# Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

Authors: Can-Yun Huang; Min Zhao; Hai-Feng Huo
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.


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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 Beddington-DeAngelis and Holling II functional
response:(1.1)x1′(t)=a(t)x2(t)−b(t)x1(t)−d(t)x12(t)−p1(t)x1(t)k1(t)+m(t)y1(t)+n(t)x1(t)y1(t),x2′(t)=c(t)x1(t)−f(t)x22(t)−p2(t)x2(t)k2(t)+x2(t)y2(t),y1′(t)=y1(t)[−g1(t)+h1(t)x1(t)k1(t)+m(t)y1(t)+n(t)x1(t)−q1(t)y1(t)],y2′(t)=y2(t)[−g2(t)+h2(t)x2(t)k2(t)+x2(t)−q2(t)y2(t)],where a(t),b(t),c(t),d(t),f(t),gi(t),hi(t),ki(t),m(t),n(t),pi(t),
and qi(t)(i=1,2) are all continuous positive ω-periodic functions. Here, 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, and y2(t) represents the density of the other predator
that preys on mature prey at time t.The birth rate into the immature population is given
bya(t)x2(t), 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, that is, it is proportional to the square
of the population with a proportionality f(t). The transition rate from the immature
individuals to the mature individuals is assumed to be proportional to the
existing immature population, with a proportionality coefficient c(t). Similarly, −g1(t)y1(t)−q1(t)y12(t) and −g2(t)y2(t)−q2(t)y22(t) give the density dependent death rate of the
two predators, respectively. p1(t) and p2(t) are the capturing rate of the two predators,
respectively. h1(t)/p1(t) and h2(t)/p2(t) are the rate of conversion of nutrients into
the reproduction of the two mature predators, respectively.The functional response of predator speciesy1(t) to immature prey species takes the
Beddington-DeAngelis form, that is, x1(t)/(k1(t)+m(t)y1(t)+n(t)x1(t)). It was introduced by Beddington [1] and DeAngelis et al. [2] independently in 1975. It is
similar to the well-known Holling type II functional response but has an extra
term m(t)y1(t) in the denominator which models mutual
interference between predators. The Beddington-DeAngelis form of functional
response has some of the same qualitative features as the ratio-dependent
models form but avoids some of the same behaviors of ratio-dependent models at
low densities which have been the source of controversy. The function x2(t)/(k2(t)+x2(t)) represents the functional response of predator y2(t) to mature prey, which is called Holling type II function or Michaelis-Menten function. Holling type II is the second
function that 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.Cui and Song [3] proposed the following predator-prey model with stage
structure for prey:(1.2)x1′(t)=a(t)x2(t)−b(t)x1(t)−d(t)x12(t)−p(t)x1(t)y(t),x2′(t)=c(t)x1(t)−f(t)x22(t),y′(t)=y(t)(−g(t)+h(t)x1(t)−q(t)y(t)).They obtained a set of sufficient and necessary
conditions which guarantee the permanence of the
system. For more back ground and the relevant work on system (1.2), one could
refer to [3–6] and the references cited therein. Recently, Chen
[7, 8] and Yang [9] consider the functional
response of the predator to immature prey species. Lin and Hong [10] consider a biological model
for two predators and one prey with periodic delays.In reality, mature prey was also consumed by some
predators. Different predator usually consumes prey in different stage
structure. Some predators only prey on immature prey, and some predators only prey
on mature prey. There is different functional response in different predator.
So, we add a predator species which consumes mature prey to the model (1.2). By
assuming that one predator consumes immature prey according to the
Beddington-DeAngelis functional response while the other predator consumes
mature prey according to Holling II functional response, we get model (1.1).
