# Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response

Authors: Lili Wang; Rui Xu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

A Holling type II predator-prey model with time delay and stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. The existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and the sufficient conditions are obtained for the global stability of the coexistence equilibrium.


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## Body

## 1. Introduction

In population dynamics, the functional response of predator to prey density refers to the change in the density of prey attacked per unit time per predator as the prey density changes [1]. Based on experiments, Holling [2] suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. The most popular functional response used in the modelling of predator-prey systems is Holling type II with φ(x)=fx/(1+mx) which takes into account the time a predator uses in handing the prey being captured. There has been a large body of work about predator-prey systems with Holling type II functional responses, and many good results have been obtained (see, e.g., [1, 3, 4]).Time delays of one type or another have been incorporated into biological models by many researchers. We refer to the monographs of Gopalsamy [5], Kuang [6], and Wangersky and Cunningham [7] on delayed predator-prey systems. In these research works, it is shown that a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. Hence, delay differential equations exhibit more complex dynamics than ordinary differential equations. Time delay due to gestation is a common example, since generally the consumption of prey by the predator throughout its past history decides the present birth rate of the predator. In [7], Wangersky and Cunningham proposed and studied the following non-Kolmogorov-type predator-prey model:
(1)x˙(t)=x(t)(r1-ax(t)-a1y(t)),y˙(t)=a2x(t-τ)y(t-τ)-r2y(t).
In this model, it is assumed that a duration of τ time units elapses when an individual prey is killed and the moment when the corresponding addition is made to the predator population.In natural world, there are many species whose individuals have a history that can be  divided into two stages immature and mature. Usually the dynamics-eating habits of predator are often quite different in different stages. Generally speaking, the immature predators are raised by their parents and do not have the ability to attack prey, so the rate at which they attack prey and the reproductive rate can be ignored. Hence, it is of ecological importance to investigate predator-prey models with stage structure. In recent years, the predator-prey population models with stage structure have received much attraction (see, e.g., [8–10]). In [10], it was assumed that feeding on prey can only make contribution to the increasing of the physique of the predator and does not make contribution to the reproductive ability, and the following strengthen type predator-prey model with stage structure was studied:
(2)x˙(t)=x(t)(r-ax(t)-a1y2(t)1+mx(t)),y˙1(t)=ey2(t)-(r1+D)y1(t),y˙2(t)=Dy1(t)-r2y2(t)+a2x(t)y2(t)1+mx(t),
where x(t) represents the density of the prey at time t and y1(t) and y2(t) represent the densities of the immature and the mature predator at time t, respectively. a1x/(1+mx) describes the Holling type II functional response; a1 and m represent the effects of capturing rate and handling time, respectively. a denotes the intraspecific competition rate of the prey, e denotes the birth rate of immature predator, r denotes the intrinsic growth rate of the prey, a2/a1 denotes the rate of conversing prey into new mature predator, r1 denotes the death rate of the immature predator, r2 denotes the death rate of the mature predator, and D denotes the rate of immature predator becoming mature predator. All parameters are positive constants. In [10], sufficient conditions were derived in for the global asymptotic stability of nonnegative equilibria of the model by constructing suitable Lyapunov functions.Motivated by the work of Wangersky and Cunningham [7] and Tian and Xu [10], we are concerned with the combined effects of the stage structure for the predator and time delay due to the gestation of mature predator on the global dynamics of a predator-prey model with Holling type II functional response. To this end, we consider the following delay differential system:
(3)x˙(t)=x(t)(r-ax(t)-a1y2(t)1+mx(t)),y˙1(t)=ey2(t)-(r1+D)y1(t),y˙2(t)=Dy1(t)-r2y2(t)+a2x(t-τ)y2(t-τ)1+mx(t-τ),
where the meanings of the variables x, y1, y2, and the parameters a, r, e, a1, a2, r1, r2, D, are the same as those of system (2) and the constant τ≥0 denotes the time delay due to the gestation of the mature predator. This is based on the assumption that the change rate of predators depends on the number of prey and of mature predators present at some previous time.The initial conditions for system (3) take the form
(4)x(θ)=ϕ(θ),y1(θ)=ψ1(θ),y2(θ)=ψ2(θ),000θ∈[-τ,0),ϕ(0)>0,ψ1(0)>0,ψ2(0)>0,
where
(5)00(ϕ(θ),ψ1(θ),ψ2(θ))∈C([-τ,0],R+03),R+03={(x1,x2,x3):x1≥0,x2≥0,x3≥0}.By the fundamental theory of functional differential equations [11], it is well known that system (3) has a unique solution (x(t),y1(t),y2(t)) satisfying initial conditions (4). Further, it is easy to show that all solutions of system (3) are defined on [0,+∞) and remain positive for all t≥0.The paper is organized as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (3) is discussed and the existence of Hopf bifurcations at the coexistence equilibrium is established. In Section 3, permanence of the system (3) is proved by means of the persistence theory on infinite dimensional systems. In Section 4, by using Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are received for the global asymptotic stability of the predator-extinction equilibrium and the coexistence equilibrium.


