=0,
if d2>b, then
(3.7)λ=b-d2<0,
that is, the trivial solution E0 of system (3.2) is stable when there is no delay τ. If the increasing of τ leads to instability of system (3.2), then there is an ω>0 such that G(ωi,τ)=0 for some τ>0. Note that
(3.8)d2=be-d1τcos⁡⁡ωτ,ω=-be-d1τsin⁡⁡ωτ.
Hence,
(3.9)ω2=(be-d1τ)2-d22.
But, from hypothesis d2>b, we know that (3.9) has no real root. Hence, the equilibrium E0 of system (3.2) is uniformly asymptotic stable for any delay τ>0.
(2) If d2<b, then the trivial solution E0 of system (3.2) is unstable when there is no delay τ.
(i) Whenτ∈[0,τ1), we know that (3.9) has a positive root ω(τ). For τ∈[0,τ1), let θ(τ)∈(0,2π) be defined by
(3.10)cos⁡⁡θ(τ)=d2be-d1τ,sin⁡⁡θ(τ)=-ω(τ)be-d1τ,
where, jointly with (3.9), we define the following maps:
(3.11)Sn(τ)=τ-θ(τ)+2nπω(τ),n∈N.
IfSn(τ)=0 has no roots in [0,τ1) when n=0, then we know that the equilibrium E0 of system (3.2) is unstable through using geometric stability switch criteria [1]. Otherwise, let τ*=min{τ:τ∈[0,τ1),S0(τ)=0}, then
(3.12)sign⁡{dReλdτ|λ=ω(τ*)i}=sign⁡{dSn(τ)dτ|τ=τ*}>0(becauseofS0(0)<0),
which straightforwards implies (i).
(ii) Whenτ=τ1, we have G(λ,τ1)=λ+d2-d2e-λτ1, and λ=0 is always a simple eigenvalue. Suppose that G(λ,τ1)=0 has a root λ=u+iv, where u>0, then
(3.13)(u+d2)2+v2=d22e-2uτ1≤d22,
a contradiction. Hence, (ii) is true.
(iii) Whenτ∈(τ1,+∞), (3.9) has no positive root, which means that the stability of E0 does not change as τ increases. Let τ10=τ1+ε, where ε is a positive constant; we consider the following characteristic equation with delay-independent coefficients:
(3.14)λ+d2-be-d1τ10e-λτ=0,d2>be-d1τ10.
By Theorem3.4.1 [2, page 66] we know that (3.14) has only negative real-part eigenvalues; thus, (iii) is valid. The proof is complete.Remark 3.3.
From Theorem3.2 we know that when the birth rate of prey species is less than death rate of itself, then the prey species tend to be extinct; therefore, the predator species reduces to zero since there is no enough foods, which accords with the basic evolution law of biology. We also see that time delay can stabilize an unstable system and keep this state to τ large enough. This phenomenon cannot appear in the case that the coefficients of a corresponding characteristic equation are independent of the delay.


## 3.2.E1=(x1,0)

Now, we analyze the stability ofE1. The characteristic equation of system (3.2) at E1 takes the form(3.15)(λ-d2+2be-d1τ-be-λτ-d1τ)(λ+d-a2x11+cx1)=0.DenoteH1(λ,τ)=λ-d2+2be-d1τ-be-λτ-d1τ and H2(λ)=λ+d-a2(x1)/(1+cx1).Obviously, the root of equationH2(λ)=0 is positive when τ∈[0,τ2) and negative when τ∈(τ2,τ1); hence, we only need to consider H1(λ,τ)=0. By the same method with the proof of Theorem 3.2, we know that H1(λ,τ)=0 has only negative real-part characteristic roots when τ∈[0,τ1).Summarizing the discussion above, we get the following conclusion.Theorem 3.4.
If (H1) is satisfied, then the equilibrium E1 of system (3.2) is
(1) unstable when τ∈[0,τ2) (see Remark 3.1),
(2) stable, but not asymptotically stable when τ=τ2,
(3) asymptotically stable when τ∈(τ2,τ1).


