# Permanence and Extinction of a Generalized Gause-Type Predator-Prey System with Periodic Coefficients

Authors: Jinhuo Luo
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We study the permanence and extinction of a generalized Gause-type predator-prey system with periodic coefficients. We provide a sufficient and necessary condition to guarantee
the predator and prey species to be permanent and a sufficient condition for the existence of a
periodic solution. In addition we prove that when the predator population tends to extinction,
the prey population keeps oscillating above a positive population level.


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## Body

## 1. Introduction

Permanence of a dynamical system has always been a hot issue in the past few decades. The concept of permanence has been introduced and investigated by several authors, each using his own terminology: “cooperativity” in the earlier papers of Schuster et al. [1], and Hofbauer [2], “permanent coexistence” by Hutson and Vickers [3], “uniform persistence” in Butler et al. [4], and “ecological stability” by Svirezhev and Logofet [5–7] (for more detailed statements of the concept see [8]).Many important results have been found in recent years [1–37]. Some authors (see [21, 28, 31, 33]) have considered the following two species periodic Lotka-Volterra predator-prey system (1.1)ẋ1(t)=x1(t)(b1(t)-a11(t)x1(t))-a12(t)x1(t)x2(t),ẋ2(t)=x2(t)(-b2(t)-a22(t)x2(t))+a21(t)x1(t)x2(t),
where bi(t) and aij(t)(i,j=1,2) are periodic functions on R with common period ω>0 and aij(t)≥0 for all t∈R. They have established sufficient and necessary conditions for the existence of positive ω-periodic solutions of the system by using different methods, respectively. Teng [31] has given sufficient and necessary conditions for the uniform persistence of the system.Cui [16] has considered the permanence of the following Lotka-Volterra predator-prey model with periodic coefficients: (1.2)ẋ=x(a(t)-b(t)x-c(t)yp(t)+x),ẏ=y(-d(t)+e(t)xp(t)+x-f(t)y).
He provided a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In Theorems 2.2 and 2.3 he set a precondition that f(t)≢0. This restricts the application of the theorems more or less, since many researchers often neglect the logistic term in the predator equation when the population level of the predator is relatively low and the competition between predators can be ignored, and it proved to be an unnecessary precondition in our paper. However, the research methods in his work inspired me, and many proofs, especially in the first half of this paper, are analogous to [16].In this paper we consider the permanence of the following generalized Gause-type predator-prey system,(1.3)ẋ=x(f(t,x)-g(t,x)y),ẏ=y(γ(t)g(t,x)x-μ(t)-h(t)y),
where f(t+ω,x)=f(t,x), g(t+ω,x)=g(t,x) for all t and γ(·), μ(·), h(·) are all periodic continuous functions with common period ω>0; γ(·), μ(·) are positive, and h(·) is nonnegative. We emphasize that our model includes the case when h(t)≡0.In the absence of predators, system (1.3) becomes(1.4)ẋ=xf(t,x),
where f is a real-valued function defined on (1.5)R+02={(t,x)∈R2:t≥0,x≥0}.Vance and Coddington [35] have studied system (1.4) and proved the existence of a unique periodic solution under some assumptions. Apart from the assumption we mentioned above that f(·,x) is ω-periodic, the other assumptions with a mild modification are as follows. (A1)
Functionf is continuous and differentiable with respect to x on R+02, and ∂f/∂x is continuous on R+02.(A2)
There are continuous functionsp and λ with p(x)>0 for x>0 and λ(t)≥0 for t≥0, such that
(1.6)∂f∂x≤-p(x)λ(t)fort≥0,x≥0,∫0∞λ(s)ds=∞.(A3)
There exist constantsβ≥0 and K>0, such that
(i)
f(t,K)≤β for t≥0,(ii)
∫tt+ωf(s,K)ds≤0 for t≥0.(A4)
There exist constantsα≥0 and0<δ≤K, such that
(i)
f(t,δ)≥-αfor t≥0,(ii)
∫tt+ωf(s,δ)ds>0 for t≥0.In addition, we assume that theω-periodic function g(·,x) satisfies the following. (A5)
Functiong(t,x) is continuous with respect to t, and G(t,x):=xg(t,x) is strictly monotonely increasing with respect to x,(A6)
g(t,x) is nonnegative and is positive when x>0,(A7)
g(t,x) is bounded on R+02, that is there is a constant Mg such that g(t,x)≤Mg for (t,x)∈R+02.TraditionallyG(t,x) represents a grazing rate. Usually when the amount of prey increases, the grazing rate increases and eventually tends to a maximal value as the prey population tends to infinity. But here we do not emphasize that G(t,x) is bounded because its unnecessary theoretically. Assumption (A5) means that there is a higher capture rate when there is a larger amount of prey. g(t,x) is directly proportional to the mean possibility density for each individual prey being captured, and (A6), (A7) suggest that there is a possibility, but its not definitely for each individual prey being captured. In ecology G(t,x) is called the functional response or grazing function, for example, the Holling-type grazing function:(1.7)Rmxnα+xn,n≥1,
where Rm,α>0 or the generalized form (1.8)Rm(t)xnα(t)+xn,n≥1,
where Rm(t),α(t)>0; the Ivlev grazing function (1.9)Rm(1-e-λx),
where Rm,λ>0. Obviously all of these functions satisfy (A5)–(A7).In this paper we will establish sufficient and necessary conditions for the permanence of system (1.3). In the next section we state our main results. These results are proved in Section 3. Two applications are given in Section 4.