## 2. Local Stability and Hopf Bifurcation

In this section, we discuss the local stability of each of feasible equilibria of system (3) and the existence of Hopf bifurcations at the coexistence equilibrium.It is easy to show that system (3) always has a trivial equilibrium E0(0,0,0) and a predator-extinction equilibrium E1(r/a,0,0). Furthermore, if the following holds.(H1)a2r(r1+D)>(a+mr)[r2(r1+D)-eD]>0, then system (3) has a coexistence equilibrium E*=(x*,y1*,y2*), where
(6)x*=r2(r1+D)-eD(a2-mr2)(r1+D)+meD,y1*=er1+Dy2*,y2*=a2(r1+D)(r-ax*)x*a1[r2(r1+D)-eD].We now study the local stability of the trivial equilibriumE0(0,0,0) and the predator-extinction equilibrium E1(r/a,0,0).The characteristic equation of system (3) at E0(0,0,0) is
(7)(λ-r)[λ2+(r1+r2+D)λ+r2(r1+D)-eD]=0.
Hence, E0(0,0,0) is always unstable since (7) has a positive root λ=r.The characteristic equation of system (3) at E1(r/a,0,0) is
(8)(λ+r)[P(λ)+Q(λ)e-λτ]=0,
where P(λ)=λ2+(r1+r2+D)λ+r2(r1+D)-eD, Q(λ)=-a2rλ/(a+mr)-a2r(r1+D)/(a+mr).Noting that (8) has a negative root λ=-r, the other roots of (8) are determined by the following:
(9)P(λ)+Q(λ)e-λτ=0.Denote(10)f(λ)=P(λ)+Q(λ)e-λτ.
If a2r(r1+D)>(a+mr)[r2(r1+D)-eD], then
(11)f(0)=-a2r(r1+D)-(a+mr)[r2(r1+D)-eD]a+mr<0,limλ→+∞f(λ)=+∞.
Hence, f(λ)=0 has at least one positive root; the predator-extinction equilibrium E1(r/a,0,0) is unstable.Ifa2r(r1+D)<(a+mr)[r2(r1+D)-eD], it is readily seen from (9) that E1 is locally asymptotically stable when τ=0. Denote
(12)F(y)=|P(iy)|2-|Q(iy)|2=y4+h1y2+h2,
where
(13)h1=(r1+r2+D)2-2[r2(r1+D)-eD]-a22r2(a+mr)2>(r1+D)2+r22+2eD-[r2(r1+D)-eD]2(r1+D)2=[(r1+D)2+eD]2+2eD[r2(r1+D)-eD](r1+D)2>0,h2=(a+mr)2[r2(r1+D)-eD]2-a22r2(r1+D)2(a+mr)2>0.
Hence, F(y)=0 has no positive root. By Theorem 3.4.1 in [6], we know that if a2r(r1+D)<(a+mr)[r2(r1+D)-eD], the predator-extinction equilibrium E1(r/a,0,0) is locally asymptotically stable for all τ≥0.Concluding the above discussions, we obtain the following results.Theorem 1.
For system (3), one has the following.(i) The trivial equilibriumE0(0,0,0) is always unstable.(ii) Ifa2r(r1+D)>(a+mr)[r2(r1+D)-eD], the predator-extinction equilibrium E1(r/a,0,0) is unstable; if a2r(r1+D)<(a+mr)[r2(r1+D)-eD], E1 is locally asymptotically stable for all τ≥0.In the following, we discuss the local stability of the coexistence equilibriumE*=(x*,y1*,y2*) and the existence of Hopf bifurcations at E*.The characteristic equation of system (3) at E*=(x*,y1*,y2*) is
(14)λ3+p2λ2+p1λ+p0+(q2λ2+q1λ+q0)e-λτ=0,
where
(15)p0=(-r+2ax*+a1y2*(1+mx*)2)[r2(r1+D)-eD],p1=r2(r1+D)-eD-(r-2ax*-a1y2*(1+mx*)2)(r1+r2+D),p2=r1+r2+D-r+2ax*+a1y2*(1+mx*)2,q0=a2x*1+mx*(r1+D)(r-2ax*),q1=a2x*1+mx*(r-2ax*-r1-D),q2=-a2x*1+mx*.Whenτ=0, (14) becomes
(16)λ3+(p2+q2)λ2+(p1+q1)λ+p0+q0=0.
