# Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment

Authors: Huilan Wang; Zhengqiu Zhang; Weiping Zhou
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

By using continuation theorem of coincidence degree theory, sufficient conditions of the existence of positive periodic solutions are obtained for a generalized predator-prey system with diffusion and delays. In this paper, we construct a V-function to make the prior estimation for periodic solutions, which makes the discussion more concise. Moreover, to compute the mapping's topological degree, a polynomial function matrix is constructed straightforwardly as a homotopic mapping for the generalized one, which improves the methods of computation on topological degree for a generalized mapping.


---

## Body

## 1. Introduction

In the study of dynamic population models, which are represented by differential equations, to elaborate the realistic factors into models, sometimes one needs to consider the effects caused by various of factors such as delays, diffusion, and others. The global characteristics (including persistent, stable or attractive, oscillatory, and chaotic behavior) of such differential system have been an intensive subject in biomathematics (see e.g., [1–6]). One important aspect is the existence of positive periodic solutions ([7–17]), which imply the system is periodic.In the literature available, the methods of studying the existence of periodic solutions of a periodic system often fall into one of the following three categories: (1) combining the results of persistence of system with Horn fixed point theorem (usually for a system with delay) or Brower fixed point theorem (usually for a system without delay), the existence results of periodic solutions are obtained ([7, 8]); (2) employing other kinds of fixed point theorems (usually for one dimension system), the differential model is investigated by transforming it into equivalent integral one ([9–11]); (3) by virtue of the theory of topological degree, especially by using Mawhin’s Continuation theorem of coincidence degree (usually for high dimension system), much significant work has been done ([12–17]). However, for a generalized high-dimension system, no matter by fixed point theorems or by Mawhin’s continuation theorem of coincidence degree, it is very difficult to accomplish the work. The relevant literature is seldom to find. Zhang considered a generalized prey-predator system with delay ([16]), where the generalized mapping’s degree is computed step by step and a new technique of computation on topological degree is presented.It is well known that the key and the difficult points are the prior estimation for solutions, and the topological degree’s computation of generalized mapping as Mawhin’s continuation theorem is applied. In this paper, we investigate a generalized model of one predator competing for two preys, which takes the nonautonomous form:(1.1)ẋ1(t)=x1(t)f1(t,x1(t),x3(t))+D1(t)(x2(t)-x1(t)),ẋ2(t)=x2(t)f2(t,x2(t),x3(t))+D2(t)(x1(t)-x2(t)),ẋ3(t)=x3(t)[-g(t,x3(t))+c1(t)p1(x1(t-τ1))+c2(t)p2(x2(t-τ2))],
where xi denotes the density of the prey in the ith patch; x3 represents the total predator population for both patches; τi>0 is a constant and Di(t) is a positive continuous function and denotes the dispersal rate; pi is the functional response of the predator population on the prey in the ith patch, and ci is the conversion ratio of prey into predator; Di(t),ci(t),fi(t,xi,x3) and g(t,x3) are continuous functions in t∈[0,+∞) with a common period ω>0; fi(t,xi,x3) and g(t,x3) are differentiable with the other variables; i=1,2. The model is transformed from an autonomous one (see [6]). In [6], sufficient conditions for the permanence and the existence of positive attractive equilibrium are derived.Our purpose is to obtain sufficient conditions for the existence of positive periodic solutions associated with system (1.1) and to consider whether delays and diffusion have effect on the results. What is more important is to present some new techniques in prior estimation of solutions and computation on topological degree. In this paper, we construct a V-function to make the prior estimation, which makes the proof more concise. Furthermore, we select a polynomial function matrix as a homotopic mapping to the generalized one, which plays a key role in the computation of topological degree.In a biological sense, we take the initial conditions:(1.2)xi(s)=φi(s),s∈[-τ,0],φi(0)>0,i=1,2,3,
where τ=max{τ1,τ2}.In system (1.1), we assume that (H1)
fort≥0,∂fi(t,xi,x3)/∂xi<0,∂fi(t,xi,x3)/∂x3<0,i=1,2, and (∂g(t,x3))/∂x3>0;(H2)
pi(x) is continuous and pi(0)=0,pi′(x)>0,i=1,2.For a positive continuousω-periodic function f(t), we define(1.3)fu=max⁡t∈[0,ω]f(t),fl=min⁡t∈[0,ω]f(t),f¯=1ω∫0ωf(t)dt.In the following sections, we derived the sufficient conditions for the existence of positive periodic solutions and show that the delays and diffusion have no effect on the result.