## 2. Main Results

Throughout this paper, we will assume that all the functionsf(·,x), g(·,x), γ(·), μ(·), and h(·) are continuous and periodic with common period ω>0. For any continuous ω-periodic function u(t) defined on R, we denote(2.1)u(t)¯=1ω∫0ωu(t)dt,uM=max⁡t∈[0,ω]u(t),uL=min⁡t∈[0,ω]u(t).
In order to describe our main results, we first introduce a lemma.Lemma 2.1 (see [35]).
Suppose thatf satisfies (A1)–(A4). Then system (1.4) possesses a unique ω-periodic positive solution x*(t) which is globally asymptotically stable with respect to the positive x-axis.Theorem 2.2.
Suppose thatf satisfies (A1)–(A4), g satisfies (A5)–(A7). Then system (1.3) is permanent provided that
(2.2)γ(t)g(t,x*(t))x*(t)-μ(t)¯>0,
where x*(t) is the unique periodic solution of (1.4) given by Lemma 2.1.Theorem 2.3.
Suppose thatf satisfies (A1)–(A4), g satisfies (A5)–(A7), and
(2.3)γ(t)g(t,x*(t))x*(t)-μ(t)¯≤0,
then(i)
limt→∞y(t)=0,(ii)
limt→∞infx(t)>0, for any solution (x(t),y(t)) of system (1.3) with positive initial conditions, where x*(t) is the unique periodic solution of (1.4) given by Lemma 2.1.By Theorems2.2 and 2.3, we have the following corollary.Corollary 2.4.
Suppose thatf satisfies (A1)–(A4), g satisfies (A5)–(A7). Then system (1.3) is permanent if and only if (2.2) holds.Lemma 2.5 (see Theorem 15.5 in [30, 37]).
Consider a periodic system(2.4)ẋ=F(t,x),F(t+ω,x)=F(t,x),ω>0,
where F(t,x)∈C(R×R2,R2) satisfies a local Lipschitz condition. If all solutions of the above system exist in the future and one of them is bounded, then there exists a periodic solution of period ω.By Theorem2.2 and Lemma 2.5, we have the following corollary (see proof in Section 3).Corollary 2.6.
Supposef satisfies (A1)–(A4), g satisfies (A5)–(A7), and condition (2.2) holds. If in addition g(t,x) satisfies a local Lipschitz condition with respect to x, then system (1.3) has a positive ω-periodic solution.


## 3. Proof of Our Main Results

Lemma 3.1 (see [31]).
Ifa(t), b(t) are ω-periodic functions, b(t)≥0, for all t∈Rand b(t)¯>0, then
(3.1)u̇=u(a(t)-b(t)u)
has a unique nonnegative ω-periodic solution u* which is globally asymptotically stable with respect to the positive u-axis. Moreover, if a(t)¯>0, then u*(t)>0, for all t∈R and if a(t)¯≤0, then u*(t)=0.Lemma 3.2.
Suppose thatf satisfies (A1)–(A4), then there exists an ɛ0>0, such that for any 0<ɛ≤ɛ0, system
(3.2)ẋ=x(f(t,x)-ɛ)
possesses an ω-periodic positive solution xɛ*(t) which is globally asymptotically stable with respect to the positive x-axis.Proof.
It suffices to show that iff satisfies (A1)–(A4), then there exists an ɛ0>0 such that for any 0<ɛ≤ɛ0, f̃=f-ɛ satisfies (A1)–(A4). Obviously f̃ satisfies (A1)–(A3). Taking ɛ0=(1/2ω)∫0ωf(s,δ)ds, its easy to see f̃ satisfies (A4).From a comparison theorem (see Theorem 1.1 [37]) and Lemma 2.1, we have the following lemma.Lemma 3.3.
Suppose thatf satisfies (A1)–(A4), x(t) is a solution of system (1.4), and x*(t) is the periodic solution of system (1.4) given by Lemma 2.1. Let v(t) be a right maximal (right minimal) solution of a scalar differential equation v̇=w(t,v) on [t0,+∞)⊂[0,+∞), where w(t,v) is continuous and satisfies w(t,x)≤(≥)xf(t,x) for all (t,x)∈R+02. Then the following conclusions hold.(i)
Ifv(t0)≤(≥)x(t0), then v(t)≤(≥)x(t) for t≥t0.(ii)
For allɛ>0 there exist a τ≥t0 such that v(t)<x*(t)+ɛ(v(t)>x*(t)-ɛ) for t≥τ.Proposition 3.4.