By calculations, we obtain that
(17)p2+q2=r1+D+2ax*-r+eDr1+D+a1y2*(1+mx*)2,p0+q0=a1y2*(1+mx*)2[r2(r1+D)-eD].
If (H1) holds, p0+q0>0. Hence, by the Routh-Hurwitz criterion, we know that the coexistence equilibrium E*=(x*,y1*,y2*) is locally asymptotically stable provided the following.(H2)p2+q2>0,  (p2+q2)(p1+q1)-(p0+q0)>0, and E* is unstable if one of the inequality in (H2) is reversed.Clearly,λ=iω (ω>0) is a root of (14) if and only if
(18)-iω3-p2ω2+ip1ω+p0+(-q2ω2+iq1ω+q0)(cosωτ-isinωτ)=0.
Separating real and imaginary parts, we have
(19)-ω3+p1ω=(q0-q2ω2)sinωτ-q1ωcosωτ,-p2ω2+p0=-q1ωsinωτ-(q0-q2ω2)cosωτ.
Squaring and adding the two equations of (19), we obtain
(20)ω6+l2ω4+l1ω2+l0=0,
where
(21)l0=p02-q02=(p0+q0)(p0-q0)=-a1y2*(1+mx*)2[r2(r1+D)-eD]2×[2(r-2ax*)-a1y2*(1+mx*)2],l1=p12-2p0p2+2q0q2-q12=a1y2*(1+mx*)2[(D+r1)2+r22+2eD]×[a1y2*(1+mx*)2-2(r-2ax*)]+(r-2ax*)2×[(D+r1)2+2eD+eD(2r2(r1+D)-eD)(r1+D)2],l2=p22-2p1-q22=(D+r1)2+(r-2ax*-a1y2*(1+mx*)2)2+eD(r1+D)2[2(r1+D)2+2r2(r1+D)-eD].Lettingω2=z, (20) can be rewritten as
(22)z3+l2z2+l1z+l0=0.Denote(23)l(z)=z3+l2z2+l1z+l0.
If Δ=l22-3l1>0, then l′(z)=0 has two unequal roots:
(24)z1*=-l2+Δ3,z2*=-l2-Δ3.By Lemma 2.1 in [12], we can obtain the following conclusion.Lemma 2.
Suppose that  (H1) is satisfied andz1* is defined by (24).(i) Ifp0<q0, then (22) has at least one positive root.(ii) Ifp0≥q0 and Δ<0, then (22) has no positive roots.(iii) Ifp0≥q0, then (22) has positive roots if and only if z1*>0, l(z1*)≤0.Without loss of generality, we assume that (22) has three positive roots, denoted by zk(k=1,2,3), and ωk=zk(k=1,2,3) are the three positive roots of (20) correspondingly.Denote(25)τk(j)=1ωk[arcsin((q22ωk4-(2q0q2-q12)ωk2+q02)-1(q2ωk5+(p2q1-p1q2-q0)ωk30000000000000+(p1q0-p0q1)ωkq2ωk5+(p2q1-p1q2-q0)ωk3)000000000000×(q22ωk4-(2q0q2-q12)ωk2+q02)-1)+2jπ],000000000000000000000000000k=1,2,3;j=0,1,2,…;
then ±iωk is a pair of purely imaginary roots of (14) with τ=τk(j), k=1,2,3; j=0,1,2,….Defineτ0=τk0(0)=min1≤k≤3{τk(0)}, ω0=ωk0. By Lemma 2.2 in [12], the following result can be obtained.Lemma 3.
Suppose that (H1) and (H2) are satisfied.(i) Ifp0≥q0 and Δ<0, then all roots of (14) have negative real parts for all τ≥0.(ii) Ifp0<q0, or p0≥q0, z1*>0, l(z1*)≤0, then all roots of (14) have negative real parts when τ∈[0,τ0).Letλ(τ)=α(τ)+iω(τ) be the root of (14) satisfying α(τ0)=0, ω(τ0)=ω0. The conditions in the following lemma ensure that the transversality condition holds.Lemma 4.
