# On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

Authors: Xiangzeng Kong; Zhiqin Chen; Li Xu; Wensheng Yang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We propose and study the permanence of the following periodic Holling
III predator-prey system with stage structure for prey and both two predators which
consume immature prey. Sufficient and necessary conditions which guarantee the
predator and the prey species to be permanent are obtained.


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## Body

## 1. Introduction

The aim of this paper is to investigate the permanence of the following periodic stage-structure predator-prey system with Holling III functional response:(1.1)ẋ1(t)=a(t)x2(t)-b(t)x1(t)-d(t)x12(t)-p1(t)x12(t)k1(t)+x12(t)y1(t)-p2(t)x1(t)k2(t)+m(t)x1(t)+n(t)y2(t)y2(t),ẋ2(t)=c(t)x1(t)-f(t)x22(t)-p3(t)x22(t)k3(t)+x22(t)y2(t),ẏ1(t)=y1(t)(-g1(t)+h1(t)x12(t)k1(t)+x12(t)-q1(t)y1(t)),ẏ2(t)=y2(t)(-g2(t)+h2(t)x1(t)k2(t)+m(t)x1(t)+n(t)y2(t)+h3(t)x22(t)k3(t)+x22(t)-q2(t)y2(t)),
where a(t), b(t), c(t), d(t), f(t), pi(t), gi(t), qi(t), i=1,2 and hi(t), ki(t), i=1,2,3, are all continuous positive ω-periodic functions. Here x1(t) and x2(t) denote the density of immature and mature prey species, respectively, and yi is the density of the predators.The periodic functions in (1.1) have the following biological meanings. The birth rate into the immature population is given by a(t)x2; that is, it is assumed to be proportional to the existing mature population, with a proportionality coefficient a(t). The death rate of the immature population is proportional to the existing immature population and to its square with coefficients b(t) and d(t), respectively. The death rate of the mature population is of a logistic nature, with proportionality coefficient f(t). The transition rate from the immature individuals to the mature individuals is assumed to be proportional to the existing immature population,  with proportionality coefficient c(t).  Similarly, -gi(t)yi-qi(t)yi gives the density-dependent death rate of the predators. pi(t) and hi(t) give the coefficients that relate to the conversion rate of the immature prey biomass into predator biomass. More details about the biological background for (1.1) can be found in [1–10].The functionpi(t)x12(t)/(ki(t)+x12(t)) represents the functional response of predator to immature prey. Let φi(t,x1)=pi(t)x12(t)/(ki(t)+x12(t)), i=1,2, then we have(1.2)∂∂x1φi(t,x1)≥0,x1(t)>0,i=1,2.The functional response of predator speciesyi(t) to immature prey species takes the Holling type III, that is, pi(i)x12(t)/(ki(t)+x12(t)). Holling type III is the third function in which  Holling proposed three kinds of functional response of the predator to prey based on numerous experiments for different species. The Holling type form of functional response is intituled prey-dependent model form. It is applied to almost invertebrate that is one of the most extensive applied functional responses.In [2], Cui and Takeuchi considered the following periodic predator-prey system with a stage structure:(1.3)ẋ1(t)=a(t)x2(t)-b(t)x2(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)x1y(t)-q(t)y(t),
where(1.4)0<ϕ(t,x1)<L,∂∂x1(ϕ(t,x1)x1)≥0(x1>0).Different predators usually consume prey in different stage structures. Some predators only prey on immature prey, and some predators only prey on mature prey  [5]. Based on system (1.3), we also consider another predator species which also consumes immature prey. Assuming that the predator consumes immature prey according to Holling III functional response while the other predator consumes mature prey also according to the Holling III functional response, we get model (1.1).To the best of the authors' knowledge, for the nonautonomous case of predator-prey systems with two predators which consume immature prey and stage structure for prey, whether one could obtain the sufficient and necessary conditions which insure the permanence of the system or not is still an open problem.The aim of this paper is, by further developing the analysis technique of Cui and Takeuchi [2], to derive a set of sufficient and necessary conditions which ensure the permanence of the system (1.1). 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.Throughout this paper, for a continuousω-periodic function f(t), we set(1.5)Aω(f)=ω-1∫0ωf(t)dt.