Under assumptions (A1)–(A7), there existMx and My, such that
(3.3)lim⁡t→∞sup⁡x(t)≤Mx,lim⁡t→∞sup⁡y(t)≤My
for all solutions (x(t),y(t)) of system (1.3) with positive initial values.Proof.
Obviously,R+2={(x,y):x≥0,y≥0}is a positively invariant set of system (1.3). Given any solution (x(t),y(t)) of (1.3) with positive initial values, from system (1.3) we have
(3.4)ẋ≤xf(t,x).
The following equation
(3.5)v̇=vf(t,v),
has a globally asymptotically stable positive ω-periodic solution v*(t) by Lemma 2.1. By Lemma 3.3, there exists T1>0, such that
(3.6)x(t)<v*(t)+1,fort>T1.
Let Mx=max0≤t≤ω{v*(t)+1}. We have
(3.7)lim⁡t→∞sup⁡x(t)≤Mx.
By (1.3),
(3.8)γMẋ+ẏ=γMxf(t,x)-(γM-γ(t))g(t,x)xy-μ(t)y-h(t)y2≤γMxf(t,x)-μ(t)y≤γMxf(t,x)-μLy=γMx(f(t,x)+μL)-μL(γMx+y).
Denote w(t)=γMx(t)+y(t). Then we have
(3.9)ẇ+μLw≤γMx(f(t,x)+μL).
Since x(t)≤max0≤t≤ω{v*(t)+1}, fort≥T1, there is a constant B>0, such that
(3.10)γMx(f(t,x)+μL)≤Bfort≥T1,
by the continuity of f. It follows that
(3.11)ẇ+μLw≤B,fort≥T1.
Notice that μL>0. It's easy to show that there is an A>0, such that
(3.12)lim⁡t→∞sup⁡w(t)≤A.
By the notation w=γMx+y, taking Mx=A/γM and My=A, Proposition 3.4 is proved.Proposition 3.5.
Under assumption (A1)–(A7), there exists a positive constantηx, such that
(3.13)lim⁡t→∞sup⁡x(t)≥ηx,
for all solutions (x(t),y(t)) of (1.3) with positive initial values.Proof.
Suppose that (3.13) is not true. Then there is a sequence {zm}⊂R+2, such that
(3.14)lim⁡t→∞sup⁡x(t,zm)<1m,m=1,2,…,
where (x(t,zm),y(t,zm)) is the solution of (1.3) with (x(0,zm),y(0,zm))=zm. By assumption (A5), we can choose sufficiently small positive numbers ɛx<1 and ɛy<1, such that
(3.15)(γ(t)g(t,ɛx)ɛx-μ(t))¯<0,(3.16)ϕɛ(t)̅>0,
where
(3.17)ϕɛ(t)=f(t,ɛx)-Mgɛyexp⁡(αω),α=max⁡0≤t≤ω(μ(t)+γ(t)g(t,ɛx)ɛx+h(t)ɛy).
By (3.14), for the given ɛx>0, there exists a positive integer N0, such that(3.18)lim⁡t→∞sup⁡x(t,zm)<1m<ɛx,m>N0.
For the rest of this proof, we assume that m>N0. Equation (3.18) implies there exists τ1(m)>0, such that
(3.19)x(t,zm)<ɛx,t≥τ1(m),
and further
(3.20)ẏ(t,zm)≤y(t,zm)(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)y(t,zm)),fort≥τ1(m).
By (3.15), and Lemma 3.1, any solution v(t) of the following equation, (3.21)v̇=v(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)v)
with positive initial conditions satisfies
(3.22)lim⁡t→∞v(t)=0.
Hence,
(3.23)lim⁡t→∞y(t,zm)=0
by Lemma 3.3. So there is a τ2(m)>τ1(m), such that
(3.24)y(t,zm)<ɛy,fort≥τ2(m).
It follows that
(3.25)ẋ(t,zm)≥x(t,zm)(f(t,x(t,zm))-g(t,ɛx)ɛy),t≥τ2(m).
By (3.16), the equation
(3.26)ẋ=x(f(t,x(t,zm))-g(t,ɛx)ɛy)
has an ω-periodic positive solution x*(t) which is globally asymptotically stable. Hence,
(3.27)x(t,zm)>x*(t)2,
for sufficiently large t>0 and m>N0, which is a contradiction with (3.14). This completes the proof of Proposition 3.5.Proposition 3.6.
Under assumption (A1)–(A7), there exists a positive constantγx, such that
(3.28)lim⁡t→∞inf⁡x(t)≥γx
for all solutions (x(t),y(t)) of system (1.3) with positive initial values.Proof.
Suppose that (3.28) is not true, then there exists a sequence {zm}⊂R+2 such that
(3.29)lim⁡t→∞inf⁡x(t,zm)<ηx2m2,m=1,2,….
On the other hand, by Proposition 3.5, we have
(3.30)lim⁡t→∞sup⁡x(t,zm)>ηx,m=1,2,….