Ifl′(z0)≠0 and the conditions (ii) in Lemma 3 are satisfied, then
(26)dReλ(τ)dτ|τ=τ0>0.Proof.
Differentiating (14) with respect τ, it follows that
(27)[(3λ2+2p2λ+p1)+(2q2λ+q1)e-λτ0-(q2λ2+q1λ+q0)τe-λτ]dλdτ=(q2λ2+q1λ+q0)λe-λτ,
which yields
(28)(dλ(τ)dτ)-1=3λ2+2p2λ+p1-λ(λ3+p2λ2+p1λ+p0)+2q2λ+q1λ(q2λ2+q1λ+q0)-τλ.
On substituting τ=τ0 into (28), then
(29)(dReλ(τ)dτ)-1|τ=τ0=l′(z0)q12ω02+(q0-q2ω02)2.
Sincel′(z0)≠0,
(30)dReλ(τ)dτ|τ=τ0≠0.
If dReλ(τ)/dτ|τ=τ0<0, there exists a root λ(τ)=α(τ)+iω(τ) satisfying α(τ)>0 for τ<τ0 and close to τ0, which contradicts (ii) in Lemma 3. The proof is complete.Applying Lemmas3 and 4, we obtain the following results.Theorem 5.
For system (3), suppose that (H1) and (H2) are satisfied.(i) Ifp0≥q0 and Δ<0, then the coexistence equilibrium E*=(x*,y1*,y2*) is locally asymptotically stable for all τ≥0.(ii) Ifp0<q0, or p0≥q0, z1*>0, l(z1*)≤0, then E* is locally asymptotically stable for 0<τ<τ0 and is unstable for τ>τ0.(iii) If the conditions in (ii) are satisfied, andl′(z0)≠0, then system (3) exhibits the Hopf bifurcation at E* when τ=τ0.Example 6.
In (2), let a=0.5, a1=0.4, a2=0.4, e=0.1, D=0.1, r=1, r1=0.1, r2=0.25, and m=0.5. System (3) with above coefficients has a unique coexistence equilibrium E*(0.67,1.11,2.22). A direct calculation shows that p2+q2≈0.42>0, (p2+q2)(p1+q1)-(p0+q0)≈0.04>0, p0-q0≈-0.01<0, τ0≈5.67. By Theorem 5, we see that the positive E* is locally asymptotically stable if 0<τ<τ0 and is unstable if τ>τ0. Numerical simulation illustrates our result (see Figure 1).The numerical solution of system (3) with a=0.5, a1=0.4, a2=0.4, e=0.1, D=0.1, r=1, r1=0.1, r2=0.25, m=0.5; (a) τ=3, (b) τ=6 and (ϕ,ψ1,ψ2)=(1;1;1).
(a)
(b)


## 3. Permanence

In the following, we show that system (3) is permanent.Definition 7.
System (3) is said to be permanent if there are positive constants m1, m2, M1, and M2 such that each positive solution (x(t),y1(t),y2(t)) of system (3) satisfies
(31)m1≤liminft→+∞x(t)≤limsupt→+∞x(t)≤M1,m2≤liminft→+∞yi(t)≤limsupt→+∞yi(t)≤M2,i=1,2.
Firstly, we prove that system (3) is ultimately bounded.Lemma 8.
Ifr2(r1+D)-eD>0, the arbitrary solution (x(t),y1(t),y2(t)) of system (3) is ultimately bounded.Proof.
Let(x(t),y1(t),y2(t)) be any positive solution of system (3). Define
(32)L(t)=a2x(t-τ)+ky1(t)+a1y2(t),
where k is a constant satisfying a1D/(r1+D)<k<a1r2/e.
Calculating the derivative ofL(t) along positive solutions of system (3), we get
(33)L˙(t)=a2x(t-τ)[r-ax(t-τ)]-(r1+D-a1Dk)ky1-(r2-eka1)a1y2≤a2x(t-τ)[r+δ-ax(t-τ)]-δ[a2x(t-τ)+ky1+a1y2]≤a2(r+δ)24a-δL(t),
where δ=min{r1+D-a1D/k,r2-ek/a1}.
It is easy to know thatδ>0 for a1D/(r1+D)<k<a1r2/e. According to (33), we get
(34)limsupt→+∞L(t)≤a2(r+δ)24aδ.