## 2. Main Results

Definition 2.1.
The system(2.1)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.The following lemma can be found in [4].Lemma 2.2.
Ifa(t), b(t), c(t), d(t), and f(t) are all ω-periodic, then system
(2.2)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}.Lemma 2.3 (see [11]).
Ifb(t) and a(t) are all ω-periodic, and if Aω(b)>0 and Aω(a)>0 for all t∈R, then the system
(2.3)ẋ=x[b(t)-a(t)x]
has a positive ω-periodic solution which is globally asymptotically stable.Theorem 2.4.
Suppose that(2.4)Aω(-g1(t)+h1(t)(x1*(t))2k1(t)+(x1*(t))2)>0,Aω(-g2(t)+h2(t)x1*(t)k2(t)+m(t)x1*(t)+h3(t)(x2*(t))2k3(t)+(x2*(t))2)>0
holds  then system (1.1) is permanent, where (x1*(t),x2*(t)) is the unique positive periodic solution of system (2.2) given by Lemma 2.2.Theorem 2.5.
System (1.1) is permanent if and only if (2.4) holds.


## 3. Proof of the Main Results

We need the following propositions to prove Theorems2.4 and 2.5. The hypothesis of the lemmas and theorems of the preceding section are assumed to hold in what follows.Proposition 3.1.
There exist positive constantsMx and My such that
(3.1)limsupt→+∞xi(t)≤Mx,limsupt→+∞yi(t)≤My,i=1,2.Proof.
Obviously,R+4 is a positively invariant set of system (1.1). Given any positive solution (x1(t),x2(t),y1(t),y2(t)) of (1.1), we have
(3.2)ẋ1(t)≤a(t)x2(t)-b(t)x1(t)-d(t)x12(t),ẋ2(t)≤c(t)x1(t)-f(t)x22(t).
By Lemma2.2, the following auxiliary equation
(3.3)u̇1(t)=a(t)u2(t)-b(t)u1(t)-d(t)u12(t),u̇2(t)=c(t)u1(t)-f(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
(3.4)xi(t)≤ui(t),i=1,2
for t≥0. By (2.4), we can choose a positive ɛ>0 small enough such that
(3.5)Aω(-g1(t)+h1(t)(x1*(t)+ɛ)2k1(t))>0,Aω(-g2(t)+h2(t)(x1*(t)+ɛ)k2(t)+h3(t)(x2*(t)+ɛ)2k3(t))>0.
Thus, from the global attractivity of (x1*(t),x2*(t)), for the above given ɛ>0, there exists a T0>0 such that
(3.6)|ui(t)-xi*(t)|<ɛ,t≥T0,
Equation (3.4) combined with (3.6) leads to
(3.7)xi(t)<xi*(t)+ɛ,t>T0.
In addition, for t≥T0, from the third and fourth equations of (1.1)  and (3.7) we get
(3.8)ẏ1(t)≤y1(t)[-g1(t)+h1(t)x12(t)k1(t)-q1(t)y1(t)]≤y1(t)[-g1(t)+h1(t)(x1*(t)+ɛ)2k1(t)-q1(t)y1(t)],ẏ2(t)≤y2(t)[-g2(t)+h2(t)x1(t)k2(t)+h3(t)x22(t)k3(t)-q2(t)y2(t)]≤y2(t)[-g2(t)+h2(t)(x1*(t)+ɛ)k2(t)+h3(t)(x2*(t)+ɛ)2k3(t)-q2(t)y2(t)].
Consider the following auxiliary equation:
(3.9)v̇1(t)=v1(t)[-g1(t)+h1(t)(x1*(t)+ɛ)2k1(t)-q1(t)v1(t)],v̇2(t)=v2(t)[-g2(t)+h2(t)(x1*(t)+ɛ)k2(t)+h3(t)(x2*(t)+ɛ)2k3(t)-q2(t)v2(t)].
It follows from (3.5) and Lemma 2.3 that   (3.9) has a unique positive ω-periodic solution yi*(t)>0 which is globally asymptotically stable. Similarly to the above analysis, there exists a T1>T0 such that for the above ɛ, one has
(3.10)yi(t)<yi*(t)+ɛ,t≥T1.
Let Mx=max0≤t≤ω{xi*(t)+ɛ:i=1,2},   My=max0≤t≤ω{yi*(t)+ɛ:i=1,2}, then we have
(3.11)limsupt→+∞xi(t)≤Mx,limsupt→+∞yi(t)≤My.
This completes the proof of Proposition 3.1.Proposition 3.2.
There exist positive constantsδi<Mx, i=1,2, such that
(3.12)liminft→+∞xi(t)≥δi,i=1,2.Proof.
By Proposition3.1, there exists T1>0 such that
(3.13)0<xi(t)≤Mx;0<yi(t)≤My;t>T1.
Hence, from the first and second equations of system (1.1), we have
(3.14)ẋ1(t)≥a(t)x2(t)-(b(t)+(p1(t)k1(t)+p2(t)k2(t))My)x1(t)-d(t)x12(t),ẋ2(t)≥c(t)x1(t)-(f(t)+p3(t)k3(t)My)x22
for t>T1. By Lemma 2.2, the following auxiliary equation
(3.15)u̇1(t)=a(t)u2(t)-(b(t)+(p1(t)k1(t)+p2(t)k2(t))My)u1(t)-d(t)u12(t),u̇2(t)≥c(t)u1(t)-(f(t)+p3(t)k3(t)My)u22
has a globally asymptotically stable positive ω-periodic solution (x̃1*(t),x̃2*(t)). Let (u1(t),u2(t)) be the solution of (3.15) with (u1(T1),u2(T1))=(x1(T1),x2(T1)); by comparison, we have
(3.16)xi(t)≥ui(t)(i=1,2),t>T1.
Thus, from the global attractivity of (x̃1*(t),x̃2*(t)), there exists a T2>T1, such that
(3.17)|ui(t)-x̃i*(t)|<x̃i*(t)2(i=1,2),t>T2.
Equation (3.17) combined with (3.16) leads to
(3.18)xi(t)>δi=min0≤t≤ω{x̃i*(t)2},i=1,2,t>T2.
That is, we have
(3.19)liminft→+∞xi(t)≥δi,i=1,2.
This completes the proof of Proposition 3.2.Proposition 3.3.
There exists a positive constantδy such that
(3.20)limsupt→+∞yi(t)≥δy,i=1,2.Proof.
By assumption (2.4), we can choose a constant ɛ0>0 and the same constant ɛ as in Proposition 3.1 such that
(3.21)Aω(ψiɛ0(t))>0,i=1,2,
where
(3.22)ψ1ɛ0(t)=-g1(t)+h1(t)(x1*(t)-ɛ0)2k1(t)+(x1*(t)-ɛ0)2-q1(t)ɛ0,ψ2ɛ0(t)=-g2(t)+h2(t)(x1*(t)-ɛ0)k2(t)+m(t)(x1*(t)+ɛ)+n(t)ɛ0+h3(t)(x2*(t)-ɛ0)2k3(t)+(x2*(t)-ɛ0)2-q2(t)ɛ0.
Consider the following equation with a parameter β>0:
(3.23)ẋ1(t)=a(t)x2(t)-(b(t)+2β(p1(t)k1(t)+p2(t)k2(t))My)x1(t)-d(t)x12(t),ẋ2(t)=c(t)x1(t)-(f(t)+p3(t)k3(t)2β)x22.
By Lemma 2.2, (3.23) has a unique positive ω-periodic solution (x1β(t),x2β(t)), which is globally asymptotically stable. Let (x¯1β(t),x¯2β(t)) be the solution of (3.23) with initial condition x¯iβ(0)=xi*(0), i=1,2;  then, for the above ɛ0, there exists a T3>T2, such that