Hence, there are time sequences {sq(m)} and {tq(m)} satisfying
(3.31)0<s1(m)<t1(m)<s2(m)<t2(m)<⋯<sq(m)<tq(m)<⋯,sq(m)⟶∞,tq(m)⟶∞,asq⟶∞,x(tq(m),zm)=ηxm2<x(t,zm)<ηxm=x(sq(m),zm),t∈(sq(m),tq(m)).
By Proposition 3.4, for a given positive integer m, there is a T(m)>0, such that
(3.32)x(t,zm)≤Mx,y(t,zm)≤My,fort≥T1(m).
Because of sq(M)→∞ as q→∞, there is a positive integer K(m) such that sq(m)>T1(m) since q≥K(m), and hence
(3.33)ẋ≥x(t,zm)(f(t,Mx)-MgMy):=ζ(t)x(t,zm)
for q≥K(m). Integrating (3.33) from sq(m) to tq(m) yields
(3.34)x(tq(m),zm)≥x(sq(m),zm)exp⁡∫sq(m)tq(m)ζ(t)dt
or
(3.35)-∫sq(m)tq(m)ζ(t)dt≥ln⁡m,forq≥K(m).
If ζ(t)¯≥0, it leads to a contradiction. Otherwise, ζ(t)¯<0, we have
(3.36)tq(m)-sq(m)⟶∞,asm⟶∞,q≥K(m),
since ζ(t) is bounded. By (3.15) and (3.16), there are constants P>0 and N0>0, such that
(3.37)ηxm<ɛx,tq(m)-sq(m)>2P,(3.38)Myexp⁡∫0P(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)ɛy)dt<ɛy,∫0Pϕɛ(t)dt>0
for m≥N0, q≥K(m), and a≥P. Equation (3.37) implies
(3.39)x(t,zm)<ɛx,t∈[sq(m),tq(m)]
for m≥N0, q≥K(m). For the positive ɛy satisfying (3.16) and (3.38), we have the following two cases:(i)
y(t,zm)≥ɛy for all t∈[sq(m),sq(m)+P];(ii)
there existsτq1(m)∈[sq(m),sq(m)+P], such that y(τq1(m),zm)<ɛy.If (i) holds, by (3.38) and (3.39) we have
(3.40)ɛy≤y(sq(m)+P,zm)≤y(sq(m),zm)exp⁡∫sq(m)sq(m)+P(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)ɛy)dt≤Myexp⁡∫0P(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)ɛy)dt<ɛy.
This is a contradiction.
If (ii) holds, we now claim that(3.41)y(t,zm)≤ɛyexp⁡(αω),t∈(τq1(m),tq(m)].
Otherwise, there exists τq2(m)∈(τq1(m),tq(m)] such that
(3.42)y(τq2(m),zm)>ɛyexp⁡(αω).
By the continuity ofy(t,zm), there must exist τq3(m)∈(τq1(m),τq2(m)) such that(3.43)y(τq3(m),zm)=ɛy,y(t,zm)>ɛy,fort∈(τq3(m),τq2(m)).
Denote P(m) as the nonnegative integer, such that τq2(m)∈(τq3(m)+P(m)ω,τq3(m)+(P(m)+1)ω]. By (3.15), we have
(3.44)ɛyexp⁡(αω)<y(τq2(m),zm)<y(τq3(m),zm)exp⁡∫τq3(m)τq2(m)(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)ɛy)dt=ɛyexp⁡{∫τq3(m)τq3(m)+P(m)ω+∫τq3(m)+P(m)ωτq2(m)}(γ(t)g(t,ɛx)ɛx-μ(t)-h(t)ɛy)dt<ɛyexp⁡(αω).
This contradiction establishes that (3.41) is true, particularly (3.41) holds for t∈[sq(m)+P,tq(m)].
By (3.31) and (3.38), we have(3.45)ηxm2=x(tq(m),zm)≥x(sq(m)+P,zm)exp⁡∫sq(m)+Ptq(m)(f(t,ɛx)-Mgɛyexp⁡(αω))dt=x(sq(m)+P,zm)exp⁡∫sq(m)+Pτq(m)ϕɛ(t)dt>ηxm2,
which is also a contradiction. This completes the proof of Proposition 3.6.Proposition 3.7.
Supposef satisfies (A1)–(A4), g satisfies (A5)–(A7), and (2.2) holds. Then there exists a positive constant ηy such that
(3.46)lim⁡t→∞sup⁡y(t)>ηy
for all solutions (x(t),y(t)) of (1.3) with positive initial values.Proof.
By assumption (A5) and (2.2) we can choose a constant ɛ1>0 such that
(3.47)ψɛ1(t)¯>0,
where
(3.48)ψɛ1(t)=γ(t)g(t,x*(t)-ɛ1)(x*(t)-ɛ1)-μ(t)-h(t)ɛ1.
Consider the following equation with positive parameter α:
(3.49)ẋ=x(f(t,x)-2Mgα).