## 3.3.E=(x*,y*)

Finally, we analyze the stability of positive equilibriumE of system (3.2). The linearizing equation of system (3.2) at E is(3.16)x.(t)=be-d1τx(t-τ)-[d2+a1y*(1+cx*)2+2βx*]x(t)-a1x*1+cx*y(t),y.(t)=a2y*(1+cx*)2x(t)+(a2x*1+cx*-d2)y(t),
hence, the characteristic equation takes the form(3.17)D(λ,τ)=λ2+p(τ)λ+q(τ)+r(τ)λe-λτ=0,
where(3.18)p(τ)=d2+a1y*(1+cx*)2+2βx*,q(τ)=a1dy*(1+cx*)2,r(τ)=-d2-βx*-a1y*1+cx*.Note that these are dependent on time delayτ since y* includes τ. In order to apply the geometric criterion of Beretta and Kuang [1], we rewrite D(λ,τ)=0 into(3.19)D(λ,τ)=P(λ,τ)+Q(λ,τ)e-λτ=0,
where(3.20)P(λ,τ)=λ2+p(τ)λ+q(τ),Q(λ,τ)=r(τ)λ.For convenience, letf(τ)=βa2-c((be-d1τ-d2)(a2-cd)-βd),τ∈[0,τ2), then f(τ) is strictly monotone increasing function in τ. If f(0)>0, hence, f(τ)>0 for all τ∈[0,τ2). Otherwise, that is, if f(0)<0, since f(τ2)=βa2>0, there is a unique τ3∈(0,τ2) such that f(τ3)=0.Lemma 3.5.
(1) If (H2) and (H3): βa2-c[(b-d2)(a2-cd)-βd]>0 hold, then q(τ)>0 and p(τ)+r(τ)>0 for all τ∈[0,τ2).
(2) If (H2) and (H4): βa2-c[(b-d2)(a2-cd)-βd]<0 hold, then q(τ)>0 for all τ∈[0,τ2), p(τ)+r(τ)<0 when τ∈[0,τ3) and p(τ)+r(τ)>0 when τ∈(τ3,τ2).Proof.
Obviously,q(τ)>0. From (3.18), we have
(3.21)p(τ)+r(τ)=βx*-ca1y*x*(1+cx*)2=x*(1+cx*)2[β(1+cx*)2-ca1y*]=x*(1+cx*)(a2-cd)[βa2-c((be-d1τ-d2)(a2-cd)-βd)].(1)
Iff(0)>0, then f(τ)>0 for all τ∈[0,τ2); hence, p(τ)+r(τ)>0 for all τ∈[0,τ2).(2)
Iff(0)<0, then f(τ)<0 when τ∈[0,τ3) and f(τ)>0 when τ∈(τ3,τ2); hence, (2) is true. The proof is complete.Step 1.
Whenτ=0, (3.17) becomes
(3.22)D(λ,0)=λ2+(p(0)+r(0))λ+q(0)=0,
by Routh-Hurwitz Criterion we know that D(λ,0)=0 has only negative real-part eigenvalue if (H3) holds and has positive real-part eigenvalue if (H4) holds. Hence, we obtain the following theorem.Theorem 3.6.
(1) If (H2) and (H3) are satisfied, then the equilibrium E of system (3.2) is asymptotically stable when τ=0.
(2) If (H2) and (H4) are satisfied, then the equilibrium E of system (3.2) is unstable when τ=0.Step 2.
SupposeD(ωi,τ)=0 for some τ∈[0,τ2) and ω>0, we get
(3.23)sin⁡⁡ωτ=(ω2-q)ωr,cos⁡⁡ωτ=-pr.
Hence,(3.24)F(ω,τ)=|P(ωi,τ)|2-|Q(ωi,τ)|2=ω4+A(τ)ω2+B(τ)=0,
where
(3.25)A(τ)=p2-2q-r2,B(τ)=q2.
Now, we take hypotheses (H3) or (H4) into account, respectively.Case 1 (βa2-c[(b-d2)(a2-cd)-βd]>0).