(3.24)|x¯iβ(t)-xiβ(t)|<ɛ04,i=1,2,t>T3.
By continuity of the solution in the parameter, we have (x¯1β(t),x¯2β(t))→(x1*(t),x2*(t)) uniformly in [T3,T3+ω] as β→0. Hence, for ɛ0>0 there exists  β0=β0(ɛ0)>0 such that
(3.25)|x¯iβ(t)-xi*(t)|<ɛ04,t∈[T3,T3+ω],0<β<β0.
So we have
(3.26)|xiβ(t)-xi*(t)|≤|x¯iβ(t)-xiβ(t)|+|x¯iβ(t)-xi*(t)|<ɛ02,t∈[T3,T3+ω].
Since xiβ(t) and xi*(t) are all ω-periodic, we have
(3.27)|xiβ(t)-xi*(t)|<ɛ02,t≥0,0<β<β0.
Choose a constant β1(0<β1<β0,2β1<ɛ0) and
(3.28)xiβ1(t)≥xi*(t)-ɛ02,t≥0.
Suppose that the conclusion (3.20) is not true. Then there exists a Z∈R+4  such that, for the positive solution (x1(t),x2(t),y1(t),y2(t)) of (1.1) with an initial condition (x1(0),x2(0),y1(0),y2(0))=Z, we have
(3.29)limsupt→+∞yi(t)<β1.
So there exists a T4>T3 such that
(3.30)yi(t)<2β1,t≥T4.
By applying (3.30), from the first and second equations of system (1.1) it follows that for all t≥T4,
(3.31)ẋ1(t)≥a(t)x2(t)-(b(t)+2β1p1(t)k1(t)+2β1p2(t)k2(t))x1(t)-d(t)x12(t),ẋ2(t)≥c(t)x1(t)-(f(t)+p3(t)k3(t)2β1)x22(t).
Let (u1(t),u2(t)) be the solution of (3.23) with β=β1 and ui(T4)=xi(T4),  i=1,2; we know that xi(t)≥ui(t), t≥T4, i=1,2.
By the global asymptotic stability of(x1β1(t),x2β1(t)), for the given ɛ=ɛ0/2, there exists T5≥T4 such that
(3.32)|ui(t)-xiβ1(t)|<ɛ02,t≥T5.
So we have
(3.33)xi(t)≥ui(t)>xiβ1(t)-ɛ02,t≥T5
and hence
(3.34)xi(t)>xi*(t)-ɛ0,t≥T5.
From (3.7) and (3.34), we have
(3.35)xi*(t)-ɛ0<xi(t)<xi*(t)+ɛ,t≥T5.
By (3.35) and (1.2), from the third and fourth equations of system (1.1) we have
(3.36)ẏi(t)≥ψiɛ0(t)yi(t),t≥T5.
Integrating (3.36) from T5 to t yields
(3.37)yi(t)≥yi(T5)exp{∫T5tψiɛ0(t)dt},t≥T5.
By (3.21), we know that yi(t)→+∞ as t→+∞, i=1,2, which is a contradiction. This completes the proof.Proof of Theorem2.4.
By Propositions3.2 and 3.3, system (1.1) is uniform weak persistent [2]. From   [12, Propositions 3.1 and Theorem 1.3.3], system (1.1) is permanent. This completes the proof of Theorem 2.4.Proof of Theorem2.5.
The sufficiency of Theorem2.5 now follows from Theorem 2.4. We thus only need to prove the necessity of Theorem 2.5. Suppose that
(3.38)Aω(-g1(t)+h1(t)(x1*(t))2k1(t)+(x1*(t))2)≤0,Aω(-g2(t)+h2(t)x1*(t)k2(t)+m(t)x1*(t)+h3(t)(x2*(t))2k3(t)+(x2*(t))2)≤0.
We will show that
(3.39)limt→+∞yi(t)=0,i=1,2.
