# Global Stability for a Delayed Predator-Prey System with Stage Structure for the Predator

Authors: Xiao Zhang; Rui Xu; Qintao Gan
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

A delayed predator-prey system with stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of equilibria of the system is discussed. The existence of Hopf bifurcation at the positive equilibrium is established. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and two boundary equilibria of the
system. Numerical simulations are carried out to illustrate the theoretical results.


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## Body

## 1. Introduction

Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. As is common, the dynamics-eating habits, susceptibility to predators, and so forth. are often quite different in these two subpopulations. Hence, it is of ecological importance to investigate the effects of such a subdivision on the interaction of species. In [1], Chen et al. introduced the following stage-structured single-species population model:(1.1)Ṅi(t)=B(t)-Di(t)-W(t),Ṅm(t)=αW(t)-Dm(t),
where Ni(t) and Nm(t) denote the immature and mature population densities at time t, respectively; B(t) is the birth rate of the immature population at time t; Di(t) and Dm(t) are the death rates of the immature and mature at time t; W(t) represents the transformation rate of the immature into the mature; α is the probability of the successful transformation of the immature into the mature. If the birth rate of model (1.1) obeys the Malthus rule, that is, B(t)=aNm, the death rates of the immature and mature populations are logistic, that is,(1.2)Di(t)=riNi(t)+biNi2(t),Dm(t)=rmNm(t)+bmNm2(t),
and the transformation rate of the immature into mature is proportional to the immature population, that is, W(t)=bNi(t), then model (1.1) becomes(1.3)Ṅi(t)=aNm-riNi(t)-biNi2(t)-bNi(t),Ṅm(t)=αbNi(t)-rmNm(t)-bmNm2(t).
Based on the idea above, many authors studied different kinds of stage-structured models, and a significant body of work has been carried out (see, for example, [2–8]).In [3], Gao et al. considered the following predator-prey model with stage structure:(1.4)ẋ(t)=x(t)(r-a11x(t)-a12y1(t-τ1)),ẏ1(t)=y1(t)[-r1+a21x(t-τ2)-a22y1(t)]+θy2(t),ẏ2(t)=αby1(t)-(θ+r2)y2(t),
where x(t) represents the density of the prey at time t; y1(t) and y2(t) represent the densities of the mature and the immature predator at time t, respectively. The parameters r,r1,r2,a11,a12,a21,a22,b,θ are positive constants in which r is the intrinsic growth of the prey, r1 is the death rate of the mature predator population, r2 is the death rate of the immature predator population, a11 is the intra-specific competition rate of the prey population, a12 is the capturing rate of the predator population, a21/a12 is the conversion rate of nutrients into the reproduction of the predator, a22 is the intra-specific competition rate of the mature predator, b is the birth rate of the immature predator, θ is the transformation rate from the immature predator individuals to mature predator individuals. The predation decreases the average growth rate of prey linearly with a certain time delay τ1≥0, this assumption corresponds to the fact that predators cannot hunt prey when the predators are infant; predators have to mature for a duration of τ1 units of time before they are capable of decreasing the average growth rate of the prey species; τ2≥0 is the time delay due to gestation, the delay in time for prey biomass to increase predator number. In [3], Gao et al. studied the global stability of the positive equilibrium and boundary equilibria of model (1.4) by constructing Liapunov functionals and comparison argument, respectively.We note that most of the predator-prey models with time delays studied in the literature are all of the Kolmogorov-type. In [9], Wangersky and Cunningham proposed delayed predator-prey models that are not of the Kolmogorov-type. They considered the following delayed system:(1.5)ẋ(t)=x(t)(r-a11x(t)-a12y(t)),ẏ(t)=a21x(t-τ)y(t-τ)-r1y(t),
where the delay τ is a constant based on the assumption that the change rate of predators depends on the number of both the prey and the predators present at some previous time.Motivated by the work of Gao et al. [3] and Wangersky and Cunningham [9], in the present paper, we consider the following predator-prey model with stage structure and time delay:(1.6)ẋ(t)=x(t)(r-a11x(t)-a12y1(t)),ẏ1(t)=a21x(t-τ)y1(t-τ)-r1y1(t)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
The meanings of the positive parameters r,r1,r2,a11,a12,a21,a22,b,θ are the same as those in system (1.4). The meaning of time delay τ≥0 is the same as in system (1.5).The initial conditions for system (1.6) take the form(1.7)x(θ)=ϕ(θ),y1(θ)=ψ1(θ),y2(θ)=ψ2(θ),ϕ(θ)≥0,ψ1(θ)≥0,ψ2(θ)≥0,θ∈[-τ,0],ϕ(0)>0,ψ1(0)>0,ψ2(0)>0,
where (ϕ1(θ),ψ1(θ),ψ2(θ))∈C([-τ,0],R+03), the Banach space of continuous functions mapping the interval [-τ,0] into R+03, where R+03={(x1,x2,x3):xi≥0,i=1,2,3}.This paper is organized as follows. In the next section, we introduce some notations and state several lemmas which will be essential to our proofs. In Section3, we discuss the local stability of a positive equilibrium and boundary equilibria of system (1.6). The existence of Hopf bifurcation is studied. In Section 4, by means of an iterative technique and comparison argument, sufficient conditions are derived for the global stability of the positive equilibrium and boundary equilibria of system (1.6). Some numerical examples are given to illustrate the results above. A brief discussion is given in Section 6 to end this work.