By Lemma 3.2, (3.49) has a unique positive ω-periodic solution xα*(t) which is globally asymptotically stable since α<ɛ0/(2Mg). Let xα(t) be the solution of (3.49) with initial condition xα(0)=x*(0) in which x*(t) is the unique periodic solution of (1.4) given by Lemma 2.1. Hence, for the above ɛ1, there exists sufficiently large T2>T1, such that
(3.50)|xα(t)-xα*(t)|<ɛ14,fort≥T2.
By the continuity of the solution in the parameter, we have xα(t)→x*(t) uniformly in [T2,T2+ω] as α→0. Hence, for ɛ1>0 there exists α0>0, such that
(3.51)|xα(t)-x*(t)|<ɛ14,fort∈[T2,T2+ω],0<α<α0.
So, we have
(3.52)|xα*(t)-x*(t)|<ɛ12,t∈[T2,T2+ω].
Notice that xα(t) and x*(t) are all ω-periodic, hence
(3.53)|xα*(t)-x*(t)|<ɛ12,t≥0,0<α<α0.
Choosing a constant α1(0<α1<α0,2α1<ɛ1), we have
(3.54)xα1*(t)≥x*(t)-ɛ12,t≥0.
Suppose that (3.46) is not true. Then there exists z∈R+2, such that(3.55)lim⁡t→∞sup⁡y(t,z)<α1,
where (x(t,z),y(t,z)) is the solution of (1.3) with (x(0,z),y(0,z))=z. So, there exist T3≥T2, such that
(3.56)y(t,z)<2α1<ɛ1,t≥T3
and hence
(3.57)ẋ≥x(t,z)(f(t,x)-2Mgα1).
Let u(t) be the solution of (3.49) with α=α1 and u(T3)=x(T3,z), then
(3.58)x(t,z)≥u(t),t≥T3.
By the global asymptotic stability of xα1*(t), for the given ɛ=ɛ1/2, there exists T4≥T3, such that
(3.59)|u(t)-xα1*(t)|<ɛ12,t≥T4.
Hence
(3.60)x(t,z)≥u(t)>xα1*(t)-ɛ12,t≥T4
and hence
(3.61)x(t,z)>x*(t)-ɛ1,t≥T4
from (3.54). This implies
(3.62)ẏ(t,z)≥y(t,z)(γ(t)g(t,x*(t)-ɛ1)(x*-ɛ1)-μ(t)-h(t)ɛ1)=ψɛ1y(t,z),t≥T4.
Integrating the above inequality from T4 to t yields
(3.63)y(t,z)≥y(T4,z)exp⁡∫T4tψɛ1(t)dt.y(t,z)→∞ as t→∞ from (3.47) which is a contradiction. This completes the proof of Proposition 3.7.Proposition 3.8.
Supposef satisfies (A1)–(A4), g satisfies (A5)–(A7), and (2.2) holds. Then there exists a positive constant γy, such that
(3.64)lim⁡t→∞inf⁡y(t)≥γy
for all solutions (x(t),y(t)) of (1.3) with positive initial values.Proof.
Otherwise, there exists a sequence{zm}⊂R+2, such that
(3.65)lim⁡t→∞inf⁡y(t,zm)<ηy(m+1)2,m=1,2,….
However
(3.66)lim⁡t→∞sup⁡y(t,zm)>ηy,m=1,2,…,
from Proposition 3.7. Hence there are two time sequence {sq(m)} and {tq(m)} satisfying the following conditions:
(3.67)0<s1(m)<t1(m)<s2(m)<t1(m)<⋯<sq(m)<tq(m)<⋯,sq(m)⟶∞,tq(m)⟶∞,asq⟶∞,y(tq(m),zm)=ηy(m+1)2<y(t,zm)<ηym+1=y(sq(m),zm),t∈(sq(m),tq(m)).
By Proposition 3.4, for a given integer m>0 there is a T1(m)>0, such that
(3.68)y(t,zm)≤My,fort≥T1(m).
Because of sq(m)→∞ as q→∞, there is a positive integer K(m), such that sq(m)>T1(m) as q≥K(m), and hence
(3.69)ẏ(t,zm)≥y(t,zm)(-μ(t)-h(t)My)
for q≥K(m) and t∈[sq(m),tq(m)]. Integrating the above inequality from s(m) to tq(m), we have
(3.70)y(tq(m),zm)≥y(sq(m),zm)exp⁡∫sq(m)tq(m)(-μ(t)-h(t)My)dt
or
(3.71)∫sq(m)tq(m)(μ(t)+h(t)My)dt≥ln⁡(m+1),q≥K(m).
Because of the boundedness of the functionμ(t)+h(t)My, we know that(3.72)tq(m)-sq(m)⟶∞,asm⟶∞,q≥K(m).
By (3.47), there is a constant P>0 and a positive integer N0 such that
(3.73)ηym+1<α1<ɛ1,tq(m)-sq(m)>2P,(3.74)∫0qψɛ1(t)dt>0
for m≥N0, q≥K(m) and a≥P. Further,
(3.75)y(t,zm)<α1<ɛ1,t∈[sq(m),tq(m)],m≥N0,q≥K(m).
