We study the permanence of periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey and obtain that the predator and the prey species are permanent.

1. Introduction

An important and ubiquitous problem in predator-prey theory and related topics in mathematical ecology concerns the long-term coexistence of species. In the natural world there are many species whose individual members have a life history that takes them through two stages: immature and mature. In particular, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages. Recently, nonautonomous systems with a stage structure have been considered in [1–16]; in particular periodic predator-prey systems with a stage structure were discussed in [3, 4, 7, 13, 14].

Already, in [3], Cui and Song proposed the following predator-prey model with stage structure for prey:
ẋ1(t)=a(t)x2(t)-b(t)x1(t)-d(t)x12(t)-p(t)x1(t)y(t),ẋ2(t)=c(t)x1(t)-f(t)x22(t),ẏ(t)=y(t)(-g(t)+h(t)x1(t)-q(t)y(t)).
They obtained a set of sufficient and necessary condition which guarantee the permanence of the system.

In [4], Cui and Takeuchi considered the following periodic predator-prey system with a stage structure:
ẋ1(t)=a(t)x2(t)-b(t)x1(t)-d(t)x12(t)-p(t)ϕ(t,x1)x1y(t),ẋ2(t)=c(t)x1(t)-f(t)x22(t),ẏ(t)=y(t)(-g(t)+h(t)ϕ(t,x1)y(t)-q(t)y(t)),
where

0<ϕ(t,x1)<L,∂∂x1(ϕ(t,x1)x1)≥0(x1>0).

Recently, Huang et al. [7] studied the following periodic stage-structured three-species predator-prey system with Holling IV and Beddington-DeAngelis functional response:

ẋ1(t)=a(t)x2(t)-b(t)x1(t)-d(t)x12(t)-h1(t)x1(t)k1(t)+x12(t)y1(t),ẋ2(t)=c(t)x1(t)-f(t)x22(t)-h2(t)x2(t)k2(t)+m(t)x2(t)+n(t)y2(t)y2(t),ẏ1(t)=y1(t)(-q1(t)+p1(t)x1(t)k1(t)+x12(t)-g1(t)y1(t)),ẏ2(t)=y2(t)(-q2(t)+p2(t)x2(t)k2(t)+m(t)x2(t)+n(t)y2(t)-g2(t)y2(t)),
where a(t),b(t),c(t),d(t),f(t),m(t),n(t),hi(t),ki(t),pi(t),qi(t), and gi(t)(i=1,2) are all continuous positive ω-periodic functions; x1(t) and x2(t) denote the density of immature and mature prey species at time t, respectively; y1(t) represents the density of the predator that preys on immature prey; y1(t) represents the density of the other predator that preys on mature prey at time t.

It is assumed in the classical predator-prey model that each individual predator admits the same ability to attack prey and each individual prey admits the same risk to be attacked by predator. This assumption seems not to be realistic for many animals. On the other hand, predator-prey systems where only immature individuals are consumed by their predator are well known in nature. One example is described in [9], where the Chinese fire-bellied newt, which is unable to prey upon the mature rana chensinensis, can only prey on its immature.

To the best of the authors' knowledge, for the nonautonomous periodic case of predator-prey systems with stage structure for both predator and prey, whether one could obtain the permanence of the system or not is still an open problem.

Motivated by the above question, we consider the following periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey:
ẋ1(t)=a1(t)x2(t)-b1(t)x1(t)-d1(t)x12(t)-g(x1(t))y2(t),ẋ2(t)=c1(t)x1(t)-f1(t)x22(t),ẏ1(t)=a2(t)y2(t)-b2(t)y1(t)-d2(t)y12(t)+k(t)g(x1(t))y1(t),ẏ2(t)=c2(t)y1(t)-f2(t)y22(t),
where ai(t),bi(t),ci(t),di(t),fi(t),i=1,2, and k(t) are all continuous positive ω-periodic functions. Here x1(t) and x2(t) denote the density of immature and mature prey species, respectively, and y1(t) and y2(t) denote the density of immature and mature predator species, respectively. The function g(x) is assumed to satisfy the following assumptions which has been studied in detail by Georgescu and Morosanu in [17].

(G) g(x) of class C1 is increasing on R+,g(0)=0, and such that x↦g(x)/x is decreasing on R+,|g′(x)|≤L for x∈R+, where L≥0.

Note that hypothesis (G) is satisfied if function g(x) represents Holling type II functional response, that is, g(x)=ax/(1+bx), in which a is the search rate of the resource and of the intermediate consumer, and b represents the corresponding clearance rate, that is, search rate multiplied by the (supposedly constant) handling time.