## 2. Preliminaries

In this section, we introduce some notations and state several results which will be useful in next section. Letℝ+n be the cone of nonnegative vectors in ℝn. If x,y∈ℝn, we write x≤y(x<y) if xi≤yi(xi<yi) for 1≤i≤n. Let e1,e2,…,en denote the standard basis in ℝn. Suppose r≥0 and let C=C([-r,0],ℝn) be the Banach space of continuous functions mapping the interval [-r,0] into ℝn with supremum norm. If ϕ,ψ∈C, we write ϕ≤ψ(ϕ<ψ) when the indicated inequality holds at each point of [-r,0]. Let C+={ϕ∈C:ϕ≥0} and let denote the inclusion ℝn→C([-r,0],ℝn) by x→x̂,x̂(θ)=x,θ∈[-r,0]. Denote the space of functions of bounded variation on [-r,0] by BV[-r,0]. If t0∈ℝn,A≥0, and x∈C([-t0-r,t0+A],ℝn), then for any t∈[t0,t0+A], we let xt∈C be defined by xt(θ)=x(t+θ),-r≤θ≤0.We now consider(2.1)ẋ(t)=f(t,xt).
We assume throughout this section that f:ℝ×C→ℝn is continuous; f(t,ϕ) is continuously differentiable in ϕ; f(t+T,ϕ)=f(t,ϕ) for all (t,ϕ)∈ℝ×C+, and some T>0. Then by [10], there exists a unique solution of (2.1) through (t0,ϕ) for t0∈ℝ,ϕ∈C+. This solution will be denoted by x(t,t0,ϕ) if we consider the solution in ℝn, or by xt(t0,ϕ) if we work in the space C. Again by [10], x(t,t0,ϕ)(xt(t0,ϕ)) is continuously differentiable in ϕ. In the following, the notation xt0=ϕ will be used as the condition of the initial data of (2.1), by which we mean that we consider the solution x(t) of (2.1) which satisfies x(t0+θ)=ϕ(θ),θ∈[-r,0].To proceed further, we need the following results from [11, 12]. Let r=(r1,r2,…,rn)∈ℝn,|r|=maxi{ri}, and define(2.2)Cr=∏i=1nC([-ri,0],ℝ).
We write ϕ=(ϕ1,ϕ2,…,ϕn) for a generic point of Cr. Let Cr+={ϕ∈Cr:ϕ≥0}. Due to the ecological applications, we choose Cr+ as the state space of (2.1) in the following discussions.Fixϕ0∈Cr+ arbitrarily. Then we set L(t,·)=Dϕ0f(t,ϕ0), and Dϕ0f(t,ϕ0) denotes the Frechet derivation of f with respect to ϕ0. It is convenient to have the standard representation of L=(L1,L2,…,Ln) as(2.3)Li(t,ϕ)=∑j=1n∫-rj0ϕj(θ)dθηij(θ,t)(1≤i≤n),
in which ηij:ℝ×ℝ→ℝ satisfies(2.4)ηij(θ,t)=ηij(0,t),θ≤-rj,ηij(θ,t)=0,θ≥0,ηij(·,t)∈BV[-rj,0],
where ηij(·,t) is continuous from the left in (-rj,0).We make the following assumptions for (2.1).(h0)
Ifϕ,ψ∈C+,ϕ≤φ, and ϕi(0)=φi(0) for some i, then fi(t,ϕ)≤fi(t,ψ).(h1)
For allϕ∈Cr+ with ϕi(0)=0,Li(t,ϕ)≥0 for t∈ℝ.(h2)
The matrixA(t) defined by
(2.5)A(t)=col(L(t,ê1),L(t,ê2),…,L(t,ên))=(ηij(0,t)),
is irreducible for each t∈ℝ.(h3)
For eachj, for which rj>0, there exist i such that for all t∈ℝ and for positive constant ε sufficiently small, ηij(-rj+ε,t)>0.(h4)
Ifϕ=0, then x(t,t0,ϕ)≡0 for all t≥t0.The following result was established by Wang et al. [12].Lemma 2.1.
Let (h1)–(h4) hold. Then the hypothesis (h0) is valid and the following.(i)
Ifϕ and ψ are distinct elements of Cr+ with ϕ≤ψ and [t0,t0+σ) with n|r|<σ≤∞ is the intersection of the maximal intervals of existence of x(t,t0,ϕ) and x(t,t0,ψ), then
(2.6)0≤x(t,t0,ϕ)≤x(t,t0,ψ)fort0≤t<t0+σ,0≤x(t,t0,ϕ)<x(t,t0,ψ)fort0+n|r|≤t<t0+σ.(ii)
Ifϕ∈Cr+,ϕ≠0,t0∈ℝ and x(t,t0,ϕ) is defined on [t0,t0+σ) with σ>n|r|, then
(2.7)0<x(t,t0,ϕ)fort0+n|r|≤t<t0+σ.Lemma2.1 shows that if (h1)–(h4) hold, then the positivity of solutions of (2.1) follows.The following definitions and results are useful in proving our lemma.Definition 2.2.
System (2.1) is cooperative if ∂fi/∂xj≥0 whenever i≠j.Definition 2.3.
A square matrixA is said to be a reducible matrix if and only if for some permutation matrix P the matrix PTAP is block upper triangular. If a square matrix is not reducible, it is said to be an irreducible matrix. System (2.1) is called irreducible if the Jacobian matrix ((∂fi/∂xj)) is irreducible.Lemma 2.4 (Smith [11]).