In the resource limited environment, could the wild animals be coexistence for
long-term under the animals' law of the jungle? To
keep the biology's variety of the nature, the permanence of biotic population
is a significant and comprehensive problem in biomathematics. So, it is
meaningful to investigate the permanence of the model (1.1).The aim of this paper is, by further developing the
analysis technique of Cui [3, 11], 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. In
Section 4, we give an example which shows the feasibility of our result. The
last section is devoted to make some explanation on the biological meaning of
our result.Throughout this paper, for a continuousω-periodic function f(t), we set(1.3)Aω(f)=1ω∫0ωf(t)dt.


## 2. Main Result

In this section, we introduce a definition and some
lemmas which will be useful in subsequent sections and state the main result.Definition 2.1.
System (1.1) is said to be permanent if there exist positive
constants  m,M, and  T0,   such that each positive solution  (x1(t),x2(t),y1(t),y2(t))
of the system (1.1) with any
positive initial value  φ   fulfills  m≤xi(t)≤M,m≤yi(t)≤M,i=1,2   for all  t≥T0,  where  T0  may depend on  φ.Lemma 2.2 (see [12]).
Ifa(t),  b(t),c(t),  d(t),  and  f(t)  are all  ω-periodic, then system (2.1)x1′(t)=a(t)x2(t)−b(t)x1(t)−d(t)x12(t),x2′(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 [13]).
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 (2.2)x′=x(b(t)−a(t)x)has a positive ω-periodic solution which is globally
asymptotically stable.Now, we state the main result of this paper.Theorem 2.4.
System (1.1) is permanent if and only if (2.3)Aω(−g1(t)+h1(t)x1*(t)k1(t)+n(t)x1*(t))>0,Aω(−g2(t)+h2(t)x2*(t)k2(t)+x2*(t))>0,  where (x1*(t),x2*(t))
is the unique positive periodic solution of
system (2.1) given by Lemma 2.2.


## 3. Proof of the Main Result

We need the following propositions to prove Theorem2.4. The hypothesis of the lemmas and theorem of the preceding section is assumed to hold in what follows.Proposition 3.1.
There exist positive constantsMx  and  My  such that (3.1)lim⁡t→+∞sup⁡xi(t)≤Mx,lim⁡t→+∞sup⁡yi(t)≤My,i=1,2 for all solution of system
(1.1) with positive initial values.Proof.
Obviously,R+4={(x1,x2,y1,y2)∣xi>0,yi>0} is a positively invariant set of system (1.1).
Given any solution (x1,x2,y1,y2) of system (1.1), we have(3.2)x1′(t)≤a(t)x2(t)−b(t)x1(t)−d(t)x12(t),x2′(t)≤c(t)x1(t)−f(t)x22(t).By Lemma 2.2, the following
auxiliary equation:(3.3)u1′(t)=a(t)u2(t)−b(t)u1(t)−d(t)u12(t),u2′(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 (u1(0),u2(0))=(x1(0),x2(0)). By comparison theorem, we then
have(3.4)xi(t)≤ui(t),i=1,2,for t⩾0. By (2.3), we can choose positive ε>0 small enough such that(3.5)Aω(−gi(t)+hi(t)(xi*(t)+ε)ki(t))>0.Thus, from the global
attractivity of (x1*(t),x2*(t)), for the above given ε>0, there exists a T0>0, such that(3.6)|ui(t)−xi*(t)|<ε,t≥T0.Inequality (3.4) combined with (3.6) leads
to(3.7)xi(t)<xi*(t)+ε,t>T0.Let Mx=max0≤t≤ω{xi*(t)+ε,i=1,2}, we have(3.8)lim⁡t→+∞sup⁡xi(t)≤Mx.In addition, for t≥T0, from the third and fourth equations of (1.1) and (3.7) we get(3.9)yi′(t)≤yi(t)[−gi(t)+hi(t)xi(t)ki(t)−qi(t)yi(t)]≤yi(t)[−gi(t)+hi(t)(xi*(t)+ε)ki(t)−qi(t)yi(t)].Consider the following auxiliary
equation:(3.10)vi′(t)=vi(t)[−gi(t)+hi(t)(xi*(t)+ε)ki(t)−qi(t)vi(t)].It follows from (3.5) and Lemma 2.3 that (3.10) has a unique positive ω-periodic solution yi*(t)>0 which is globally asymptotically stable. Similar
to the above analysis, there exists a T1>T0 such that for the above ε, one has(3.11)yi(t)<yi*(t)+ε,t≥T1.Let My=max0≤t≤ω{yi*(t)+ε,i=1,2},
then we have(3.12)lim⁡t→+∞sup⁡yi(t)≤My,i=1,2.This completes the proof of  Proposition 3.1.Proposition 3.2.