If we choose
(35)M1=(r+δ)24aδ,M2=a2(r+δ)24akδ,M3=a2(r+δ)24aa1δ,
then, for ε>0 sufficiently small, there exists a T1>0 such that if t>T1,
(36)x(t)<M1+ε,y1(t)<M2+ε,y2(t)<M3+ε.
That is, the arbitrary solution (x(t),y1(t),y2(t)) of system (3) is ultimately bounded if r2(r1+D)-eD>0. This completes the proof.Next, we use the persistence theory on infinite dimensional systems introduced by Hale and Waltman in [13] to prove the permanence of system (3).LetX be a complete metric space. Suppose that X0⊂X, X0⊂X,  and X0∩X0=ϕ. Assume that T(t) is a C0 semigroup on X satisfying
(37)T(t):X0⟶X0,T(t):X0⟶X0.LetTb(t)=T(t)|X0 and Ab be the global attractor for Tb(t). The following is a small variant of Theorem 4.1 in [13].Lemma 9 (see [13]).
Suppose thatT(t) satisfies (37) and one has the following.(i) There is at0≥0 such that T(t) is compact for t>t0;(ii)T(t) is point dissipative in X;(iii)A~b=⋃x∈Ab‍ω(x) is isolated and has an acyclic covering M~, where
(38)M~={M1,M2,…,Mn};(iv)Ws(Mi)∩X0=ϕ for i=1,2,…,n.Then, X0 is a uniform repeller with respect to X0; that is, there is an ε>0 such that, for any x∈X0, liminft→+∞d(T(t)x,X0)≥ε, where d is the distance of T(t)x from X0.Now we state and prove the permanence of system (3).Theorem 10.
If (H1) holds, then system (3) is permanent.Proof.
We need only to show that the boundaries ofR+03 repel positive solutions of system (3) uniformly.
LetC+([-τ,0],R+03) denote the space of continuous functions mapping [-τ,0] into R+03. Define
(39)C1={(ϕ,ψ1,ψ2)∈C+([-τ,0],R+03):ϕ(θ)≡0,θ∈[-τ,0]},C2={(ϕ,ψ1,ψ2)∈C+([-τ,0],R+03):ϕ(θ)>0,0ψ1(θ)≡0,ψ2(θ)≡0,θ∈[-τ,0](ϕ,ψ1,ψ2)∈C+([-τ,0],R+03):ϕ(θ)>0,}.
Denote X=C+([-τ,0],R+03), C0=C1∪C2, C0=intC+([-τ,0],R+03).
Now, we verify that the conditions in Lemma9 are satisfied. According to the definition of C0 and C0, it is easy to know that C0 and C0 are positively invariant, so condition (ii) in Lemma 9 is satisfied. The solution of system (3) is ultimately bounded if (H1) holds by Lemma 8. Thus, by the smoothing property of solutions of delay differential equations introduced in [6, Theorem 2.2.8], condition (i) is satisfied.
Next, we verify condition (iii) in Lemma9. There are two constant solutions in C0:E^0∈C1, E^1∈C2 corresponding to E0(0,0,0) and E1(r/a,0,0), respectively, which satisfy
(40)E^0={(ϕ,ψ1,ψ2)∈C+([-τ,0],R+03):ϕ(θ)≡0,0ψ1(θ)≡0,ψ2(θ)≡0,θ∈[-τ,0]([-τ,0],R+03)},E^1={(ϕ,ψ1,ψ2)∈C+([-τ,0],R+03):ϕ(θ)=ra,0ψ1(θ)≡0,ψ2(θ)≡0,θ∈[-τ,0]ra}.
If (x(t),y1(t),y2(t)) is a solution of system (3) initiating from C1, then
(41)y˙1(t)=ey2(t)-(r1+D)y1(t),y˙2(t)=Dy1(t)-r2y2(t).
Obviously, if (H1) holds, y1(t)→0 and y2(t)→0 as t→+∞. If (x(t),y1(t),y2(t)) is a solution of system (3) initiating from C2, then x˙(t)=x(t)[r-ax(t)], which yields x(t)→r/a, as t→+∞. Noting that C1∩C2=ϕ, we know the invariant E^0 and E^1 are isolated. So, {E^0,E^1} is isolated and is an acyclic covering satisfying conditions (iii) in Lemma 9.
We now show thatWs(E^0)∩C0=ϕ and Ws(E^1)∩C0=ϕ.