If (2.1) is cooperative and irreducible in D, where D is an open subset of C, and the solution with positive initial data is bounded, then the trajectory of (2.1) tends to some single equilibrium.We now consider the following delay differential system:(2.8)ẏ1(t)=-ay1(t)+by1(t-τ)-cy12(t)+θy2(t),ẏ2(t)=dy1(t)-(θ+e)y2(t),
with initial conditions(2.9)yi(s)=ψi(s)≥0,s∈[-τ,0),ψi∈C([-τ,0),ℝ+),ψi(0)>0,(i=1,2).System (2.8) always has a trivial equilibrium E0(0,0). If a(θ+e)<dθ+b(θ+e), then system (2.8) has a unique positive equilibrium E+(y1*,y2*), where(2.10)y1*=(-a+b)(θ+e)+θdc(θ+e),y2*=d(-a+b)(θ+e)+θd2c(θ+e)2.The characteristic equation of system (1.6) at the equilibrium E0 is of the form(2.11)λ2+(a+e+θ)λ+a(θ+e)-dθ-(bλ+b(θ+e))e-λτ=0.
Let(2.12)f1(λ)=λ2+(a+e+θ)λ+a(θ+e)-dθ-(bλ+b(θ+e))e-λτ.
If a(θ+e)<dθ+b(θ+e), then it is easy to see that, for λ real,(2.13)f1(0)=a(θ+e)-dθ-b(θ+e)<0,limλ→+∞f1(λ)=+∞.
Hence, f1(λ)=0 has a positive real root. Therefore, the equilibrium E0 is unstable. If a(θ+e)>dθ+b(θ+e), when τ=0, it is easy to see that the equilibrium E0 is stable. Therefore, if a(θ+e)-dθ>b(θ+e), by Kuang and So [13, Lemma  B], we see that the equilibrium E0 is locally stable for all τ>0.Ifa(θ+e)<dθ+b(θ+e), the characteristic equation of the positive equilibrium E+ takes the form(2.14)λ2+(a+e+θ+2cy1*)λ+(a+2cy1*)(θ+e)-dθ-(bλ+b(θ+e))e-λτ=0.
When τ=0, it is easy to see that the equilibrium E+ is stable. Therefore, if a(θ+e)<dθ+b(θ+e), by Kuang and So [13, Lemma  B], we see that the equilibrium E+ is locally stable for all τ>0.Lemma 2.5.
For system (2.8), one has the following.  (i)
Ifa(θ+e)<dθ+b(θ+e), then the positive equilibrium E+ of system (2.8) is globally stable.(ii)
Ifa(θ+e)>dθ+b(θ+e), the equilibrium E0 of system (2.8) is globally stable.Proof.
We represent the right-hand side of (2.8) by f(t,xt)=(f1(t,xt),f2(t,xt)) and set L(t,·)=Dϕf(t,ϕ). By direct calculation we have
(2.15)L1(t,h)=-ah1(0)+bh1(-τ)-2cϕ1(0)h1(0)+θh2(0),L2(t,h)=dh1(0)-(θ+e)h2(0).
We now claim that the hypotheses (h1)–(h4) hold for system (2.8). It is easily seen that (h1) and (h4) hold for system (2.8). We need only to verify that (h2) and (h3) hold.
The matrixA(t) takes the form
(2.16)(-a+b-2cϕ1(0)θd-(θ+e)).
Clearly, the matrix A(t) is irreducible for each t∈ℝ.
From the definition ofA(t) and ηij, it is readily seen that η12(θ,t)=η12(0,t)=θ,η21(θ,t)=η21(0,t)=d, for θ≥0; and ηij(θ,t)=0,i≠j for θ≤-τ; and ηij(·,t)∈BV[-τ,0], where ηij is a positive Borel measure on [-τ,0]. Therefore, ηij(·,t)>0. Thus, for each j, there is i≠j such that ηij(-rj+ε,t)=ηij(-τ+ε,t)>0 for all t∈ℝ and ε>0 sufficiently small, i=1,2. Hence, (h3) holds.
Thus, the conditions of Lemma2.1 are satisfied. Therefore, the positivity of solutions to system (2.8) follows. It is easy to see that system (2.8) is cooperative. By Lemma 2.4 we see that any solution starting from D=Cτ+ converges to some single equilibrium. However, system (2.8) has only two equilibria: E0 and E+. Note that if a(θ+e)<dθ+b(θ+e), the equilibrium E+ is locally stable. Hence, any solution starting from D converges to E+. Using a similar argument one can show the global stability of the equilibrium E0 when a(θ+e)>dθ+b(θ+e). This completes the proof.By a similar argument one can show that all solutions of system (1.6) with initial conditions (1.7) are defined on [0,+∞) and remain positive for all t≥0.