There exist positive constantsδix<Mx,  i=1,2,  such that (3.13)lim⁡t→+∞inf⁡xi(t)≥δix,i=1,2.Proof.
By
Proposition3.1, there exists T1>0 such that(3.14)0<xi(t)≤Mx,0<yi(t)≤My,t≥T1.Hence, from the first and second
equations of system (1.1), we have(3.15)x1′(t)≥a(t)x2(t)−(b(t)+p1(t)k1(t)My)x1(t)−d(t)x12(t),x2′(t)≥c(t)x1(t)−(f(t)+p2(t)k2(t)My)x22(t),for t≥T1. By Lemma 2.2, the following auxiliary
equation:(3.16)u1′(t)=a(t)u2(t)−(b(t)+p1(t)k1(t)My)u1(t)−d(t)u12(t),u2′(t)=c(t)u1(t)−(f(t)+p2(t)k2(t)My)u22(t)has a globally asymptotically
stable positive ω-periodic solution (x∼1*(t),x∼2*(t)). Let (u1(t),u2(t)) be the solution of (3.16) with (u1(T1),u2(T2))=(x1(T1),x2(T2)), by comparison theorem, we have(3.17)xi(t)≥ui(t)(i=1,2),t>T1.Thus, from the global
attractivity of (x∼1*(t),x∼2*(t)), there exists a T2>T1, such that(3.18)|ui(t)−x∼i*(t)|<x∼i*(t)2(i=1,2),t>T2.Inequality (3.18) combined with (3.17) leads
to(3.19)xi(t)>δix=min⁡0≤t≤ω{x∼i*(t)2}(i=1,2),t>T2.And so(3.20)lim⁡t→+∞inf⁡xi(t)≥δix,i=1,2.The proof of Proposition 3.2 is
complete.Proposition 3.3.
Suppose that (2.3) holds, then there exist positive constants δiy,  i=1,2, such that any solution  (x1(t),x2(t),y1(t),y2(t))
of system (1.1) with positive initial value
satisfies (3.21)lim⁡t→+∞sup⁡yi(t)≥δiy,i=1,2.Proof.
By Assumption (2.3), we can choose constant ε0>0 (without loss of generality, we may assume
that ε0<(1/2)min0≤t≤ω{xi*(t)}, where (x1*(t),x2*(t)) is the unique positive periodic solution of
system (2.1)) such that(3.22)Aω(φε0(t))>0,Aω(ψε0(t))>0,where(3.23)φε0(t)=−g1(t)+h1(t)(x1*(t)−ε0)k1(t)+m(t)ε0+n(t)(x1*(t)−ε0)−q1(t)ε0,ψε0(t)=−g2(t)+h2(t)(x2*(t)−ε0)k2(t)+(x2*(t)−ε0)−q2(t)ε0.Consider the following
equations with a parameter β>0:  (3.24)x1′(t)=a(t)x2(t)−(b(t)+2βp1(t)k1(t))x1(t)−d(t)x12(t),x2′(t)=c(t)x1(t)−(f(t)+2βp2(t)k2(t))x22(t).By Lemma 2.2, the system (3.24)
has a unique positive ω-periodic solution (x1β*(t),x2β*(t)), which is globally attractive. Let (x¯1β(t),x¯2β(t)) be the solution of (3.24) with initial
