AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 843178 10.1155/2012/843178 843178 Research Article Positive Periodic Solutions of Nicholson-Type Delay Systems with Nonlinear Density-Dependent Mortality Terms Chen Wei 1 Wang Lijuan 2 Peterson Allan 1 School of Mathematics and Information Shanghai Lixin University of Commerce Shanghai 201620 China lixin.edu.cn 2 College of Mathematics Physics and Information Engineering Jiaxing University Zhejiang Jiaxing 314001 China zjxu.edu.cn 2012 11 12 2012 2012 03 10 2012 09 11 2012 09 11 2012 2012 Copyright © 2012 Wei Chen and Lijuan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the periodic solutions for a class of Nicholson-type delay systems with nonlinear density-dependent mortality terms. By using coincidence degree theory, some criteria are obtained to guarantee the existence of positive periodic solutions of the model. Moreover, an example and a numerical simulation are given to illustrate our main results.

1. Introduction

In the last twenty years, the delay differential equations have been widely studied both in a theoretical context and in that of related applications . As a famous and common delay dynamic system, Nicholson’s blowflies model and its modifications have made remarkable progress that has been collected in  and the references cited there in. Recently, to describe the dynamics for the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics which belong to the Nicholson-type delay differential systems, Berezansky et al. , Wang et al. , and Liu  studied the problems on the permanence, stability, and periodic solution of the following Nicholson-type delay systems: (1.1)N1(t)=-α1(t)N1(t)+β1(t)N2(t)+j=1mc1j(t)N1(t-τ1j(t))e-γ1j(t)N1(t-τ1j(t)),N2(t)=-α2(t)N2(t)+β2(t)N1(t)+j=1mc2j(t)N2(t-τ2j(t))e-γ2j(t)N2(t-τ2j(t)), where αi,βi,cij,γij,τijC(R,(0+)), and i=1,2, j=1,2,,m.

In , Berezansky et al. also pointed out that a new study indicates that a linear model of density-dependent mortality will be most accurate for populations at low densities and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates. Consequently, Berezansky et al.  presented an open problem: to reveal the dynamics of the following Nicholson’s blowflies model with a nonlinear density-dependent mortality term: (1.2)N(t)=-D(N(t))+PN(t-τ)e-aN(t-τ), where P is a positive constant and function D might have one of the following forms: D(N)=aN/(N+b) or D(N)=a-be-N with positive constants a,b>0.

Most recently, based upon the ideas in , Liu and Gong  established the results on the permanence for the Nicholson-type delay system with nonlinear density-dependent mortality terms. Consequently, the problem on periodic solutions of Nicholson-type system with D(N)=a-be-N has been studied extensively in . However, to the best of our knowledge, there exist few results on the existence of the positive periodic solutions of Nicholson-type delay system with D(N)=aN/(N+b). Motivated by this, the main purpose of this paper is to give the conditions to guarantee the existence of positive periodic solutions of the following Nicholson-type delay system with nonlinear density-dependent mortality terms: (1.3)N1(t)=-D11(t,N1(t))+D12(t,N2(t))+j=1lc1j(t)N1(t-τ1j(t))e-γ1j(t)N1(t-τ1j(t)),N2(t)=-D22(t,N2(t))+D21(t,N1(t))+j=1lc2j(t)N2(t-τ2j(t))e-γ2j(t)N2(t-τ2j(t)), under the admissible initial conditions (1.4)xt0=φ,φC+=C([-r1,0],R+1)×C([-r2,0],R+1),φi(0)>0, where Dij(t,N)=aij(t)N/(bij(t)+N), aij,bij,cik,γik:R(0,+) and τik:R[0,+) are all bounded continuous functions, and ri=max1kl{suptR1τik(t)}, i,j=1,2, k=1,,l.

For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function g defined on R1, let g+ and g- be defined as (1.5)g-=inftR1g(t),g+=suptR1g(t). We also assume that aij,bij,cik,γik:R(0,+) and τik:R[0,+) are all ω-periodic functions, ri=max1kl{τik+}, and i,j=1,2, k=1,,l.