## 3. Local Stability

In this section, we discuss the local stability of each equilibria and the existence of Hopf bifurcation of system (1.6).It is easy to show that system (1.6) always has two equilibria E0(0,0,0) and E1(r/a11,0,0). If the following holds:(H1)
θb>r1(θ+r2),then system (1.6) has another boundary equilibrium E2(0,ỹ1,ỹ2), where
(3.1)ỹ1=1a22(θbθ+r2-r1),ỹ2=ba22(θ+r2)(θbθ+r2-r1).
Further, if the following holds:(H2)
r(θ+r2)(a12a21+a11a22)>a12[θba11+(a21r-a11r1)(θ+r2)]>0,then system (1.6) has a unique positive equilibrium E*=(x*,y1*,y2*), where(3.2)x*=ra11-a12[θba11+(θ+r2)(a21r-a11r1)](θ+r2)a11(a12a21+a11a22),y1*=θba11+(θ+r2)(a21r-a11r1)(θ+r2)(a12a21+a11a22),y2*=b[θba11+(θ+r2)(a21r-a11r1)](θ+r2)2(a12a21+a11a22).We now study the local stability of each of the nonnegative equilibrium of system (1.6).LetÊ=(x̂,ŷ1,ŷ2) be any arbitrary equilibrium. Then the characteristic equation of system (1.6) at the equilibrium Ê is given by(3.3)|r-2a11x̂-a12ŷ1-λ-a12x̂0a21ŷ1e-λτ-r1-2a22ŷ1+a21x̂e-λτ-λθ0b-θ-r2-λ|=0.
The characteristic equation of system (1.6) at the equilibrium E0(0,0,0) reduces(3.4)(λ-r)[(r1+λ)(θ+r2+λ)-bθ]=0.
Clearly, λ=r is a positive real root. Hence, E0(0,0,0) is always unstable.The characteristic equation of system (1.6) at the equilibrium E1(r/a11,0,0) reduces(3.5)(λ+r)[λ2+(r1+r2+θ)λ+r1(θ+r2)-bθ-a21ra11(λ+θ+r2)e-λτ]=0.
Clearly, λ=-r is a negative real root of (3.5). All other roots are give by the roots of equation(3.6)λ2+(r1+r2+θ)λ+r1(θ+r2)-bθ-a21ra11(λ+θ+r2)e-λτ=0.
Let(3.7)f(λ)=λ2+(r1+r2+θ)λ+r1(θ+r2)-bθ-a21ra11(λ+θ+r2)e-λτ.
If (θ+r2)(r1-a21r/a11)<bθ, then it is easy to see that for λ real,(3.8)f(0)=(θ+r2)(r1-a21ra11)-bθ<0,limλ→+∞f(λ)=+∞.
Hence, f(λ)=0 has a positive real root. Therefore, the equilibrium E1(r/a11,0,0) is unstable. If (θ+r2)(r1-a21r/a11)>bθ, when τ=0, it is easy to see that the equilibrium E1(r/a11,0,0) is stable. Therefore, if (θ+r2)(r1-a21r/a11)>bθ, by Kuang and So [13, Lemma  B], we see that the equilibrium E1(r/a11,0,0) is locally stable for all τ>0.The characteristic equation of system (1.6) at the equilibria E2(0,ỹ1,ỹ2) reduces to(3.9)(λ-r+a12ỹ1)[λ2+(r1+2a22ỹ1+θ+r2)λ+(r1+2a22ỹ1)(θ+r2)-bθ]=0.
Clearly, if r>a12ỹ1, λ=r-a12ỹ1 is a positive real root. Hence, if r>a12ỹ1, then E2(0,ỹ1,ỹ2) is unstable. Noting that(3.10)(r1+2a22ỹ1)(θ+r2)-bθ=bθ-r1(θ+r2)>0,
if r<a12ỹ1, then (3.9) only has negative real root, and E2(0,ỹ1,ỹ2) is stable.The characteristic equation of system (1.6) at the positive equilibria E*(x*,y1*,y2*) is(3.11)λ3+p2λ2+p1λ+p0+(q2λ2+q1λ+q0)e-λτ=0,
where(3.12)p0=a11x*(θ+r2)(a21x*+a22y1*),p1=a11x*(r1+2a22y1*)+(θ+r2)(a21x*+a22y1*+a11x*),p2=a11x*+2a22y1*+r1+r2+θ,q0=a21(θ+r2)x*(a12y1*-a11x*),q1=a21x*(a12y1*-a11x*-θ-r2),q2=-a21x*.
It is easy to show that(3.13)p0+q0=(a11a22+a21a12)(θ+r2)x*y1*>0,p1+q1=(a11a22+a12a21)x*y1*+(θ+r2)(a22y1*+a11x*)+bθa11x*(θ+r2)>0,p2+q2=a11x*+a22y1*+r2+θ+bθ(θ+r2)>0.
It is easy to see that (p2+q2)(p1+q1)>p0+q0. Hence, by the Routh-Hurwitz theorem, when τ=0, the positive equilibrium E* of system (1.6) is locally asymptotically stable.Ifiω(ω>0) is a root of (3.11), separating the real and imaginary parts, we obtain(3.14)p0-p2ω2=q2ω2cosωτ-q1ωsinωτ-q0cosωτ,-ω3+p1ω=-q2ω2sinωτ-q1ωcosωτ+q0sinωτ.
Squaring and adding the two equations of (3.14), it follows that(3.15)ω6+Q1ω4+Q2ω2+Q3=0,
where(3.16)Q1=p22-2p1-q22,Q2=p12+2q2q0-2p2p0-q12,Q3=p02-q02.
It is easy to show that(3.17)Q1=p22-q22-2p1=a112(x*)2+(r2+θ)2+(r1+2a22y1*+a21x*)(a22y1*+bθθ+r2)+2bθ>0,Q2=p12+2q2q0-2p2p0-q12=[(a11a22+a21a12)x*y1*+a11bθx*(r2+θ)][2a11a21(x*)2-(a12a21-a11a22)x*y1*+a11bθx*(r2+θ)]+(θ+r2)2[a222(y1*)2+a112(x*)2]+2a21a22(θ+r2)2x*+2bθa112(x*)2.Noting that ifp0>q0, 2a11a21x*>(a12a21-a11a22)y1*, hence Q2>0,Q3>0. The positive equilibrium E* of system (1.6) is locally asymptotically stable for all τ>0. If p0<q0, we know that Q3<0, (3.15) has a unique positive root ω0. Define(3.18)τj=1ω0arccos(q1-q2p2)ω04+(p0q2+p2q0-p1q1)ω02-q0p0q22ω04+(q12-2q0q2)ω02+q02+2jπω0,j=0,1,….
Then (τj,ω0) solves (3.11). This means that when τ=τj, (3.11) has a pair of purely imaginary roots ±iω0. Noting that the positive equilibrium E* is locally stable for τ=0, by the general theory on characteristic equations of delay differential equations from [14, Theorem  4.1], E* remains stable for τ<τ0.Letp0<q0 and τ0 be defined in (3.18). Denoting(3.19)λ(τ)=α(τ)+iω(τ),
the root of (3.11) is such that(3.20)α(τ0)=0,ω(τ0)=ω0.
In the following we claim that(3.21)d(Reλ)dτ|τ=τ0>0.
This will signify that there exists at least one eigenvalue with positive real part for τ>τ0. Moreover, the conditions for the existence of a Hopf bifurcation [10] are then satisfied yielding a periodic solution. To this end, differentiating equation (3.11) with respect τ, we obtain that(3.22)(dλ(τ)dτ)-1=-3λ2+2p2λ+p1λ(λ3+p2λ2+p1λ+p0)+2q2λ+q1λ(q2λ2+q1λ+q0)-τλ,
which leads to(3.23)sign{Re(dλdτ)|τ=τj}=sign{Re(dλdτ)|τ=τj-1}=sign{3ω04+(2p22-4p1-2q22)ω02+p12+2q2q0-2p2p0-q12}=sign{3ω04+2Q1ω02+Q2}.
If Q2>0, 3ω04+2Q1ω02+Q2>0. Therefore, system (1.6) undergoes a Hopf bifurcation.We therefore obtain the following results.Theorem 3.1.
For system (1.6), let τ0 be defined as in (3.18), one has the following.(i)
The positive equilibriumE0 of system (1.6) is always unstable.(ii)
If(θ+r2)(r1-a21r/a11)>bθ, the equilibrium E1(r/a11,0,0) of system (1.6) is locally stable for all τ>0; and if (θ+r2)(r1-a21r/a11)<bθ, E1 is unstable for all τ.(iii)
Let(H1) hold. If r>a12ỹ1, the equilibrium E2(0,ỹ1,ỹ2) of system (1.6) is unstable; if  r<a12ỹ1, E2 is stable for all τ≥0.(iv)
Let (H2) hold. Ifp0>q0, then the positive equilibrium E*(x*,y1*,y2*) of system (1.6) is locally asymptotically stable for all τ>0. If p0<q0, then E* is locally stable for τ<τ0; and E* is unstable for τ>τ0; if Q2>0 system (1.6) undergoes a Hopf Bifurcation at the positive equilibrium E* when τ=τ0.