The aim of this paper is, by further developing the analysis technique of Cui and Song [3] and Cui and Takeuchi [4], to obtain a set of sufficient and necessary conditions which ensure the permanence of the system (1.5). The rest of the paper is arranged as follows. In Section 2, we introduce some lemmas and then state the main result of this paper. The result is proved in Section 3.

2. Main ResultsDefinition 2.1.

The system
ẋ=F(t,x),x∈Rn
is said to be permanent if there exists a compact set K in the interior of R+n={(x1,x2,…,xn)∈Rn∣xi≥0,i=1,2,…,n}, such that all solutions starting in the interior of R+n ultimately enter K and remain in K.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

If a(t),b(t),c(t),d(t), and f(t) are all ω-periodic, then system
ẋ1(t)=a(t)x2(t)-b(t)x1(t)-d(t)x12(t),ẋ2(t)=c(t)x1(t)-f(t)x22(t)
has a positive ω-periodic solution (x1*(t),x2*(t)) which is globally asymptotically stable with respect to R+2={(x1,x2):x1>0,x2>0}.

Theorem 2.3.

System (1.5) is permanent.

3. Proof of the Main Results

We need the following propositions to prove Theorem 2.3. The hypotheses of the lemmas and theorems of the preceding section are assumed to hold in what follows.

Proposition 3.1.

There exists a positive constant Mx such that
limsupt→∞xi(t)≤Mx,i=1,2.

Proof.

Obviously, R4+ is a positively invariant set of system (1.5). Given any positive solution (x1(t),x2(t),y1(t),y2(t)) of (1.5), from the first and second equations of system(1.5), we have
ẋ1(t)≤a1(t)x2(t)-b1(t)x1(t)-d1(t)x12(t),ẋ2(t)≤c1(t)x1(t)-f1(t)x22(t).

By Lemma 2.2, the following auxiliary equation
u̇1(t)=a1(t)u2(t)-b1(t)u1(t)-d1(t)u12(t),u̇2(t)=c1(t)u1(t)-f1(t)u22(t)
has a globally asymptotically stable positive ω-periodic solution (x1*(t),x2*(t)). Let (u1(t),u2(t)) be the solution of (3.3) with ui(0)=xi(0). By comparison, we then have
xi(t)≤ui(t),i=1,2
for t≥0. From the global attractivity of (x1*(t),x2*(t)), for any positive ε>0 small enough, there exists a T0>0 such that
|ui(t)-xi*(t)|<ε,t≥T0
(3.4) combined with (3.5) leads to
xi(t)<xi*(t)+ε,t>T0.

Let Mx=max0≤t≤ω{xi*(t)+ε:i=1,2}, then we have
limsupt→∞xi(t)≤Mx.
This completes the proof of Proposition 3.1.

Proposition 3.2.

There exists a positive constant my such that
liminft→∞yi(t)≥my,i=1,2.

Proof.

Given any positive solution (x1(t),x2(t),y1(t),y2(t)) of (1.5), from the third and fourth equations of system (1.5), we have
ẏ1(t)≥a2(t)y2(t)-b2(t)y1(t)-d2(t)y12(t),ẏ2(t)≥c2(t)y1(t)-f2(t)y22(t).

By Lemma 2.2, the following auxiliary equation
v̇1(t)=a2(t)v2(t)-b2(t)v1(t)-d2(t)v12(t),v̇2(t)=c2(t)v1(t)-f2(t)v22(t)
has a globally asymptotically stable positive ω-periodic solution (y1*(t),y2*(t)). Let (v1(t),v2(t)) be the solution of (3.10) with vi(0)=yi(0). By comparison, we then have
yi(t)≥vi(t),i=1,2
for t≥0. From the global attractivity of (y1*(t),y2*(t)), for any positive ε>0 small enough (ε<min0≤t≤ω{yi*(t):i=1,2}), there exists a T1>0 such that
|vi(t)-yi*(t)|<ε,t≥T1
(3.11) combined with (3.12) leads to
yi(t)>yi*(t)-ε,t>T1.

Let my=min0≤t≤ω{yi*(t)-ε:i=1,2}, then we have
liminft→∞yi(t)≥my.
This completes the proof of Proposition 3.2.

Proposition 3.3.

There exists a positive constant My such that
limsupt→+∞yi(t)≤My,i=1,2.

Proof.

Given any positive solution (x1(t),x2(t),y1(t),y2(t)) of (1.5), by Proposition 3.1, there exists a T2>0 such that
0<xi(t)≤Mx,t>T2.

From the third and fourth equations of system (1.5), we have
ẏ1(t)≤a2(t)y2(t)-(b2(t)-k(t)g(Mx))y1(t)-d2(t)y12(t),ẏ2(t)≤c2(t)y1(t)-f2(t)y22(t).