## 2. Main Result and Proof

We first introduce some notion of the continuation theorem of coincidence degree theory and the lemma formulated in [18].LetX,Z be Banach spaces, let L:DomL⊂X→Z be a linear mapping, and let N:X→Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimkerL=codimimL<+∞ and ImL is closed in Z. Let P:X→X and Q:Z→Z be two projectors such that ImP=KerL and ImL=KerQ=Im(I-Q). It follows that L/(DomL∩KerP):(I-P)X→ImL is invertible. We denote the inverse of that map by KP. For an open bounded subset Ω of X, the mapping N will be called L-compact on Ω¯ if QN(Ω¯) is bounded and KP(I-P)N:Ω¯→X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.Lemma 2.1.
LetL be a Fredholm mapping of index zero and  N be  L-compact on  Ω¯. Suppose (a)
for eachλ∈(0,1) and x∈∂Ω, Lx≠λNx;(b)
for eachx∈KerL⋂∂Ω, QNx≠0;(c)
Brower degreedegB(JQN,Ω∩KerL,0)≠0.Then Lx=Nx has at least one solution in DomL∩Ω¯.Theorem 2.2.
Assume that there exist four positive constantsM,N,m,n such that (1)
fort∈R,fi(t,M,0)≤0 and -g(t,N)+c1(t)p1(M)+c2(t)p2(M)<0,i=1,2;(2)
fort∈R,fi(t,m,N)≥0 and -g(t,n)+c1(t)p1(m)+c2(t)p2(m)>0,i=1,2.Then system (1.1) has at least one positive ω-periodic solution.Proof.
Denotexi(t)=exp[ui(t)](i=1,2,3); then system (1.1) can be rewritten as
(2.1)u̇1(t)=f1(t,eu1(t),eu3(t))+D1(t)(eu2(t)-u1(t)-1),u̇2(t)=f2(t,eu2(t),eu3(t))+D2(t)(eu1(t)-u2(t)-1),u̇3(t)=-g(t,eu3(t))+c1(t)p1(eu1(t-τ1))+c2(t)p2(eu2(t-τ2)).
Clearly, if system (2.1) has an ω-periodic solution (u1*(t),u2*(t),u3*(t))T, then system (1.1) has a positive ω-periodic solution (exp[u1*(t)],exp[u2*(t)],exp[u3*(t)])T. We define
(2.2)X=Z={u(t)=(u1(t),u2(t),u3(t))T∈C(R,R3),u(t+ω)=u(t)},‖u‖=‖(u1(t),u2(t),u3(t))T‖=∑i=13max⁡t∈[0,ω]|ui(t)|,u∈X(orZ).
Then X and Z are Banach spaces with the norm ∥·∥.
Let(2.3)N(u,λ)=[f1(t,eu1(t),eu3(t))+λD1(t)(eu2(t)-u1(t)-1)f2(t,eu2(t),eu3(t))+λD2(t)(eu1(t)-u2(t)-1)-g(t,eu3(t))+c1(t)p1(eu1(t-τ1))+c2(t)p2(eu2(t-τ2))],u∈X,λ∈(0,1),Lu=u′=du(t)dt,Pu=1ω∫0ωu(t)dt,u∈X,Qz=1ω∫0ωz(t)dt,z∈Z.
Then, L is a Fredholm mapping of index zero and the generalized inverse of L is KP:ImL→KerP∩DomL, which is given by
(2.4)Kp(z)=∫0tz(s)ds-1ω∫0ω∫0tz(s)dsdt.
Hence
(2.5)QN(u,λ)=[1ω∫0ωF1(s)ds1ω∫0ωF2(s)ds1ω∫0ωF3(s)ds],
where
(2.6)F1(s)=f1(s,eu1(s),eu3(s))+λD1(s)(eu2(s)-u1(s)-1),F2(s)=f2(s,eu2(s),eu3(s))+λD2(s)(eu1(s)-u2(s)-1),F3(s)=-g(s,eu3(s))+c1(s)p1(eu1(s-τ1))+c2(s)p2(eu2(s-τ2)).
Consequently,
(2.7)Kp(I-Q)N(u,λ)=[∫0tF1(s)ds-1ω∫0ω∫0tF1(s)dsdt+(12-tω)∫0ωF1(s)ds∫0tF2(s)ds-1ω∫0ω∫0tF2(s)dsdt+(12-tω)∫0ωF2(s)ds∫0tF3(s)ds-1ω∫0ω∫0tF3(s)dsdt+(12-tω)∫0ωF3(s)ds].