In addition,
(3.76)ẋ(t,zm)≥x(t,zm)(f(t,x(t,zm))-Mgα1).
Let u(t) be the solution of (3.49) with α=α1 and u(sq(m))=x(sq(m),zm). Then
(3.77)x(t,zm)≥u(t),t∈[sq(m),tq(m)]
by Lemma 3.3. Further, by Propositions 3.4 and 3.6, we can choose K1(m)>K(m) such that
(3.78)γx≤x(sq(m),zm)≤Mx
for q≥K1(m). For α=α1, (3.49) has a unique positive ω-periodic solution xα1*(t) which is globally asymptotically stable. In addition, by the periodicity of (3.49), the periodic solution xα1(t) is uniformly asymptotically stable with respect to the compact set Ω={x:γx≤x≤Mx}. Hence, for the given ɛ1 in Proposition 3.7, there exists T0>P which is independent of m and q, such that
(3.79)u(t)≥xα1*(t)-ɛ12,t≥T0+sq(m).
Thus
(3.80)u(t)≥x*(t)-ɛ1,t≥T0+sq(m)
from (3.54). By (3.72), there exists a positive integer N1≥N0 such that tq(m)>sq(m)+2T0>sq(m)+2P for m≥K1(m) and q≥K1(m). So we have
(3.81)x(t,zm)≥x*(t)-ɛ1,t∈[sq(m)+T0,tq(m)]
since m≥N1 and q≥K1(m). Hence,
(3.82)ẏ(t,zm)≥ψɛ1(t)y(t,zm),t∈[sq(m)+T0,tq(m)]
from (3.75) and (3.81). Integrationg the above inequality from sq(m)+T0 to tq(m) yields
(3.83)y(tq(m),zm)≥y(sq(m)+T0,zm)exp⁡∫sq(m)+T0tq(m)ψɛ1(t)dt,
and hence
(3.84)ηy(m+1)2≥ηy(m+1)2exp⁡∫sq(m)+T0tq(m)ψɛ1(t)dt>ηy(m+1)2
from (3.74). This is a contradiction. This completes the proof of Proposition 3.8. The result of Theorem 2.2 follows from Propositions 3.4–3.8.Proof of Theorem2.3.
Supposex*(t) is the periodic solution of (1.4). For any solution (x(t),y(t)) of (1.3) with a positive initial value, there are two possible cases:(i)
for allt≥0, x(t)>x*(t);(ii)
there is aT̂>0, such that x(T̂)≤x*(t).Now, in the above two cases, we prove limt→0y(t)=0, respectively.
(i) Supposev(t) is a solution of (1.4) with v(0)=x(0). Comparing (1.3) and (1.4) we have v(t)≥x(t)≥x*(t) and limt→0(v(t)-x*(t))=0. Hence(3.85)lim⁡t→∞(x(t)-x*(t))=0.
From system (1.3), for all t0∈[0,∞), we have
(3.86)x(t0+ω)-x(t0)=∫t0t0+ωx(s)f(s,x(s))ds-∫t0t0+ωx(s)g(s,x)ydsx*(t0+ω)-x*(t0)=∫t0t0+ωx*(s)f(s,x*)ds=0.
So
(3.87)x(t0+ω)-x(t0)=∫t0t0+ω[xf(s,x)-x*f(s,x*)]ds-∫t0t0+ωxg(s,x)yds=∫t0t0+ω{(x-x*)f(s,x)+x*[f(s,x)-f(s,x*)]}ds-∫t0t0+ωxg(s,x)yds=∫t0t0+ω(x-x*)[f(s,x)+x*∂f∂x(s,x*+θ(x-x*))]ds-∫t0t0+ωxg(s,x)yds,
where 0<θ<1. By the boundedness of x, x*, and the continuity of f and ∂f/∂x, there must exist a constant A>0, such that
(3.88)|f(s,x)+x*∂f∂x(s,x*+θ(x-x*))|≤A.
For all ɛ>0, by (3.85), there exists a T0>0, such that 0<x(t)-x*(t)<ɛ, for all t≥T0. Hence
(3.89)∫t0t0+ωxg(s,x)yds≤-x(t0+ω)+x(t0)+Aωɛ=-[x(t0+ω)-x*(t0+ω)]+[x(t0)-x*(t0)]+Aωɛ<0+ɛ+Aωɛ=(Aω+1)ɛ
for t0≥T0. Denote GL=min0≤t≤ωx*(t)g(t,x*(t)), yt0L=mint0≤t≤t0+ωy(t). Then
(3.90)0<GL∫t0t0+ωyds≤∫t0t0+ωxg(s,x)yds≤(Aω+1)ɛ.
Hence
(3.91)0<yt0L≤1ω∫t0t0+ωy(s)ds≤(AGL+1(ωGL))ɛ.
This implies
(3.92)lim⁡t0→∞yt0L=lim⁡t0→∞∫t0t0+ωy(s)ds=0.