Assume thatWs(E^0)∩C0≠ϕ. Then there is a positive solution (x(t),y1(t),y2(t)) of system (3) satisfying
(42)limt→+∞x(t)=0,limt→+∞y1(t)=0,limt→+∞y2(t)=0.
Then for ε>0 sufficiently small such that r-a1ε>0, there exists a T2>0 such that if t>T2,
(43)-ε<y2(t)<ε.
Together with (43) and the first equation of system (3), we derive that
(44)x˙(t)>x(t)[r-ax(t)-a1ε].
Thus, liminft→+∞x(t)≥(r-a1ε)/a, which contradicts limt→+∞x(t)=0. Then Ws(E^0)∩C0=ϕ.
Assume thatWs(E^1)∩C0≠ϕ. Then, there is a positive solution (x(t),y1(t),y2(t)) of system (3) satisfying
(45)limt→+∞x(t)=ra,limt→+∞y1(t)=0,limt→+∞y2(t)=0.
Since (H1) holds, we can choose ε>0 small enough such that
(46)r2(r1+D)-eD<a2(r/a-ε)(r1+D)1+m(r/a-ε).
Since limt→+∞x(t)=r/a, for ε>0 satisfying (46), there exists a T3>0 such that if t>T3,
(47)ra-ε<x(t)<ra+ε.
From the last two equations of system (3), it is easy to know that if t>T3+τ,
(48)y˙1(t)=ey2(t)-(r1+D)y1(t),y˙2(t)≥Dy1(t)-r2y2(t)+a2(r/a-ε)y2(t-τ)1+m(r/a-ε).
Let us consider the following auxiliary system:
(49)z˙1=ez2(t)-(r1+D)z1(t),z˙2=Dz1(t)-r2z2(t)+a2(r/a-ε)z2(t-τ)1+m(r/a-ε),
with initial conditions (4).
Consider the following matrixAε defined by
(50)Aε=(-(r1+D)eD-r2+a2(r/a-ε)1+m(r/a-ε)).
Since Aε admits positive off-diagonal elements, the Perron-Frobenius theorem implies that there is a positive eigenvector v=(v1,v2) for the maximum root α of Aε. Since (46) holds, it is shown that the maximum root α>0 by a simple computation.
Letz(t)=(z1(t),z2(t)) be a solution of system (49) through (lv1,lv2) for t0-τ≤t≤t0, where l>0 satisfies lv1<y1(t0+θ), lv2<y2(t0+θ) for θ∈[-τ,0]. Since the semiflow of (49) is monotone and Aεv>0, it follows from [14] that zi(t) is strictly increasing and zi(t)→+∞ as t→+∞, i=1,2. By comparison, y1(t)→+∞, y2(t)→+∞ as t→+∞, contradicting Lemma 8. Hence, Ws(E^1)∩C0=ϕ.
By Lemma9, we conclude that C0 repels positive solutions of system (3) uniformly. Hence, system (3) is permanent. This proof is complete.


## 4. Global Stability

In this section, we study the global stability of the predator-extinction equilibriumE1(r/a,0,0) and the coexistence equilibrium E*=(x*,y1*,y2*), respectively, by means of Lyapunov functionals and LaSalle's invariance principle.First, we discuss the global stability of the predator-extinction equilibriumE1(r/a,0,0).Theorem 11.
Ifa2r(r1+D)<(a+mr)[r2(r1+D)-eD], then the predator-extinction equilibrium E1(r/a,0,0) is globally asymptotically stable.Proof.
Let(x(t),y1(t),y2(t)) be any positive solution of system (3) with initial conditions (4). By Theorem 1, we know that E1 is locally asymptotically stable if a2r(r1+D)<(a+mr)[r2(r1+D)-eD].
Define(51)V11(t)=a21+mx0(x-x0-x0lnxx0)+cy1+a1y2,
where x0=r/a, c=a1D/(D+r1).
Calculating the derivative ofV11(t) along positive solutions of system (3), we obtain that
(52)ddtV11(t)=a21+mx0(x(t)-x0)[r-ax(t)-a1y2(t)1+mx]+c[ey2(t)-(r1+D)y1(t)]+a1[Dy1(t)-r2y2(t)+a2x(t-τ)y2(t-τ)1+mx(t-τ)].
On substituting r=ax0 into (52), we derive that
(53)ddtV11(t)=-aa21+mx0(x(t)-x0)2-a1a2y2[x(1+mx0)-x0(1+mx)](1+mx)(1+mx0)+c[ey2(t)-(r1+D)y1(t)]+a1[Dy1(t)-r2y2(t)+a2x(t-τ)y2(t-τ)1+mx(t-τ)]=-aa21+mx0(x(t)-x0)2-a1a2xy21+mx+(a1eDD+r1-a1r2+a1a2x01+mx0)y2+a1a2x(t-τ)y2(t-τ)1+mx(t-τ).