condition x¯iβ(0)=xi*(0),i=1,2. Hence, for above ε0, there exists a sufficiently large T3>T2 such that(3.25)|x¯iβ(t)−xiβ*(t)|<ε04(i=1,2),t>T3.By the continuity of the
solution in the parameter, we have x¯iβ(t)→xi*(t) uniformly in [T3,T3+ω] as β→0. Hence, for ε0>0, there exists a β0=β0(ε0)>0 such that(3.26)|x¯iβ(t)−xi*(t)|<ε04(i=1,2),t∈[T3,T3+ω],0<β<β0.So, we have(3.27)|xiβ*(t)−xi*(t)|≤|x¯iβ(t)−xiβ*(t)|+|x¯iβ(t)−xi*(t)|<ε02,t∈[T3,T3+ω].Since xiβ*(t) and xi*(t) are all ω-periodic, we have(3.28)|xiβ*(t)−xi*(t)|<ε02(i=1,2),t≥0,0<β<β0.Choosing a constant β1(0<β1<β0,2β1<ε0), we have(3.29)xiβ1*(t)≥xi*(t)−ε02(i=1,2),t≥0.Suppose that Conclusion (3.21)
is not true. Then, there exists F∈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))=F, we have(3.30)lim⁡t→+∞sup⁡yi(t)<β1,i=1,2.So, there exists T4>T3 such that(3.31)yi(t)<2β1<ε0,t≥T4.By applying (3.31), from the
first and second equations of system (1.1) it follows that for all t≥T4,(3.32)x1′(t)≥a(t)x2(t)−(b(t)+2β1p1(t)k1(t))x1(t)−d(t)x12(t),x2′(t)≥c(t)x1(t)−(f(t)+2β1p2(t)k2(t))x22(t).Let (u1(t),u2(t)) be the solution of (3.24) with β=β1 and ui(T4)=xi(T4),i=1,2, then(3.33)xi(t)≥ui(t)(i=1,2),t≥T4.By the global asymptotic
stability of (x1β1*(t),x2β2*(t)), for the given ε=ε0/2, there exists T5≥T4, such that(3.34)|ui(t)−xiβ1*(t)|<ε02(i=1,2),t≥T5.So,(3.35)xi(t)≥ui(t)>xiβ1*(t)−ε02(i=1,2),t≥T5,and hence, by using (3.29), we
get(3.36)xi(t)>xi*(t)−ε0(i=1,2),t≥T5.Therefore, by (3.31) and (3.36),
we have(3.37)y1′(t)≥y1(t)(−g1(t)+h1(t)(x1*(t)−ε0)k1(t)+m(t)ε0+n(t)(x1*(t)−ε0)−q1(t)ε0)=φε0(t)y1(t),y2′(t)≥y2(t)(−g2(t)+h2(t)(x2*(t)−ε0)k2(t)+(x2*(t)−ε0)−q2(t)ε0)=ψε0(t)y2(t),for t≥T5. Integrating (3.37) from T5 to t yields(3.38)y1(t)≥y1(T5)exp⁡{∫T5tφε0(t)dt},y2(t)≥y2(T5)exp⁡{∫T5tψε0(t)dt}.Thus, from (3.22) we know that φε0(t)>0,ψε0(t)>0. It follows that y1(t)→+∞,y2(t)→+∞ as t→+∞. It is a contradiction. This completes the
proof.Proposition 3.4.
Suppose that (2.3) holds, then there exist positive constants  ηiy,  i=1,2,  such that any solution  (x1(t),x2(t),y1(t),y2(t))
of system (1.1) with positive initial value
satisfies (3.39)lim⁡t→+∞inf⁡yi(t)>ηiy,i=1,2.Proof.