By Arzela-Ascoli theorem, we can verify that the map KP(I-Q)N(Ω¯) is compact. Thus, N is L- compact on Ω¯ for any open bounded Ω⊂X. Since ImQ=KerL, the isomorphism J is the identical mapping.Corresponding to the operator equation Lx=λN(x,λ),λ∈(0,1), we have
(2.8)u̇1(t)=λ[f1(t,eu1(t),eu3(t))+λD1(t)(eu2(t)-u1(t)-1)],u̇2(t)=λ[f2(t,eu2(t),eu3(t))+λD2(t)(eu1(t)-u2(t)-1)],u̇3(t)=λ[-g(t,eu3(t))+c1(t)p1(eu1(t-τ1))+c2(t)p2(eu2(t-τ2))].
Assume that u=u(t)∈X is a solution of (2.8) for a certain λ∈(0,1). Then there exist ξi,ηi∈[0,ω](i=1,2,3) such that
(2.9)ui(ξi)=max⁡t∈[0,ω]ui(t),ui(ηi)=min⁡t∈[0,ω]ui(t),ui′(ξi)=0,ui′(ηi)=0.
Therefore, we obtain the equations
(2.10)f1(ξ1,eu1(ξ1),eu3(ξ1))+λD1(ξ1)(eu2(ξ1)-u1(ξ1)-1)=0,f2(ξ2,eu2(ξ2),eu3(ξ2))+λD2(ξ2)(eu1(ξ2)-u2(ξ2)-1)=0,-g(ξ3,eu3(ξ3))+c1(ξ3)p1(eu1(ξ3-τ1))+c2(ξ3)p2(eu2(ξ3-τ2))=0
and equations
(2.11)f1(η1,eu1(η1),eu3(η1))+λD1(η1)(eu2(η1)-u1(η1)-1)=0,f2(η2,eu2(η2),eu3(η2))+λD2(η2)(eu1(η2)-u2(η2)-1)=0,-g(η3,eu3(η3))+c1(η3)p1(eu1(η3-τ1))+c2(η3)p2(eu2(η3-τ2))=0.
To make estimation of ui(ξi)(i=1,2,3), we define a V-function as V¯(t)=max{u1(t),u2(t)}. Obviously, V¯(ξi)=ui(ξi) and V¯̇(ξi)=0,i=1,2. According to the hypothesis and equations (2.10), we have
(2.12)fi(ξi,eV¯(ξi),0)>fi(ξi,eV¯(ξi),eu3(ξi))=fi(ξi,eui(ξi),eu3(ξi))≥0≥fi(ξi,M,0),i=1,2,
which implies that eV¯(ξi)<M, that is,
(2.13)ui(ξi)<ln⁡M,i=1,2.
The third equation of (2.10) gives
(2.14)g(ξ3,eu3(ξ3))=c1(ξ3)p1(eu1(ξ3-τ1))+c2(ξ3)p2(eu2(ξ3-τ2))≤c1(ξ3)p1(eu1(ξ1))+c2(ξ3)p2(eu2(ξ2))<c1(ξ3)p1(M)+c2(ξ3)p2(M)<g(ξ3,N).
Thus
(2.15)eu3(ξ3)<N,
or
(2.16)u3(ξ3)<ln⁡N.
Similarly, we define V̲(t)=min{u1(t),u2(t)}. Then V̲(ηi)=ui(ηi), and V̲̇(ηi)=0,i=1,2. From (2.11), we have
(2.17)fi(ηi,eV̲(ηi),N)=fi(ηi,eui(ηi),N)<fi(ηi,eui(ηi),eu3(ηi))≤0<fi(ηi,m,N),
which implies that
(2.18)eui(ηi)>m,
that is,
(2.19)ui(ηi)>ln⁡m.
From (2.19), it follows that
(2.20)g(η3,eu3(η3))=c1(η3)p1(eu1(η3-τ1))+c2(η3)p2(eu2(η3-τ2))≥c1(η3)p1(eu1(η1))+c2(η3)p2(eu2(η2))≥c1(η3)p1(m)+c2(ξ3)p2(m)>g(η3,n).
Hence
(2.21)eu3(η3)>n,
that is,
(2.22)u3(η3)>ln⁡n.
Denote
(2.23)R1=max⁡{|ln⁡m|,|ln⁡M|},R2=max{|ln⁡n|,|ln⁡N|}.
In view of (2.13)–(2.22), we have
(2.24)|u1(t)|<R1,|u2(t)|<R1,|u3(t)|<R2,∀t∈R.
Clearly, Ri(i=1,2) are independent of λ. Denote M̃=2R1+R2+R0, where R0 is taken sufficiently large such that the solution (α*,β*,γ*)T of the following system:
(2.25)1ω∫0ωf1(s,eα,eγ)ds=0,1ω∫0ωf2(s,eβ,eγ)ds=0,1ω∫0ω[-g(s,eγ)+c1(s)p1(eα)+c2(s)p2(eβ)]ds=0
satisfies ∥(α*,β*,γ*)T∥<M̃, provided that system (2.25) has one or a number of solutions. Let Ω={u=(u1(x),u2(x),u3(x))T∈X;∥u∥<M̃}. It is easy to see that the condition (a) of Lemma 2.1 is satisfied.
Whenu∈∂Ω∩KerL=∂Ω∩R3, u is a constant vector in R3 with ∑i=13|ui|=M̃. If system (2.25) has one or a number of solutions, then