## 4. Global Stability

In this section, we are concerned with the global stability of the equilibriaE1,E2,E* of system (1.6). The strategy of proofs is to use an iteration technique and comparison arguments, respectively.Theorem 4.1.
Let (H2) hold. Then the positive equilibriumE* of system (1.6) is globally asymptotically stable provided that  (H3)
a11a22>a12a21.Proof.
Let(x(t),y1(t),y2(t)) be any positive solution of system (1.6) with initial conditions (1.7). Let
(4.1)U=lim supt→+∞x(t),V=lim inft→+∞x(t),Ui=lim supt→+∞yi(t),Vi=lim inft→+∞yi(t)(i=1,2).
We now claim that U=V=x*,U1=V1=y1*,U2=V2=y2*. The strategy of the proof is to use an iteration technique.
We derive from the first equation of system (1.6) that
(4.2)ẋ(t)≤x(t)(r-a11x(t)).
A standard comparison argument shows that
(4.3)U=lim supt→+∞x(t)≤ra11:=M1x.
Hence, for ε>0 sufficiently small there exists a T1>0 such that if t>T1, x(t)≤M1x+ε.
We derive from the second and the third equations of system (1.6) that for t>T1+τ,
(4.4)ẏ1(t)≤-r1y1(t)+a21(M1x+ε)y1(t-τ)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Consider the following auxiliary equations:
(4.5)u̇1(t)=-r1u1(t)+a21(M1x+ε)u1(t-τ)-a22u12(t)+θu2(t),u̇2(t)=bu1(t)-(θ+r2)u2(t).
Since (H2) holds, by Lemma 2.5 it follows from (4.5) that
(4.6)limt→+∞u1(t)=(-r1+a21(M1x+ε))(θ+r2)+bθa22(θ+r2),limt→+∞u2(t)=b(-r1+a21(M1x+ε))(θ+r2)+b2θa22(θ+r2)2.
By comparison, we obtain that
(4.7)U1=lim supt→+∞y1(t)≤(-r1+a21(M1x+ε))(θ+r2)+bθa22(θ+r2),U2=lim supt→+∞y2(t)≤b(-r1+a21(M1x+ε))(θ+r2)+b2θa22(θ+r2)2.
Since these inequalities are true for arbitrary ε>0 sufficiently small, it follows that U1≤M1y1,U2≤M1y2, where
(4.8)M1y1=(-r1+a21M1x)(θ+r2)+bθa22(θ+r2),M1y2=b(-r1+a21M1x)(θ+r2)+b2θa22(θ+r2)2.
Hence, for ε>0 sufficiently small, there is a T2≥T1+τ such that if t>T2,  y1(t)≤M1y1+ε, y2(t)≤M1y2+ε.
Forε>0 sufficiently small, we derive from the first equation of system (1.6) that, for t>T2,
(4.9)ẋ(t)≥x(t)(r-a11x(t)-a12(M1y1+ε)).
By comparison it follows that
(4.10)V=lim inft→+∞x(t)≥r-a12(M1y1+ε)a11.
Since this is true for arbitrary ε>0 sufficiently small, we conclude that V≥N1x, where
(4.11)N1x=r-a12M1y1a11.
Therefore, for ε>0 sufficiently small, there is a T3≥T2 such that if t>T3, x(t)≥N1x-ε.
Forε>0 sufficiently small, we derive from the second and the third equations of system (1.6) that, for t>T3+τ,
(4.12)ẏ1(t)≥-r1y1(t)+a21(N1x-ε)y1(t-τ)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Consider the following auxiliary equations:
(4.13)u̇1(t)=-r1u1(t)+a21(N1x-ε)u1(t-τ)-a22u12(t)+θu2(t),u̇2(t)=bu1(t)-(θ+r2)u2(t).
Since (H2) and (H3) hold, by Lemma 2.5 it follows from (4.13) that
(4.14)limt→+∞u1(t)=(-r1+a21(N1x-ε))(θ+r2)+bθa22(θ+r2),limt→+∞u2(t)=b(-r1+a21(N1x-ε))(θ+r2)+b2θa22(θ+r2)2.
By comparison, we obtain that
(4.15)V1=lim inft→+∞y1(t)≥(-r1+a21(N1x-ε))(θ+r2)+bθa22(θ+r2),V2=lim inft→+∞y2(t)≥b(-r1+a21(N1x-ε))(θ+r2)+b2θa22(θ+r2)2.
Since these two inequalities hold for arbitrary ε>0 sufficiently small, we conclude that V1≥N1y1,V2≥N1y2, where
(4.16)N1y1=(-r1+a21N1x)(θ+r2)+bθa22(θ+r2),N1y2=b(-r1+a21N1x)(θ+r2)+b2θa22(θ+r2)2.
Therefore, for ε>0 sufficiently small, there exists a T4≥T3+τ such that if t>T4, y1(t)≥N1y1-ε, y2(t)≥N1y2-ε.
Forε>0 sufficiently small, it follows from the first equation of system (1.6) that, for t>T4,
(4.17)ẋ(t)≤x(t)(r-a11x(t)-a12(N1y1-ε)).
A comparison argument yields