In fact, by (3.38), we know that, for any given 0<ɛ<1, there exist ɛ1>0 and ɛ0>0 such that
(3.40)Aω(-g1(t)+h1(t)(x1*(t)+ɛ1)2k1(t)+(x1*(t)+ɛ1)2-q1(t)ɛ)≤-ɛ2Aω(q1(t))≤-ɛ0,Aω(-g2(t)+h2(t)(x1*(t)+ɛ1)k2(t)+m(t)(x1*(t)+ɛ1)+n(t)ɛ+h3(t)(x2*(t)+ɛ1)2k3(t)+(x2*(t)+ɛ1)2-q2(t)ɛ)≤-ɛ2Aω(q2(t))≤-ɛ0.
Note that qi(t)>0 for t≥0. Since
(3.41)ẋ1(t)≤a(t)x2(t)-b(t)x1(t)-d(t)x12(t),ẋ2(t)≤c(t)x1(t)-f(t)x12(t),
we know that, for the given ɛ1, there exists T(1)>0 such that
(3.42)xi(t)≤xi*(t)+ɛ1,t≥T(1),i=1,2.
By (3.40), (1.2), and (3.42), we have
(3.43)Aω(-g1(t)+h1(t)x12(t)k1(t)+x12(t)-q1(t)ɛ)≤-ɛ0,t≥T(1),Aω(-g2(t)+h2(t)x1(t)k2(t)+m(t)x1(t)+n(t)ɛ+h3(t)x22(t)k3(t)+x22(t)-q2(t)ɛ)≤-ɛ0,t≥T(1).
We now show that there must exist T(2)(>T(1)) such that yi(T(2))<ɛ. Otherwise, by (3.43), we have
(3.44)ɛ≤y1(t)≤y1(T(1))exp{∫T(1)t[-g1(s)+h1(s)(x1(s))2k1(s)+(x1(s))2-q1(s)ɛ]ds}→0,t→+∞,ɛ≤y2(t)≤y2(T(1))exp{∫T(1)t[-g2(s)+h2(s)x1(s)k2(s)+m(s)x1(s)+n(s)ɛ+h3(s)x22(s)k3(s)+x22(s)-q2(s)ɛ]ds}→0,t→+∞.
This implies ɛ≤0, which is a contradiction.
Let(3.45)M1(ɛ)=max0≤s≤ω{-g1(s)+h1(s)(x1(s))2k1(s)+(x1(s))2-q1(s)ɛ},M2(ɛ)=max0≤s≤ω{-g2(s)+h2(s)x1(s)k2(s)+m(s)x1(s)+n(s)ɛ+h3(s)x22(s)k3(s)+x22(s)-q2(s)ɛ}.
We know that Mi(ɛ) is bounded (Proposition 3.1 given). We now show that
(3.46)yi(t)≤ɛexp{Mi(ɛ)ω},t≥T(2),i=1,2.
Otherwise, there exists T(3)>T(2) such that
(3.47)yi(T(3))>εexp{Mi(ɛ)ω}.
By the continuity of yi(t), there must exist T(4)∈(T(2),T(3)) such that yi(T(4))=ɛ and yi(t)>ɛ for t∈(T(4),T(3)]. Let P1 be the nonnegative integer such that T(3)∈(T(4)+P1ω,T(4)+(P1+1)ω]; by (3.43) we have
(3.48)ɛexp{M1(ɛ)ω}<y1(T(3))<y1(T(4))exp{∫T(4)T(3)[-g1(t)+h1(t)(x1(t))2k1(t)+(x1(t))2-q1(t)ɛ]dt}=ɛexp{∫T(4)T(4)+P1ω+∫T(4)+P1ωT(3)}[-g1(t)+h1(t)(x1(t))2k1(t)+(x1(t))2-q1(t)ɛ]dt≤ɛexp{∫T(4)+P1ωT(3)[-g1(t)+h1(t)(x1(t))2k1(t)+(x1(t))2-q1(t)ɛ]dt}<ɛexp{M1(ɛ)ω},ɛexp{M2(ɛ)ω}<y2(T(3))<y2(T(4))exp{∫T(4)T(3)[-g2(t)+h2(t)x1(t)k2(t)+m(t)x1(t)+n(t)ɛ+h3(t)x22(t)k3(t)+x22(t)-q2(t)ɛ]dt}=ɛexp{∫T(4)T(4)+P1ω+∫T(4)+P1ωT(3)}[-g2(t)+h2(t)x1(t)k2(t)+m(t)x1(t)+n(t)ɛ+h3(t)x22(t)k3(t)+x22(t)-q2(t)ɛ]dt≤ɛexp{∫T(4)+P1ωT(3)[-g2(t)+h2(t)x1(t)k2(t)+m(t)x1(t)+n(t)ɛ+h3(t)x22(t)k3(t)+x22(t)-q2(t)ɛ]dt}<ɛexp{M2(ɛ)ω},
which is a contradiction. This implies that  (3.46) holds. We then conclude, by the arbitrariness of ɛ, that yi(t)→0 as t→+∞, i=1,2. This completes the proof of Theorem 2.5.



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*Source: 161978-2010-03-17.xml*