(2.26)QN(u,0)=[1ω∫0ωf1(s,eu1,eu3)ds1ω∫0ωf2(s,eu2,eu3)dse1ω∫0ω[-g(s,eu3)+c1(s)p1(eu1)+c2(s)p2(eu2)]ds]≠(0,0,0)T.
If system (2.25) has no solution, then naturally QN(u,0)≠(0,0,0)T. Hence, the condition (b) of Lemma 2.1 is satisfied.
Finally, we will prove that condition (c) of Lemma2.1 is satisfied. To this end, we define a mapping ψ:DomL×[0,1]→X by
(2.27)ψ(u,ν)=ν[1ω∫0ωf1(s,eu1,eu3)ds1ω∫0ωf2(s,eu2,eu3)ds1ω∫0ω[-g(s,eu3)+c1(s)p1(eu1)+c2(s)p2(eu2)]ds]+(1-ν)[a-eu1a-eu2b-eu3],
where ν∈[0,1] is a parameter and a,b are two chosen numbers as follows:
(2.28)m<a<M,n<b<N.
In the following, we will show that when u∈∂Ω∩KerL,ψ(u,ν)≠(0,0,0)T.
We consider three possible cases: (1) either ofeui satisfies eui≥M(i=1,2), (2) either of eui satisfies eui≤m(i=1,2), and (3) both of eui satisfy m<eui<M(i=1,2).
Case 1.
Either ofeui(i=1,2) satisfies eui≥M: since there exists a constant t1 such that
(2.29)1ω∫0ωfi(s,eui,eu3)ds=fi(t1,eui,eu3),
then
(2.30)νfi(t1,eui,eu3)+(1-ν)(a-eui-beu3)<νfi(t1,eui,0)+(1-ν)(a-eui)≤νfi(t1,M,0)+(1-ν)(a-M)<0.Case 2.
Either ofeui(i=1,2) satisfies eui≤m: we consider two subcases as follows.Subcase 1 ..
eu3<N. There exists a constant t2 and ui(i=1or2) such that
(2.31)1ω∫0ωfi(s,eui,eu3)ds=fi(t2,eui,eu3),
then
(2.32)νfi(t2,eui,eu3)+(1-ν)(a-eui)>νfi(t2,m,N)+(1-ν)(a-m)≥νfi(t2,m,N)+(1-ν)(a-m)>0Subcase 2.
eu3≥N. Because the other uj(j=2or1) must satisfy either euj≤M or euj≥M. The later one is discussed in Case 1. Then we only consider the subcase when eui≤m(i=1or2) and euj≤M(j=2or1). Under these conditions, there exists a constant t3 such that
(2.33)ν[-g(t3,eu3)+c1(t5)p1(eu1)+c2(t5)p2(eu2)]+(1-ν)(b-eu3)<[-g(t3,N)+c1(t5)p1(M)+c2(t5)p2(M)]+(1-ν)(b-N)<0.Case 3.
Both ofeui(i=1,2) satisfy m<eui<M: we also consider the following two subcases.Subcase 3.
eu3≥N. There exists a constant t4 such that
(2.34)1ω∫0ω[-g(s,eu3)+c1(s)p1(eu1(s))+c2(s)p2(eu2(s))]ds=-g(t4,eu3)+c1(t4)p1(eu1)+c2(t4)p2(eu2).
Hence
(2.35)ν[-g(t4,eu3)+c1(t4)p1(eu1)+c2(t4)p2(eu2)]+(1-ν)(b-eu3)<ν[-g(t4,N)+c1(t4)p1(M)+c2(t4)p2(M)]+(1-ν)(b-N)<0.Subcase 4.
eu3<N. If n<eu3<N, then ∑i=13|ui|<M̃, which is a contradiction to ∑i=13|ui|=M̃. Hence, eu3≤n. When m<eui<M and eu3≤n, there exists a t5 such that
(2.36)ν[-g(t5,eu3)+c1(t3)p1(eu1)+c2(t5)p2(eu2)]+(1-ν)(b-eu3)>ν[-g(t5,n)+c1(t5)p1(m)+c2(t5)p2(m)]+(1-ν)(b-n)>0.
To sum up, ψ(u,ν)≠(0,0,0)T when u∈∂Ω∩KerL.Since the algebraic equations
(2.37)a-x=0,a-y=0,b-z=0
have a unique solution (x*,y*,z*) such that
(2.38)x*=y*=a>0,z*=b>0,
then
(2.39)deg⁡(JQN(u,0),Ω∩Ker⁡L,(0,0,0)T)=sign⁡|-x*000-y*000-z*|=sign⁡[-x*y*z*]=-1≠0.
This completes the proof of Theorem 2.2.Remark 2.3.
Theorem2.2 implies that the delays and the diffusion have no effect on the result provided (H1)-(H2) holds.