Define
(54)V1(t)=V11(t)+a1a2∫t-τt‍x(s)y2(s)1+mx(s)ds.
By (53) and (54), it follows that
(55)ddtV1(t)=-aa21+mx0(x(t)-x0)2+(a1eDD+r1-a1r2+a1a2x01+mx0)y2=-aa21+mx0(x(t)-x0)2+a1[a2r(D+r1)-(a+mr)(r2(r1+D)-eD)](a+mr)(r1+D)y2.
If a2r(r1+D)<(a+mr)[r2(r1+D)-eD], then V˙1(t)≤0. By Theorem 5.3.1 in [11], solutions limit to ℳ, the largest invariant sunset of {V˙1(t)=0}. We can see from (55) that V˙1(t)=0 if and only if x=x0, y2=0. Since ℳ is invariant, for each element in ℳ, we have x=x0, y2=0. Therefore, it follows from the last equation of system (3) that(56)0=y˙2=Dy1(t),
which yields y1(t)=0. Consequently, V˙1(t)=0 if and only if x=x0, y1=0, y2=0. Hence, by Lasalle’s invariance principle, E1(r/a,0,0) is globally asymptotically stable. The proof is complete.Next, we prove the global stability of the coexistence equilibriumE*=(x*,y1*,y2*).Theorem 12.
If  (H1) holds, then the coexistence equilibriumE*=(x*,y1*,y2*) is globally asymptotically stable provided the following.
(H
3
)  x_>r/(2a).
Here,x_>0 is the persistency constant for x satisfying liminft→+∞x≥x_.Proof.
Let(x(t),y1(t),y2(t)) be any positive solution of system (3) with initial conditions (4). Since x_>r/(2a), we know that there exists a T4>0 such that x(t)>r/(2a) for t>T4 and also that x*>r/(2a). By Theorem 5 in [10], it is shown that p2+q2>0, (p2+q2)(p1+q1)-(p0+q0)>0. If (H1) holds, according to (21), l0>0, l1>0, l2>0 for x*>r/(2a). Then, (20) has no positive roots. Hence, E* is locally asymptotically stable for all τ≥0.
Define(57)V21(t)=x-x*-x*lnxx*+c1(y1-y1*-y1*lny1y1*)+c2(y2-y2*-y2*lny2y2*),
where c1(D+r1)=c2D, c2=a1(1+mx*)/a2.
Calculating the derivative ofV21(t) along positive solutions of system (3), we obtain that
(58)ddtV21(t)=(1-x*x)[x(r-ax)-a1xy21+mx]+c1(1-y1*y1)[ey2-(r1+D)y1]+c2(1-y2*y2)[Dy1-r2y2+a2x(t-τ)y2(t-τ)1+mx(t-τ)].
On substituting x*(r-ax*)=a1x*y2*/(1+mx*) into (58), we derive that
(59)ddtV21(t)=(1-x*x)[(x-x*)(r-a(x+x*))+a1x*y2*1+mx*]-a1xy21+mx(1-x*x)+c1[ey2-ey1*y2y1+(r1+D)y1*]+c2[-r2y2-Dy2*y1y2+r2y2*+a2x(t-τ)y2(t-τ)1+mx(t-τ)-a2y2*x(t-τ)y2(t-τ)(1+mx(t-τ))y2]=(1-x*x)[(x-x*)(r-a(x+x*))+a1x*y2*1+mx*]-a1xy21+mx(1+mx*-x*(1+mx)x)+c1[ey2-ey1*y2y1+(r1+D)y1*]+c2[-r2y2-Dy2*y1y2+r2y2*+a2x(t-τ)y2(t-τ)1+mx(t-τ)-a2y2*x(t-τ)y2(t-τ)(1+mx(t-τ))y2]=(1-x*x)[(x-x*)(r-a(x+x*))+a1x*y2*1+mx*]-a1xy2(1+mx*)1+mx+a1x*y2+c1[ey2-ey1*y2y1+(r1+D)y1*]+c2[-r2y2-Dy2*y1y2+r2y2*+a2x(t-τ)y2(t-τ)1+mx(t-τ)-a2y2*x(t-τ)y2(t-τ)(1+mx(t-τ))y2].