Suppose that (3.39) is not true, then there exists a sequence {ξm}∈R+4, such that(3.40)lim⁡t→+∞inf⁡yi(t,ξm)<δiy(m+1)2,m=1,2,….On the other hand, by
Proposition 3.3, we have(3.41)lim⁡t→+∞sup⁡yi(t,ξm)>δiy,m=1,2,….Hence, there are time sequences {sq(m)} and {tq(m)} satisfying(3.42)0<s1(m)<t1(m)<s2(m)<t2(m)<⋯<sq(m)<tq(m)<⋯,sq(m)⟶+∞,tq(m)⟶+∞asq⟶+∞,yi(sq(m),ξm)=δiym+1,yi(tq(m),ξm)=δiy(m+1)2,δiy(m+1)2<yi(t,ξm)<δiym+1,t∈(sq(m),tq(m)).By Proposition 3.1, for a given
positive integer m, there is a T1(m)>0, such that for all t>T1(m)(3.43)xi(t,ξm)<Mx,yi(t,ξm)<My,i=1,2.Because of sq(m)→+∞ as q→+∞, there is a positive integer Z(m), such that sq(m)>T1(m) as q≥Z(m), hence(3.44)yi′(t,ξm)≥yi(t,ξm)(−gi(t)−qi(t)My)for t∈[sq(m),tq(m)],q≥Z(m). Integrating (3.44) from sq(m) to tq(m) yields(3.45)yi(tq(m),ξm)≥yi(sq(m),ξm)exp⁡{∫sq(m)tq(m)(−gi(t)−qi(t)My)dt},or(3.46)∫sq(m)tq(m)(gi(t)+qi(t)My)dt≥ln⁡(m+1)forq≥Z(m).Thus, from the boundedness of gi(t)+qi(t)My, we have(3.47)tq(m)−sq(m)⟶+∞asm⟶+∞,q≥Z(m).By (3.22) and (3.47), there are
constants P>0 and N0>0, such that(3.48)δiym+1<β1<ε0,tq(m)−sq(m)>2P,(3.49)∫0aφε0(t)dt>0,∫0aψε0(t)dt>0,for m≥N0,q≥Z(m),
and a≥P. Inequality (3.48)
implies that(3.50)yi(t,ξm)<β1<ε0,t∈[sq(m),tq(m)],for m≥N0,q≥Z(m). In addition, from (3.43) and (3.50) we have(3.51)x1′(t,ξm)≥a(t)x2(t,ξm)−(b(t)+2p1(t)β1k1(t))x1(t,ξm)−d(t)x12(t,ξm),x2′(t,ξm)≥c(t)x1(t,ξm)−(f(t)+2p2(t)β1k2(t))x22(t,ξm),for t∈[sq(m),tq(m)]. Let (u1(t),u2(t)) be the solution of (3.24) with β=β1 and ui(sq(m))=xi(sq(m),ξm), then by applying comparison theorem, we
have(3.52)xi(t,ξm)≥ui(t),t∈[sq(m),tq(m)].Further, by using Propositions 3.1 and 3.2, there exists an enough large Z1(m)>Z(m) such that(3.53)ηix<xi(sq(m),ξm)<Mx,for q≥Z1(m). For β=β1, (3.24) has a unique positive ω-periodic solution (x1β1*(t),x2β1*(t)) which is globally asymptotically stable. In
addition, by the periodicity of (3.24), the periodic solution (x1β1*(t),x2β1*(t)) is uniformly asymptotically stable with
respect to the compact set Ω={x∣ηix<x<Mx}. Hence, for given ε0 in Proposition 3.3, there exists T0>P, which is independent of m and q, such that(3.54)ui(t)>xiβ1*(t)−ε02,i=1,2ast>T0+sq(m).Thus, by using (3.29), we
get(3.55)ui(t)>xi*(t)−ε0,i=1,2ast>T0+sq(m).By (3.47), there exists a
positive integer N1≥N0 such that tq(m)>sq(m)+2T0>sq(m)+2P for m≥N1 and q≥Z1(m). So, we have(3.56)xi(t,ξm)≥xi*(t)−ε0,i=1,2ast∈[T0+sq(m),tq(m)],where m≥N1 and q≥Z1(m). Hence, by using (3.50) and (3.56), from the third and fourth equations of
system (1.1), we have(3.57)y1′(t,ξm)≥φε0(t)y1(t,ξm),y2′(t,ξm)≥ψε0(t)y2(t,ξm),t∈[T0+sq(m),tq(m)].Integrating the above
inequalities from T0+sq(m) to tq(m), we have(3.58)y1(tq(m),ξm)≥y1(T0+sq(m),ξm)exp⁡{∫T0+sq(m)tq(m)φε0(t)dt},y2(tq(m),ξm)≥y2(T0+sq(m),ξm)exp⁡{∫T0+sq(m)tq(m)ψε0(t)dt},that is(3.59)δ1y(m+1)2≥δ1y(m+1)2exp⁡{∫T0+sq(m)tq(m)φε0(t)dt}>δ1y(m+1)2,δ2y(m+1)2≥δ2y(m+1)2exp⁡{∫T0+sq(m)tq(m)ψε0(t)dt}>δ2y(m+1)2.These are contradictions. This
completes the proof of Proposition 3.4.Proof of Theorem2.4.