In this case, since(3.26)▵(τ)=A(τ)2-4B(τ)=(p2-2q-r2)2-4q2=(p2-4q-r2)(p-r)(p+r),B(τ)=q2(τ)>0,p-r>0,p+r>0.
Thus, if A(τ)≥0, then (3.24) has no positive real root, and if A(τ)<0, then ▵(τ)<0 and (3.24) has no real root, that is, the stability of equilibrium E cannot change for all τ∈[0,τ2). Summarizing the discussion above, we obtain the following theorem.Theorem 3.7.
Assume that (H2) and (H3) are satisfied, then the equilibrium E of system (3.2) is asymptotically stable for all τ∈[0,τ2).Case 2 (βa2-c[(b-d2)(a2-cd)-βd]<0).
In this case, whenτ∈(τ3,τ2), by the similar analysis we also obtain that the stability of equilibrium E cannot change for all τ∈(τ3,τ2). Let τ0=τ3+ε, where ε is a positive constant; we consider the following characteristic equation with delay-independent coefficients:
(3.27)D(λ,τ)=λ2+p(τ0)λ+q(τ0)+r(τ0)λe-λτ=0,
where p(τ0)+r(τ0)>0 and q(τ0)>0. By Theorem 3.4.1 [2, page 66] we know that (3.27) has only negative real-part eigenvalues, then the equilibrium E is asymptotically stable for all τ∈(τ3,τ2).
Whenτ∈[0,τ3), since p(τ)+r(τ)<0, hence, A(τ)<0,B(τ)>0, and ▵(τ)>0; we easily know that (3.24) has two positive real roots ω+(τ) and ω-(τ), where
(3.28)ω±(τ)=-A(τ)±A2(τ)-4B(τ)2.
Forτ∈[0,τ3), let θ(τ)∈(0,2π) be defined by
(3.29)sin⁡θ±(τ)=(ω±2-q)rω±,cos⁡θ±(τ)=-pr,
where, jointly with (3.23), we define the following maps:
(3.30)Sn±(τ)=τ-θ±(τ)+2nπω±(τ),n∈N.
Obviously,sinθ+(τ)=(ω+2-q)/rω+<0, sinθ-(τ)=(ω-2-q)/(rω-)>0, and cosθ±(τ)=-p/r>0, which imply
(3.31)θ+(τ)=2π-arccos-pr,θ-(τ)=arccos-pr.
Furthermore, we note thatS0-(0)<0 and limτ→τ3S0-(τ)=τ3>0 (because of limτ→τ3θ-(τ)=0); thus, equation S0-(τ)=0 has at least one root in [0,τ3); let τ*=min{τ:τ∈[0,τ3),S0-(τ)=0}, then
(3.32)δ(τ*):=sign⁡{dReλdτ|λ=ω-(τ*)i}=-sign⁡{dS0-(τ)dτ|τ=τ*}<0,
that is, the pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right. Summarizing the discussion above, we obtain the following result.Theorem 3.8.
Assume that (H2) and (H4) are satisfied; if equation S0+(0)=0 has no root in [0,τ*), then the equilibrium E of system (3.2) is unstable for all τ∈[0,τ*) and becomes asymptotically stable for τ staying in some right neighborhood of τ*; hence, system (3.2) undergoes Hopf bifurcation when τ=τ*. The equilibrium E of system (3.2) is asymptotically stable for τ∈(τ3,τ2).