Set (1.6)Ai=20ωaii(t)bii(t)dt,Bi=j=1l0ωcij(t)dt,γi+=max1jl{γij+},γi-=min1jl{γij-},D1=0ωa12(t)dt,D2=0ωa21(t)dt,Ci=0ωaii(t)dt,i=1,2.

Let Rn(R+n) be the set of all (nonnegative) real vectors; we will use x=(x1,x2,,xn)TRn to denote a column vector, in which the symbol (T) denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x|=(|x1|,|x2|,,|xn|)T and define ||x||=max1in|xi|. For matrix A=(aij)n×n, AT denotes the transpose of A. A matrix or vector A0 means that all entries of A are greater than or equal to zero. A>0 can be defined similarly. For matrices or vectors A and B, AB (resp. A>B) means that A-B0 (resp. A-B>0). We also define the derivative and integral of vector function x(t)=(x1(t),x2(t))T as x=(x1(t),x2(t))T and 0ωx(t)dt=(0ωx1(t)dt,0ωx2(t)dt)T.

The organization of this paper is as follows. In the next section, some sufficient conditions for the existence of the positive periodic solutions of model (1.3) are given by using the method of coincidence degree. In Section 3, an example and numerical simulation are given to illustrate our results obtained in the previous section.

2. Existence of Positive Periodic Solutions

In order to study the existence of positive periodic solutions, we first introduce the continuation theorem as follows.

Lemma 2.1 (continuation theorem [<xref ref-type="bibr" rid="B14">14</xref>]).

Let X and Z be two Banach spaces. Suppose that L:D(L)XZ is a Fredholm operator with index zero and N~:XZ is L -compact on Ω¯, where Ω is an open subset of X. Moreover, assume that all the following conditions are satisfied:

LxλN~x, for all xΩD(L), λ(0,1);

N~xImL, for all xΩKerL;

the Brouwer degree (2.1)deg{QN~,ΩKerL,0}0.

Then equation Lx=N~x has at least one solution in domLΩ¯.

Our main result is given in the following theorem.

Theorem 2.2.

Suppose (2.2)Ci>2Di,ln2BiAi>Ai,i=1,2,(2.3)j=1lc1j+a11-γ1j-e+a12+a11-<1,j=1lc2j+a22-γ2j-e+a21+a22-<1. Then (1.3) has a positive ω-periodic solution.

Proof.

Set N(t)=(N1(t),N2(t))T and Ni(t)=exi(t)(i=1,2). Then (1.3) can be rewritten as (2.4)x1(t)=-a11(t)b11(t)+ex1(t)+a12(t)ex2(t)-x1(t)b12(t)+ex2(t)+j=1lc1j(t)ex1(t-τ1j(t))-x1(t)-γ1j(t)ex1(t-τ1j(t)):=Δ1(x,t),x2(t)=-a22(t)b22(t)+ex2(t)+a21(t)ex1(t)-x2(t)b21(t)+ex1(t)+j=1lc2j(t)ex2(t-τ2j(t))-x2(t)-γ2j(t)ex2(t-τ2j(t)):=Δ2(x,t). As usual, let X=Z={x=(x1(t),x2(t))TC(R,R2):x(t+ω)=x(t) for all tR} be Banach spaces equipped with the supremum norm ||·||. For any xX, because of periodicity, it is easy to see that Δ(x,·)=(Δ1(x,·),Δ2(x,·))TC(R,R2) is ω-periodic. Let (2.5)L:D(L)={xX:xC1(R,R2)}xx=(x1,x2)TZ,P:Xx(1ω0ωx1(s)ds,1ω0ωx2(s)ds)TX,Q:Zz(1ω0ωz1(s)ds,1ω0ωz2(s)ds)TZ,N~:XxΔ(x,·)Z. It is easy to see that (2.6)ImL={xxZ,0ωx(s)ds=(0,0)T},KerL=R2,ImP=KerL,KerQ=ImL.