(4.18)U=lim supt→+∞x(t)≤r-a12(N1y1-ε)a11.
Since this is true for arbitrary ε>0, we conclude that U≤M2x, where
(4.19)M2x=r-a12N1y1a11.
Hence, for ε>0 sufficiently small, there exists a T5≥T4 such that if t>T5,  x(t)≤M2x+ε.
Again, we derive from the second and the third equations of system (1.6) that for t>T5+τ,
(4.20)ẏ1(t)≤-r1y1(t)+a21(M2x+ε)y1(t-τ)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Since (H2) and (H3) hold, by Lemma 2.5, a comparison argument shows that
(4.21)U1=lim supt→+∞y1(t)≤(-r1+a21(M2x+ε))(θ+r2)+bθa22(θ+r2),U2=lim supt→+∞y2(t)≤b(-r1+a21(M2x+ε))(θ+r2)+b2θa22(θ+r2)2.
Since these inequalities are true for arbitrary ε>0 sufficiently small, we conclude that U1≤M2y1,U2≤M2y2, where
(4.22)M2y1=(-r1+a21M2x)(θ+r2)+bθa22(θ+r2),M2y2=b(-r1+a21M2x)(θ+r2)+b2θa22(θ+r2)2.
Hence, for ε>0 sufficiently small, there is a T6≥T5+τ such that if t>T6, y1(t)≤M2y1+ε, y2(t)≤M2y2+ε.
Forε>0 sufficiently small, it follows from the first equation of system (1.6) that, for t>T6,
(4.23)ẋ(t)≥x(t)(r-a11x(t)-a12(M2y1+ε)).
By comparison we obtain that
(4.24)V=lim inft→+∞x(t)≥r-a12(M2y1+ε)a11.
Since this is true for arbitrary ε>0 sufficiently small, we conclude that V≥N2x, where
(4.25)N2x=r-a12M2y1a11.
Therefore, for ε>0 sufficiently small, there is a T7≥T6 such that if t>T7,  x(t)≥N2x-ε.
Forε>0 sufficiently small, we derive from the second and the third equations of system (1.6) that for t>T7+τ,
(4.26)ẏ1(t)≥-r1y1(t)+a21(N2x-ε)y1(t-τ)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Since (H2) and (H3) hold, by Lemma 2.5 and by comparison, it follows from (4.26) that
(4.27)V1=lim inft→+∞y1(t)≥(-r1+a21(N2x-ε))(θ+r2)+bθa22(θ+r2),V2=lim inft→+∞y2(t)≥b(-r1+a21(N2x-ε))(θ+r2)+b2θa22(θ+r2)2.
Since these two inequalities hold for arbitrary ε>0 sufficiently small, we conclude that V1≥N2y1,V2≥N2y2, where
(4.28)N2y1=(-r1+a21N2x)(θ+r2)+bθa22(θ+r2),N2y2=b(-r1+a21N2x)(θ+r2)+b2θa22(θ+r2)2.
Continuing this process, we derive six sequences Mnx,Mny1,Mny2,Nnx,Nny1,Nny2(n=1,2,…) such that, for n≥2,
(4.29)Mnx=r-a12Nn-1y1a11,Mny1=(-r1+a21Mnx)(θ+r2)+bθa22(θ+r2),Mny2=b(-r1+a21Mnx)(θ+r2)+b2θa22(θ+r2)2,Nnx=r-a12Mny1a11,Nny1=(-r1+a21Nnx)(θ+r2)+bθa22(θ+r2),Nny2=b(-r1+a21Nnx)(θ+r2)+b2θa22(θ+r2)2.
It is readily seen that
(4.30)Nnx≤V≤U≤Mnx,Nnyi≤Vi≤Ui≤Mnyi(i=1,2).
We derive (4.29) that
(4.31)Mn+1y1=a11a22-a12a21a112a222(a11bθθ+r2+a21r-a11r1)+a122a212a112a222Mny1.
Noting that Mny1≥y1* and a11a22>a12a21, it follows from (4.31) that
(4.32)Mn+1y1-Mny1=a11a22-a12a21a112a222(a11bθθ+r2+a21r-a11r1)+(a122a212a112a222-1)Mny1≤a11a22-a12a21a112a222(a11bθθ+r2+a21r-a11r1)+(a122a212a112a222-1)y1*=0.
Thus, the sequence Mny1 is nonincreasing. Hence, limn→+∞Mny1 exists. Taking n→+∞, it follows from (4.31) that
(4.33)limn→+∞Mny1=(θba11/(θ+r2))+a21r-a11r1a12a21+a11a22=y1*.
We therefore obtain from (4.30) and (4.33) that
(4.34)limn→+∞Mny2=y2*,limn→+∞Nny1=y1*,limn→+∞Nny2=y2*,limn→+∞Mnx=x*,limn→+∞Nnx=x*.
It follows from (4.30), (4.33) and (4.34) that
(4.35)U=V=x*,U1=V1=y1*,U2=V2=y2*.
We therefore have
(4.36)limn→+∞x(t)=x*,limn→+∞y1(t)=y1*,limn→+∞y2(t)=y2*.
Noting that if a11a22>a12a21, then p0-q0>2(θ+r2)a11a21(x*)2>0. By Theorem 3.1, the positive equilibrium E* is locally stable. We therefore conclude that E* is globally stable. The proof is complete.Theorem 4.2.