## 3. Application

Example 3.1.
Consider the system:(3.1)ẋ1(t)=x1(t)[r1(t)-a1(t)x1(t)-b1(t)x3(t)]+D1(t)(x2(t)-x1(t)),ẋ2(t)=x2(t)[r2(t)-a2(t)x2(t)-b2(t)x3(t)]+D2(t)(x1(t)-x2(t)),ẋ3(t)=x3(t)[-r3(t)-a3(t)x3+c1(t)x1(t-τ1)+c2(t)x2(t-τ2)],
where τi(i=1,2) are positive constants and all the coefficients are positive continuous functions with period ω.It is easy to see that(3.2)fi(t,xi,x3)=ri(t)-ai(t)xi-bi(t)x3,i=1,2;g(t,x3)=r3(t)+a3(t)x3;pi(x)=x.Obviously, the assumptions (H1)-(H2) are satisfied. Let(3.3)M=max⁡{r1ua1l,r2ua2l}.
Then(3.4)fi(t,M,0)≤0.Let(3.5)N=(c1u+c2u)M-r3la3l;
then -g(t,N)+c1(t)M+c2(t)M<0. While(3.6)m=min⁡{r1l-b1uNa1u,r2l-b2uNa2u}.fi(t,m,N)≥0. Let(3.7)n=(c1l+c2l)m-r3ua3u;
then -g(t,n)+c1(t)m+c2(t)m>0. Therefore, the conditions of Theorem 2.2 are satisfied. We can draw the conclusion that system (3.1) has at least one positive periodic solution.



---

*Source: 423193-2012-04-05.xml*