## 4. Global Stability

In this section, we discuss the global stability of boundary equilibria. In some of the subsequent analysis, we will need the following result.Lemma 4.1 (see [16]).
Consider the following equation:(4.1)x.(t)=ax(t-τ)-bx(t)-cx2(t)(a,b,c,τ>0,x(t)>0 for -τ≤t≤0), then one has the following.(i)
Ifa>b, then limt→∞x(t)=(a-b)/c.(ii)
Ifa<b, then limt→∞x(t)=0.Theorem 4.2.
Assume thatbe-d1τ<d2, then the equilibrium E0 of system (3.2) is globally asymptotically stable.Proof.
We know that ifbe-d1τ<d2, then E0 is locally asymptotically stable. Now we show
(4.2)lim⁡t→∞(x,y)=(0,0).
From the first equation of system (3.2), we get
(4.3)x.(t)≤be-d1τx(t-τ)-d2x(t)-βx2(t).
By Lemma4.1 and comparing principle, we obtain
(4.4)lim⁡t→∞x=0,
that is, for the ε>0, there exists a T1>0 such that
(4.5)0<x(t)<ε,t>T1,
which jointly with the second equation of system (3.2) leads to
(4.6)y.(t)<(εa2-d)y(t).
Since the arbitrariness ofε>0, we have
(4.7)lim⁡t→∞y=0.
The proof is complete.Theorem 4.3.
Assume that (H1) is satisfied, then the equilibrium E1 of system (3.2) is globally asymptotically stable for τ∈(τ2,τ1).Proof.
We need to show attractivity. From the first equation of system (3.2), we get
(4.8)x.(t)≤be-d1τx(t-τ)-d2x(t)-βx2(t).
By Lemma4.1 and comparing principle, we obtain
(4.9)lim⁡sup⁡t→∞x(t)≤be-d1τ-d2β=x1,
that is, for the ε>0, there exists a T1>0 such that
(4.10)0<x(t)<x1+ε,fort>T1,
which jointly with the second equation of system (3.2) leads to
(4.11)y.(t)<[a2(x1+ε)1+c(x1+ε)-d]y(t).
On the other hand, forτ∈(τ2,τ1), we have
(4.12)a2x11+cx1-d=(a2-cd)x1-d1+cx1<0.
Thus, forε>0 is small enough, there is
(4.13)a2(x1+ε)1+c(x1+ε)-d<0,
which leads to
(4.14)lim⁡t→∞y(t)=0,
that is, for the ε>0, there exists a T2>T1 such that
(4.15)0<y(t)<ε,fort>T2,
hence, we have
(4.16)x.(t)≥be-d1τx(t-τ)-(d2+ε)x(t)-βx2(t),
by hypothesis (H1), Lemma 4.1, and the arbitrariness of ε>0, there is
(4.17)lim⁡inf⁡t→∞x(t)≥be-d1τ-d2β=x1.
The proof is complete.