By Lemma 2.2, the following auxiliary equation
v̇1(t)=a2(t)v2(t)-(b2(t)-k(t)g(Mx))v1(t)-d2(t)v12(t),v̇2(t)=c2(t)v1(t)-f2(t)v22(t)
has a globally asymptotically stable positive ω-periodic solution (ỹ1*(t),ỹ2*(t)). Let (v1(t),v2(t)) be the solution of (3.18) with vi(0)=yi(0). By comparison, we then have
yi(t)≤vi(t),i=1,2
for t≥0. From the global attractivity of (ỹ1*(t),ỹ2*(t)), for above given positive ε>0, there exists a T3>0 such that
|vi(t)-ỹi*(t)|<ε,t≥T3
(3.19) combined with (3.20) leads to
yi(t)<ỹi*(t)+ε,t>T3.

Let My=max0≤t≤ω{ỹi*(t)+ε:i=1,2}, then we have
limsupt→∞yi(t)≤My.
This completes the proof of Proposition 3.3.

Proposition 3.4.

There exists a positive constant mx<Mx, such that
liminft→∞xi(t)≥mx,i=1,2.

Proof.

By Propositions 3.1 and 3.3, there exists a T4>0 such that
0<xi(t)≤Mx;0<yi(t)≤My;t>T4.

From the first and second equations of system (1.5), we have
ẋ1(t)≥a1(t)x2(t)-(b1(t)+g(x1(t))x1(t)My)x1(t)-d1(t)x12(t),ẋ2(t)≥c1(t)x1(t)-f1(t)x22(t).

Since g(x) is assumed to satisfy the assumptions (G), by the differential mean value theorem, we have
g(x1(t))x1(t)=g(x1(t))-g(0)x1(t)-0=g′(ξ)≤L,0<ξ<x1(t).

From (3.25) and (3.26), one has
ẋ1(t)≥a1(t)x2(t)-(b1(t)+LMy)x1(t)-d1(t)x12(t),ẋ2(t)≥c1(t)x1(t)-f1(t)x22(t).

By Lemma 2.2, the following auxiliary equation:
u̇1(t)=a1(t)u2(t)-(b1(t)+LMy)u1(t)-d1(t)u12(t),u̇2(t)=c1(t)u1(t)-f1(t)u22(t)
has a globally asymptotically stable positive ω-periodic solution (x̃1*(t),x̃2*(t)). Let (u1(t),u2(t)) be the solution of (3.27) with ui(0)=xi(0). By comparison, we then have
xi(t)≥ui(t),i=1,2
for t≥0. From the global attractivity of (x̃1*(t),x̃2*(t)), for any positive ε>0 small enough (ε<min0≤t≤ω{x̃i*(t):i=1,2}), there exists a T5>0 such that
|ui(t)-x̃i*(t)|<ε,t≥T5
(3.29) combined with (3.30) leads to
xi(t)>x̃i*(t)-ε,t>T5.

Let mx=min0≤t≤ω{x̃i*(t)-ε:i=1,2}, then we have
liminft→∞xi(t)≥mx.
This completes the proof of Proposition 3.4.

Proof of Theorem <xref ref-type="statement" rid="thm2.3">2.3</xref>.

By Propositions 3.1, 3.2, 3.3, and 3.4, system (1.5) is permanent. This completes the proof of Theorem 2.3.

AndrewsJ. F.A mathematical model for the continuous culture of microorganisms utilizing inhibitory substratesSokolW.HowellJ. A.Kinetics of phenol oxidation by washed cellsCuiJ. A.SongX. Y.Permanence of predator-prey system with stage structureCuiJ. A.TakeuchiY.A predator-prey system with a stage structure for the preyZhangX.ChenL.NeumannA. U.The stage-structured predator-prey model and optimal harvesting policyCuiJ.ChenL.WangW.The effect of dispersal on population growth with stage-structureHuangC.-Y.ZhaoM.ZhaoL.-C.Permanence of periodic predator-prey system with two predators and stage structure for preyXuR.ChaplainM. A. J.DavidsonF. A.A Lotka-Volterra type food chain model with stage structure and time delaysWangS.Research on the suitable living environment of the Rana temporaria chensinensis larvaZhangX.ChenL.NeumannA. U.The stage-structured predator-prey model and optimal harvesting policyWangW.ChenL.A predator-prey system with stage-structure for predatorBernardO.SouissiS.Qualitative behavior of stage-structured populations: application to structural validationYangW. S.LiX. P.BaiZ. J.Permanence of periodic Holling type-IV predator-prey system with stage structure for preyChenF. D.Permanence of periodic Holling type predator-prey system with stage structure for preyChenF. D.YouM. S.Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for preyCuiJ. A.SongX. Y.Permanence of predator-prey system with stage structureGeorgescuP.MorosanuG.Impulsive pertubations of a three-trophic prey-dependent food chain system