On the other hand,(3.93)y(t)=y(t0)exp⁡∫t0t0+ω[γ(s)x(s)g(s,x(s))-μ(s)-h(s)y(s)]ds.
By the boundedness of x, y, g, and the continuity of γ, μ, and h, there exists a C>0 such that
(3.94)|γ(t)g(t,x)x-μ(t)-h(t)y(t)|≤C,t≥0.
Hence
(3.95)|ln⁡y(t)y(t0)|≤C∫t0t0+ωds=Cω,fort∈[0,ω].
For all t, s∈[0,ω], we have
(3.96)|ln⁡y(t)y(s)|=|ln⁡y(t)y(t0)-ln⁡y(s)y(t0)|≤2Cω.
It follows that
(3.97)exp⁡(-2Cω)≤y(t)y(s)≤exp⁡(2Cω),|t-s|≤ω.
Considering (3.92), we have
(3.98)lim⁡t→∞y(t)=0.
(ii) Consider the second case. We claim that once there is aT̂>0 such that x(T̂)≤x*(T̂), then(3.99)x(t)<x*(t),∀t>T̂.
Otherwise, supposeT̃=mint>T̂{t:x(t)=x*(t)}. Then(3.100)ẋ(T̃)=x′(T̃-0)=lim⁡t→T̃-0x(t)-x(T̃)t-T̃=lim⁡t→T̃-0x(t)-x*(T̃)t-T̃≥lim⁡t→T̃-0x*(t)-x*(T̃)t-T̃=ẋ*(T̃).
This is a contradiction since from (1.3) and (1.4) we know ẋ(t)<ẋ*(t) at the same point (t,x)∈R+2. So (3.99) holds.
Now we show for allɛ>0, there is a T(1)>T̂≥0 such that(3.101)y(T(1))<ɛ.
Otherwise, suppose for all t≥T̂, y(t)≥ɛ. Then
(3.102)ẋ≤x(f(t,x)-g(t,x)ɛ),t≥T̂.
Suppose v(t) is a solution of (1.4) with v(T̂)=x(T̂). Then by Lemma 2.1 we have v(t)→x*(t)(t→∞), where x*(t) denotes the periodic solution of (1.4). Since ẋ(T̂)<v̇(T̂), there exists a σ>0 such that x(t)<v(t) for t∈(T̂,T̂+σ]. Denote δ(t)=v(t)-x(t). From (3.99) we know δ(t)>0,for t≥T̂. We show that there is an ɛ1>0 such that limt→∞infδ(t)≥ɛ1. In fact,
(3.103)v̇-ẋ=vf(t,v)-(xf(t,x)-g(t,x)ɛ)=v(f(t,v)-f(t,x))+f(t,x)(v-x)+xg(t,x)ɛ=(v∂f∂x(t,x+θ(v-x))+f(t,x))(v-x)+xg(t,x)ɛ
where 0<θ<1. By the boundedness of x, v, and the continuity of f and ∂f/∂x, there exists an L>0, such that |v(∂f/∂x)(t,x+θ(v-x))+f(t,x)|<L. Hence
(3.104)δ̇>-Lδ+xg(t,x)ɛ.
Choosing T(0)>T̂, such that v(t)≥x*(t)/2for t≥T(0), and letting
(3.105)ɛ1=min⁡{max⁡0≤t≤σδ(t),min⁡0≤t≤;ω(x*(t)4),min⁡0≤t≤ω⁡g(t,x*(t)/4)x*(t)ɛ4L}.
Then whenever δ(t)≤ɛ1 for t≥T(0), we have x(t)=v(t)-δ(t)≥x*(t)/2-x*(t)/4=x*(t)/4. By assumption (A6), δ̇(t)>-Lɛ1+g(t,x*(t)/4)x*(t)/4≥0, and this implies limt→∞infδ(t)>ɛ1. Choosing T′>T̂, such that v(t)<x*(t)+ɛ1/2and δ(t)≥ɛ1 for t≥T′, we have
(3.106)x(t)≤v(t)-ɛ1≤(x*(t)+ɛ12)-ɛ1=x*(t)-ɛ12,fort≥T′.
Hence, there exists an ɛ0>0 such that
(3.107)γ(t)g(t,x(t))x(t)-μ(t)-h(t)ɛ¯≤γ(t)g(t,x*(t)-ɛ12)(x*(t)-ɛ12)-μ(t)¯≤-ɛ0<0
for t≥T′, by (2.3) and assumption (A5). So
(3.108)ɛ≤y(t)≤y(T′)exp⁡∫T′t[γ(s)g(s,x(s))x(s)-μ(s)-h(s)ɛ]ds⟶0(t⟶∞).
This is a contradiction and it implies that (3.101) holds.
Second, we show that(3.109)y(t)≤ɛexp⁡(M(ɛ)ω),fort≥T(1),
where
(3.110)M(ɛ)=max⁡0≤t≤ω{γ(t)g(t,x*)(t)x*(t)+μ(t)+h(t)ɛ}.
Otherwise, there exists T(2)>T(1), such that
(3.111)y(T(2))>ɛexp⁡(M(ɛ)ω).