Ifr1(θ+r2)>bθ+a21r/a11(θ+r2), the equilibrium E1 of system (1.6) is globally asymptotically stable.Proof.
Let(x(t),y1(t),y2(t)) be any positive solution of system (1.6) with initial conditions (1.7). It follows from the first equation of system (1.6) that
(4.37)ẋ(t)≤x(t)(r-a11x(t)).
A standard comparison argument shows that
(4.38)lim supt→+∞x(t)≤ra11.
Choose ε>0 sufficiently small such that
(4.39)r1(θ+r2)>bθ+a21(ra11+ε)(θ+r2).
Hence, for ε>0 sufficiently small satisfying (4.39), there is a T1>0 such that if t>T1, then x(t)≤r/a11+ε. We derive from the second and the third equations of system (1.6) that for t>T1+τ,
(4.40)ẏ1(t)≤-r1y1(t)+a21(ra11+ε)y1(t-τ)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Consider the following auxiliary equations:
(4.41)u̇1(t)=-r1u1(t)+a21(ra11+ε)u1(t-τ)-a22u12(t)+θu2(t),u̇2(t)=bu1(t)-(θ+r2)u2(t).
Since (4.39) holds, by Lemma 2.5 it follows from (4.41) that
(4.42)limt→+∞u1(t)=0,limt→+∞u2(t)=0.
By comparison, we obtain that
(4.43)limt→+∞y1(t)=0,limt→+∞y2(t)=0.
Hence, for ε>0 sufficiently small satisfying (4.39), there is a T2>T1 such that if t>T2,y1(t)<a11ε.
Forε>0 sufficiently small satisfying (4.39) and a11ε<r, we derive from the first equation of system (1.6) that for t>T2
(4.44)ẋ(t)≥x(t)(r-a11x(t)-a11ε).
By comparison, we obtain that
(4.45)lim inft→+∞x(t)≥ra11-ε,
which, together with (4.38), yields
(4.46)limt→+∞x(t)=ra11.
By Theorem3.1, if r1(θ+r2)>bθ+a21r/a11(θ+r2), the boundary equilibrium E1 is locally stable. We therefore conclude that E1 is globally stable in this case. This completes the proof.Theorem 4.3.
Let(H1) hold. If (θ+r2)(a22r+a12r1)<a12θb, then the equilibrium E2 of system (1.6) is globally asymptotically stable.Proof.
Let(x(t),y1(t),y2(t)) be any positive solution of system (1.6) with initial conditions (1.7). We derive from the second and the third equations of system (1.6) that, for t>T1+τ,
(4.47)ẏ1(t)≥-r1y1(t)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Consider the following auxiliary equations:
(4.48)u̇1(t)=-r1u1(t)-a22u12(t)+θu2(t),u̇2(t)=bu1(t)-(θ+r2)u2(t).
Since (H1) holds, by Lemma 2.5 it follows from (4.48) that
(4.49)limt→+∞u1(t)=1a22(θbθ+r2-r1)=ỹ1,limt→+∞u2(t)=ba22(θ+r2)(θbθ+r2-r1)=ỹ2.
By comparison we derive that
(4.50)limt→+∞y1(t)≥1a22(θbθ+r2-r1)=ỹ1,limt→+∞y2(t)≥ba22(θ+r2)(θbθ+r2-r1)=ỹ2.
Hence, for ε>0 sufficiently small, there is a T1 such that if t>T1, y1(t)≥ỹ1-ε, y2(t)≥ỹ2-ε.
Forε>0 sufficiently small, it follows from the first equation of system (1.6) that for t>T1,
(4.51)ẋ(t)≤x(t)(r-a11x(t)-a12(ỹ1-ε)).
Since (θ+r2)(a22r+a12r1)<a12θb and ε>0 is sufficiently small, by comparison we derive that
(4.52)lim supt→+∞x(t)≤0.
We therefore have limt→+∞x(t)=0. Hence, for ε>0 sufficiently small, there is a T2>T1+τ such that if t>T2, x(t)≤ε.
Forε>0 sufficiently small, it follows from the first equation of system (1.6) that for t>T2+τ
(4.53)ẏ1(t)≤-r1y1(t)+a21εy1(t-τ)-a22y12(t)+θy2(t),ẏ2(t)=by1(t)-(θ+r2)y2(t).
Since (H1) holds, by Lemma 2.5, a comparison argument shows that
(4.54)limt→+∞y1(t)≤(-r1+a21ε)(θ+r2)+bθa22(θ+r2),limt→+∞y2(t)≤b(-r1+a21ε)(θ+r2)+b2θa22(θ+r2)2,
Together with (4.49), yields
(4.55)limt→+∞y1(t)=1a22(θbθ+r2-r1)=ỹ1,limt→+∞y2(t)=ba22(θ+r2)(θbθ+r2-r1)=ỹ2.
Noting that if (H1) holds and (θ+r2)(a22r+a12r1)<a12θb, the equilibrium E2 of system (1.6) is locally stable. We therefore conclude that E2 is globally stable. The proof is complete.