Noticing that a1x*+ec1=c2r2, (59) can be rewritten as
(60)ddtV21(t)=(x-x*)2x[r-a(x+x*)]+a1x*y2*1+mx*(1-x*x)-a1(1+mx*)1+mxxy2+c1[-ey2*y1*y2*y2y1+(r1+D)y1*]+c2[-Dy1*y2*y1*y1y2+r2y2*+a2x(t-τ)y2(t-τ)1+mx(t-τ)-a2y2*x(t-τ)y2(t-τ)(1+mx(t-τ))y2].
Define(61)V2(t)=V21(t)+a2c2∫t-τt‍[x(s)y2(s)1+mx(s)-x*y2*1+mx*0000000000-x*y2*1+mx*0000000000×lnx(s)y2(s)(1+mx*)x*y2*(1+mx(s))]ds.
According to (60) and (61), it follows that
(62)ddtV2(t)=(x-x*)2x[r-a(x+x*)]+a1x*y2*1+mx*(1-x*x)-a1(1+mx*)1+mxxy2-c1ey2*y1*y2*y2y1+c1(r1+D)y1*-c2Dy1*y2*y1*y1y2+c2r2y2*-a2c2y2*x(t-τ)y2(t-τ)(1+mx(t-τ))y2+a2c2xy21+mx-a2c2x*y2*1+mx*lnx(t)y2(t)(1+mx*)x*y2*(1+mx(t))+a2c2x*y2*1+mx*lnx(t-τ)y2(t-τ)(1+mx*)x*y2*(1+mx(t-τ)).
Noting that(63)c1ey2*=c1(r1+D)y1*=c2Dy1*,c2r2=a1x*+ec1,a2c2=a1(1+mx*),a1x*y2*1+mx*(1-x*x)=a1x*y2*(1-x*(1+mx)x(1+mx*)),
we derive from (62) that
(64)ddtV2(t)=(x-x*)2x[r-a(x+x*)]+c1ey2*(2-y1*y2*y2y1-y2*y1*y1y2)-a1x*y2*[(1+mx*)x(t-τ)y2(t-τ)(1+mx(t-τ))x*y2-1000000000-ln(1+mx*)x(t-τ)y2(t-τ)(1+mx(t-τ))x*y2]-a1x*y2*[x*(1+mx)x(1+mx*)-1-lnx*(1+mx)x(1+mx*)].
Ifx(t)>r/(2a) for t>T4, then
(65)(x-x*)2x[r-a(x+x*)]≤0,
and the equality holds only for x=x*. Since the arithmetic mean is greater than or equal to the geometric mean, it is shown that
(66)2-y1*y2*y2y1-y2*y1*y1y2≤0,
with equality if and only if y1*y2/(y2*y1)=y2*y1/(y1*y2). Together with (64), it follows that if x(t)>r/(2a) for t>T4, V˙2(t)≤0, with equality if and only if x=x*,y1*y2/(y2*y1)=y2*y1/(y1*y2), (1+mx*)x(t-τ)y2(t-τ)/(x*y2(1+mx(t-τ)))=1. Hence, we now look for the invariant subset ℳ within the set
(67)N={(x,y1,y2):x=x*,y1*y2*y2y1=y2*y1*y1y2,0(1+mx*)x(t-τ)y2(t-τ)(1+mx(t-τ))x*y2=1}.
Since x=x* on ℳ, then 0=x˙(t)=x*(r-ax*-a1y2/(1+mx*)), which yields y2(t)=y2*. It follows from the last equation of system (3) that 0=y˙2(t)=Dy1(t)-ry2*+a2x*y2*/(1+mx*), which yields y1(t)=y1*. Thus, the only invariant set in N is ℳ=(x*,y1*,y2*). Hence, by Lasalle's invariance principle, E* is globally asymptotically stable. The proof is complete.Remark 13.
By Theorem12, it is shown that if (H1) holds and x_>r/(2a), then the coexistence equilibrium E*=(x*,y1*,y2*) is globally asymptotically stable. We now give a sufficient condition for this inequality. By (35), y2(t) has an ultimately upper bound M3=a2(r+δ)2/(4aa1δ). Hence, we derive from the first equation of system (3) that x˙(t)≥x(t)(r-ax(t)-a1M3), which yields liminft→+∞x(t)≥x_=(r-a1M3)/a. A brief calculation shows that we need only to choose 2arδ>a2(r+δ)2; then x_>r/(2a).



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