By the continuity of y(t), there must exist T(3)∈(T(1),T(2)) such that y(T(3))=ɛ and y(t)>ɛ for t∈(T(3),T(2)]. Let P1 be the nonnegative integer, such that T(2)∈(T(3)+P1ω,T(3)+(P1+1)ω], and by (2.3), (A6) and (3.99), we have
(3.112)ɛexp⁡(M(ɛ)ω)<y(T(2))<y(T(3))exp⁡∫T(3)T(2)[γ(s)g(s,x(s))x(s)-μ(s)-h(s)ɛ]ds=ɛexp⁡{∫T(3)T(3)+P1ω+∫T(3)+P1ωT(2)}[γ(s)g(s,x)x-μ(s)-h(s)ɛ]ds<ɛexp⁡{∫T(3)T(3)+P1ω+∫T(3)+P1ωT(2)}[γ(s)g(s,x*)x*-μ(s)-h(s)ɛ]ds≤ɛexp⁡{∫T(3)+P1ωT(2)[γ(s)g(s,x*)x*-μ(s)-h(s)ɛ]ds}≤ɛexp⁡{∫T(3)+P1ωT(2)[γ(s)g(s,x*)x*+|μ(s)|+h(s)ɛ]ds}<ɛexp⁡(M(ɛ)ω),
which is a contradiction. This completes the proof of conclusion (i) of Theorem 2.3.
Now we prove the second conclusion. Since (i) holds andg(t,x) is bounded, there is a t1>0 and an ɛ>0, such that g(t,x)y<ɛ≤ɛ0 for t≥t1. Then we have(3.113)ẋ≥x(f(t,x)-ɛ),fort≥t1.
The following auxiliary equation,
(3.114)v̇=v(f(t,v)-ɛ)
has a globally asymptotically stable positive ω-periodic solution xɛ*(t) by Lemma 3.2. So by Lemma 3.3, we have
(3.115)lim⁡t→∞inf⁡x(t)≥xɛ*(t).
This completes the proof of conclusion (ii) of Theorem 2.3.Proof of Corollary2.6.
We claim that onceg(t,x) satisfies a local Lipschitz condition with respect to x, then the function
(3.116)F(t,x,y)=(x(f(t,x)-g(t,x)y)y(γ(t)g(t,x)x-μ(t)-h(t)y))
satisfies a local Lipschitz condition with respect to x and y. Considering assumptions (A1)–(A7), this is clearly the case. So the uniqueness of solutions of system (1.3) is guaranteed, and by Lemma 2.5 we know that there exsits an ω-periodic solution Z*(t)=(x*(t),y*(t)). From the proof of Theorem 15.5 in [37], the initial value of the periodic solution is the fix point of the mapping T:z0→z(ω;0,z0) which is the limit point of a subsequence of the sequence {Tnz0}1∞, where z0=(x0,y0) is the initial value of a bounded solution of system (1.3). Taking any positive x0 and y0, by Theorem 2.2 we know the solution started from this point is bounded and the limit of any subsequence of the sequence {Tnz0}1∞ is positive. So the periodic solution (x*(t),y*(t)) is positive.


## 4. Examples

Example 4.1.
Supposef(t,x)=1-(2+cost)x,g(t,x)=2xy/(2+sint+x2), γ(t)=0.5, μ(t)=(1/10)-(1/20)sint, h(t)=1. The corresponding system is
(4.1)ẋ=x(1-(2+cos⁡t)x-2xy2+sint+x2),ẏ=y(x22+sint+x2-110-120sint).
We know that the periodic solution of system (3.1) is
(4.2)x*(t)=1-exp⁡(-∫0ωa(s)ds)∫0ωb(t-s)exp⁡(-∫0ωa(t-τ)dτ)ds.
Using this formula we can compute the periodic solution of (4.1) in absence of predators
(4.3)x*(t)=21-e-2π(4e2π+cos⁡(t)e2π+sin(t)e2π-4-cos⁡(t)-sin(t))e-2π≈2(4+cos⁡(t)+sin(t)),γ(t)g(t,x*(t))x*(t)-μ(t)¯=12π∫02π((x*)22+sint+(x*)2-110-120sint)dt≈0.05313>0.
System (4.1) is permanent.Example 4.2.
Takingf(t,x), g(t,x) and γ(t) the same as in Example 4.1, and μ(t)=(1/5)-(1/10)sint, we can compute
(4.4)γ(t)g(t,x*(t))x*(t)-μ(t)¯=12π∫02π((x*)22+sint+(x*)2-15-110sint)dt≈-0.04687<0.
In this circumstance the predator population tends to extinction and the prey population keeps oscillating.
Simulation results of the two examples are shown in Figure1.(a) Simulation results of Example4.1 with initial values x(0)=0.3, y(0)=0.5. (b) Simulation results of Example 4.2 with initial values x(0)=0.3, y(0)=0.6.
(a)(b)



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*Source: 845606-2011-01-27.xml*