## 5. Numerical Examples

In this section, we give some examples to illustrate the main results.Example 5.1.
In system (1.6), we let r=2,r1=2,r2=1,a11=4,a12=3,a21=2,a22=4,b=1,θ=2,andτ=1. It is easy to show that r1(θ+r2)-bθ-a21r(θ+r2)/a11=1>0. By Theorem 4.2 we see that the equilibrium E1(0.5,0,0) of system (1.6) is globally stable. Numerical simulation illustrates our result (see Figure 1).Figure 1
The temporal solution found by numerical integration of system (1.6) with r=2,r1=2,r2=1,a11=4,a12=3,a21=2,a22=4,b=1,θ=2,τ=1 and initial value is (0.6,0.2,0.2).Example 5.2.
In system (1.6), we let r=1,r1=1,r2=1,a11=4,a12=3,a21=2,a22=1,b=4,θ=1,andτ=1. It is easy to show that bθ-r1(θ+r2)=2>0,andr-a12ỹ1=-2<0. By Theorem 4.3 we see that the equilibrium E2(0,1,2) of system (1.6) is globally stable, as depicted in Figure 2.Figure 2
The temporal solution found by numerical integration of system (1.6) with r=1,r1=1,r2=1,a11=4,a12=3,a21=2,a22=1,b=4,θ=1,τ=1 and initial value is (0.6,0.2,0.2).Example 5.3.
In system (1.6), we let r=4,r1=1,r2=1,a11=2,a12=1,a21=2,a22=1.5,b=4,θ=1,andτ=1. System (1.6) with above coefficients has a unique positive equilibrium E*(1.5,2,4). It is easy to show that a11a22-a12a21=1>0. By Theorem 4.1 we see that the positive equilibrium E*(1.5,2,4) of system (1.6) is globally stable. Numerical simulation illustrates our result (see Figure 3).Figure 3
The temporal solution found by numerical integration of system (1.6) with r=4,r1=1,r2=1,a11=2,a12=1,a21=2,a22=1.5,b=4,θ=1,τ=1 and initial value is (0.8,1.5,3.6).Example 5.4.
In system (1.6), we let r=4,r1=1,r2=1,a11=1,a12=2,a21=4,a22=2,b=4,andθ=1. System (1.6) with above coefficients has a unique positive equilibrium E*(0.6,1.7,3.4). By computing, we can get p0-q0=-6.48<0 and ω0=0.9326,τ0=1.664. By Theorem 3.1 we see that the equilibrium E* is locally stable for τ<τ0; and E* is unstable for τ>τ0. And system (1.6) undergoes a Hopf Bifurcation at the positive equilibrium E* when τ=τ0. See Figures 4 and 5.Figure 4
Whenτ=1<τ0, the positive equilibrium E* is asymptotically stable. Here r=4,r1=1,r2=1,a11=1,a12=2,a21=4,a22=2,b=4,θ=1 and initial value is (0.8,1.5,3.6).Figure 5
Whenτ=2>τ0, the positive equilibrium E* is asymptotically stable. Here r=4,r1=1,r2=1,a11=1,a12=2,a21=4,a22=2,b=4,θ=1 and the initial value is (0.8,1.5,3.6).


## 6. Discussion

In this paper, we considered a delayed predator-prey model with stage structure for the predator. By using the iteration technique and comparison argument, respectively, sufficient conditions were established for the global stability of the positive equilibrium and two boundary equilibria of system (1.6). By Theorems 4.1, 4.2 and 4.3, we see that: (i) If (H2) holds and a11a22>a12a21, system (1.6) is permanent. (ii) If r1(θ+r2)>bθ+a21r/a11(θ+r2), the prey species is persistent but the predator becomes extinct. (iii) If (H1) holds and (θ+r2)(a22r+a12r1)<a12θb, the predator species is persistent but the prey species becomes extinct.



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*Source: 285934-2009-10-07.xml*