## 5. Numerical Simulation

In this section, we present some numerical simulations of system (3.2) to illustrate our theoretical analysis.Example 5.1.
Letb=a2=0.8,d=a1=0.5,d1=c=1,d2=0.1, and β=0.2, and we consider the following system:
(5.1)x.(t)=0.8e-0.5τx(t-τ)-0.1x(t)-0.2x2(t)-0.5x(t)y(t)1+x(t),y.(t)=0.8x(t)y(t)1+x(t)-0.5y(t).
By calculating, we have that τ1=̇2.0794,τ2=̇0.6131, and hypotheses (H2) and (H3) hold. By the discussion above, we obtain following corollary.Corollary 5.2.
For system (5.1), when τ∈[0,0.6131),E is asymptotically stable, E1 is globally stable when τ∈(0.6131,2.0794), and E0 is globally stable when τ∈(2.0794,+∞).First, we takeτ=2.5 and initial value is that(5.2)x(t)=y(t)=1.8,t∈[-τ,0].Figure1 illustrates that prey and predator converge asymptotically to zero, when τ=1and initial value is that x(t)=y(t)=5,t∈[-τ,0].Figure 1
The equilibriumE0 is globally stable when τ=2.5.Figure2 illustrates that prey and predator converge asymptotically to E1, where x1=̇0.9715.Figure 2
The equilibriumE1 is globally stable when τ=1.Finally, we takeτ=0.001; system (5.1) has positive equilibrium E=̇(1.6667,1.9513).Figure3 shows that the positive solutions converge asymptotically to the steady-state E.Figure 3
The equilibriumE is asymptotically stable when τ=0.001.Example 5.3.
In order to illustrate Theorem3.8, we choose a set of parameters as b=0.8,d=0.2,a1=0.5,a2=3,d1=0.05,c=5,d2=0.1, and β=1, then system (3.2) reduces to
(5.3)x.(t)=0.8e-0.05τx(t-τ)-0.1x(t)-x2(t)-0.5x(t)y(t)1+5x(t),y.(t)=3x(t)y(t)1+5x(t)-0.2y(t).
By calculating, we have that τ2=̇27.7259,τ3=̇9.4001, and hypotheses (H2) and (H4) hold.
Figure4 shows that equation S0+(τ)=0 has no root and equation S0-(τ)=0 has a unique root τ*, where τ*=̇2.9104, which leads to the following corollary.Figure 4
The graphs ofS0+(τ) and S0-(τ).Corollary 5.4.
The equilibriumE of system (5.3) is unstable for all τ∈[0,τ*) and becomes asymptotically stable for τ staying in some right neighborhood of τ*; hence, system (5.3) undergoes Hopf bifurcation when τ=τ*.By Corollary5.4, we know that the positive equilibrium is unstable and positive solutions are oscillatory when τ=2.9<τ*, the bifurcating periodic solutions exist at least for the value of τ slightly larger than the critical value τ*, the computer simulation is given in Figure 5. We continue to choose τ=3>τ*, by Corollary 5.4 we know that the positive equilibrium becomes asymptotically stable, and the computer simulation is depicted in Figure 6. In Figures 7 and 8, we plot the relationship between delay τ and ultimate amplitudes of x(t) and y(t), which indicate the fact that the positive equilibrium E is from instability to stability as τ increases.Figure 5
The bifurcating periodic solutions fromE when τ=2.9<τ*.Figure 6
The positive equilibrium is stable whenτ=3>τ*.Figure 7
The relationship between ultimate oscillation range of mature prey and delayτ.Figure 8
The relationship between ultimate oscillation range of predator and delayτ.


## 6. Discussion

In this paper, a model which describes the Holling type predator-prey system with stage structure for prey is proposed. Sufficient conditions which ensure the stability of equilibria and the existence of Hopf bifurcation are obtained. Numerical results show the feasibility of our results.By the analysis, we find the system (1.4) has complex dynamics behavior. Our results indicate that the stability switches of the positive equilibrium may occur as time delay τ increases and delay τ can stabilize system (1.4). Under condition (H4), the positive equilibrium is stable only if delay τ is larger, consequently, y* becomes smaller, that is, predator coexists with prey in lower population densities (with respect to delay τ), which is different from [22]. In [22], predator coexists with prey only if the population densities of prey are larger and those of predator are lower (with respect to delay τ). Moreover, Liu and Zhang [6] investigated system (1.2) when functional response is Beddington-DeAngelis type, by monotone dynamic theories, they obtained that the positive equilibrium is globally asymptotically stable provided that k2 (describing the magnitude of interference among predators) is more than a positive constant; that is, their results cannot directly apply to study system (1.4). Furthermore, they did not consider d2, the death rate of mature prey population, which play an important role on stability of the equilibria and existence of Hopf bifurcation. Hypothesis (H3) means that d2>d20, where d20=b-(β(a2+cd)/c(a2-cd)), and Hypothesis (H4) means that d2<d20. Hence, from Theorem 3.7 and Theorem 3.8, we know that d2 can also stabilize system (1.4).There are still many interesting and challenging mathematical questions that need to be studied for system (1.4). For example, we do not discuss the global stability of positive equilibrium. We will leave this for future work.



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*Source: 628103-2012-03-01.xml*