The sufficiency of Theorem2.4 now follows from Propositions 3.1–3.4. We thus only need to prove the necessity of Theorem 2.4. Suppose
that(3.60)Aω(−g1(t)+h1(t)x1*(t)k1(t)+n(t)x1*(t))≤0,Aω(−g2(t)+h2(t)x2*(t)k2(t)+x2*(t))≤0.We will show
that(3.61)lim⁡t→+∞yi(t)=0,i=1,2.In fact, by (3.60), we know
that, for any given positive constant 0<ε<1, there exist ε1>0, (0<ε1<ε), ε0>0 such that(3.62)Aω(−g1(t)+h1(t)(x1*(t)+ε1)k1(t)+n(t)(x1*(t)+ε1)−q1(t)ε)≤−ε2Aω(q1(t))<−ε0,Aω(−g2(t)+h2(t)(x2*(t)+ε1)k2(t)+(x2*(t)+ε1)−q2(t)ε)≤−ε2Aω(q2(t))<−ε0.Since(3.63)x1′(t)≤a(t)x2(t)−b(t)x1(t)−d(t)x12(t),x2′(t)≤c(t)x1(t)−f(t)x22(t).We know that, for above ε1 there exists a T(1)>0 such that(3.64)xi(t)<xi*(t)+ε,t≥T(1).It follows from (3.62) and
(3.64) that for t≥T(1),(3.65)Aω(−g1(t)+h1(t)x1(t)k1(t)+n(t)x1(t)−q1(t)ε)<−ε0,Aω(−g2(t)+h2(t)x2(t)k2(t)+x2(t)−q2(t)ε)<−ε0.
First, we show that there exists aT(2)>T(1) such that yi(T(2))<ε,i=1,2. Otherwise, by (3.65), we have(3.66)ε≤y1(t)≤y1(T(1))exp⁡{∫T(1)t(−g1(s)+h1(s)x1(s)k1(s)+n(s)x1(s)−q1(s)ε)ds}≤y1(T(1))exp⁡{−ε0(t−T(1))}⟶0as t→+∞. Similarly, we have(3.67)ε≤y2(t)≤y2(T(1))exp⁡{−ε0(t−T(1))}⟶0,t⟶+∞,which are contradictions.
Second, we now show that(3.68)yi(t)≤εexp⁡{M(ε)ω},i=1,2,fort≥T(2),where(3.69)M(ε)=max⁡0≤t≤ω{g1(t)+h1(t)(x1*(t)+ε)k1(t)+n(t)(x1*(t)+ε)+q1(t)ε,g2(t)+h2(t)(x2*(t)+ε)k2(t)+(x2*(t)+ε)+q2(t)ε}is a bounded constant for 0<ε<1. Otherwise, there exists a T(3)>T(2) such that(3.70)yi(T(3))>εexp⁡{M(ε)ω},i=1,2.By the continuity of yi(t), there must exist T(4)∈(T(2),T(3)) such that yi(T(4))=ε and yi(t)>ε 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 the first inequality of (3.65), we
have(3.71)εexp⁡{M(ε)ω}<y1(T(3))<y1(T(4))exp⁡{∫T(4)T(3)(−g1(t)+h1(t)x1(t)k1(t)+n(t)x1(t)−q1(t)ε)dt}=εexp⁡{∫T(4)T(4)+P1ω+∫T(4)+P1ωT(3)}(−g1(t)+h1(t)x1(t)k1(t)+n(t)x1(t)−q1(t)ε)dt<εexp⁡{∫T(4)+P1ωT(3)(g1(t)+h1(t)x1(t)k1(t)+n(t)x1(t)+q1(t)ε)dt}<εexp⁡{∫T(4)+P1ωT(3)(g1(t)+h1(t)(x1*(t)+ε)k1(t)+n(t)(x1*(t)+ε)+q1(t)ε)dt}≤εexp⁡{M(ε)ω}.Similarly, by the second
inequality of (3.65), we have(3.72)εexp⁡{M(ε)ω}<y2(T(3))≤εexp⁡{M(ε)ω},which are contradictions. These
imply that (3.68) holds. By the arbitrariness of ε, it immediately follows that yi(t)→0 as t→+∞. This completes the proof of Theorem 2.4.


## 4. Example

Consider the
following predator-prey system:(4.1)x1′(t)=5x2(t)−2x1(t)−x12(t)−(2+sin⁡(t)/200)x1(t)5+y1(t)+x1(t)y1(t),x2′(t)=3x1(t)−x22(t)−(2+sin⁡(t)/100)x2(t)4+x2(t),y1′(t)=y1(t)[−13−sin⁡(t)100+(2+sin⁡(t)/200)x1(t)5+y1(t)+x1(t)−(4+cos⁡(t))y1(t)],y2′(t)=y2(t)[−12−sin⁡(t)100+(2+sin⁡(t)/100)x2(t)4+x2(t)−(3+cos⁡(t))y2(t)].In this case, corresponding to system (1.1), one has a(t)=5, b(t)=2, c(t)=3, d(t)=1, f(t)=1, g1(t)=1/3+sin(t)/100, g2(t)=1/2+sin(t)/100, h1(t)=p1(t)=2+sin(t)/200, h2(t)=p2(t)=2+sin(t)/100, k1(t)=5, k2(t)=4, m(t)=n(t)=1, q1(t)=4+cos(t), q2(t)=3+cos(t).One could easily see that(4.2)x1′(t)=5x2(t)−2x1(t)−x12(t),x2′(t)=3x1(t)−x22(t)has
a unique positive periodic solution (x1*(t),x2*(t))=(3,3), that is, in this case, the positive periodic
solution is the positive equilibrium. By simple computation, one
has(4.3)Aω(−g1(t)+h1(t)x1*(t)k1(t)+n(t)x1*(t))=512>0,Aω(−g2(t)+h2(t)x2*(t)k2(t)+x2*(t))=514>0.Hence, corresponding to Theorem 2.4, we know that system (4.1) is permanent.


## 5. Conclusion

In this paper, a model which describes the
nonautonomous periodic predator-prey system with Beddington-DeAngelis and
Holling II functional response and stage structure for prey is proposed. Under
Assumption (2.3), sufficient and necessary conditions which guarantee the
predator and the prey species to be permanent are obtained.The results of this paper suggest the following
biological implication. Note that(x1*(t),x2*(t)) is the globally asymptotically stable periodic
solution of system (1.1) without predation, which, as showed by Lemma 2.2, always exists. Hence, condition (2.3) implies that if the death rate of the two
predator species is all small enough and the growth
by foraging minus the death for the second predator is sufficiently high, the
system is permanent.



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*Source: 371632-2008-10-08.xml*