Thus, the operator L is a Fredholm operator with index zero. Furthermore, denoting by LP-1:ImLD(L)KerP the inverse of L|D(L)KerP, we have (2.7)LP-1y(t)=-1ω0ω0ty(s)dsdt+0ty(s)ds=(-1ω0ω0ty1(s)dsdt+0ty1(s)ds,-1ω0ω0ty2(s)dsdt+0ty2(s)ds)T. It follows that (2.8)QN~x=1ω0ωN~x(t)dt=(1ω0ωΔ1(x(t),t)dt,1ω0ωΔ2(x(t),t)dt)T,(2.9)LP-1(I-Q)N~x=0tN~x(s)ds-tω0ωN~x(s)ds-1ω0ω0tN~x(s)dsdt+1ω0ω0tQN~x(s)dsdt. Obviously, QN~ and LP-1(I-Q)N~ are continuous. It is not difficult to show that LP-1(I-Q)N~(Ω¯) is compact for any open bounded set ΩX by using the Arzela-Ascoli theorem. Moreover, QN~(Ω¯) is clearly bounded. Thus N~ is L-compact on Ω¯ with any open bounded set ΩX.

Considering the operator equation Lx=λN~x,λ(0,1), we have (2.10)x(t)=(x1(t),x2(t))T=λΔ(x,t)=(λΔ1(x,t),λΔ2(x,t))T. Suppose that x=(x1(t),x2(t))TX is a solution of (2.10) for some λ(0,1).

Firstly, we claim that there exists a positive number H such that ||x||<H. Integrating the first equation of (2.10) and in view of xX, it results that (2.11)0=0ωx1(t)dt=λ0ωΔ1(x,t)dt, which together with (2.4) implies that (2.12)0ω|j=1lc1j(t)ex1(t-τ1j(t))-x1(t)-γ1j(t)ex1(t-τ1j(t))+a12(t)ex2(t)-x1(t)b12(t)+ex2(t)|dt=0ωa11(t)b11(t)+ex1(t)dt<0ωa11(t)b11(t)dt. Similarly, we have (2.13)0ω|j=1lc2j(t)ex2(t-τ2j(t))-x2(t)-γ2j(t)ex2(t-τ2j(t))+a21(t)ex1(t)-x2(t)b21(t)+ex1(t)|dt=0ωa22(t)b22(t)+ex2(t)dt<0ωa22(t)b22(t)dt. It follows from (2.12) and (2.13) that (2.14)0ω|x1(t)|dtλ0ω|j=1lc1j(t)ex1(t-τ1j(t))-x1(t)-γ1j(t)ex1(t-τ1j(t))+a12(t)ex2(t)-x1(t)b12(t)+ex2(t)|dt+λ0ω|a11(t)b11(t)+ex1(t)|dt<20ωa11(t)b11(t)dt=A1,(2.15)0ω|x2(t)|dtλ0ω|j=1lc2j(t)ex2(t-τ2j(t))-x2(t)-γ2j(t)ex2(t-τ2j(t))+a21(t)ex1(t)-x2(t)b21(t)+ex1(t)|dt+λ0ω|a22(t)b22(t)+ex2(t)|dt<20ωa22(t)b22(t)dt=A2.

Since xX, there exist ξ1,ξ2,η1,η2[0,ω] such that (2.16)xi(ξi)=mint[0,ω]xi(t),xi(ηi)=maxt[0,ω]xi(t),xi(ξi)=xi(ηi)=0,i=1,2. It follows from (2.12) and (2.14) that (2.17)A12=0ωa11(t)b11(t)dt>0ωa11(t)b11(t)+ex1(t)dt=0ωj=1lc1j(t)ex1(t-τ1j(t))-x1(t)-γ1j(t)ex1(t-τ1j(t))dt+0ωa12(t)ex2(t)-x1(t)b12(t)+ex2(t)dt>ex1(ξ1)-x1(η1)-γ1+ex1(η1)j=1l0ωc1j(t)dt=B1ex1(ξ1)-x1(η1)-γ1+ex1(η1), which implies that (2.18)x1(ξ1)<lnA12B1+x1(η1)+γ1+ex1(η1). Using (2.14) yields (2.19)x1(t)x1(ξ1)+0ω|x1(t)|dt<lnA12B1+x1(η1)+γ1+ex1(η1)+A1,t[0,ω]. In particular, (2.20)x1(η1)<x1(ξ1)+0ω|x1(t)|dt<lnA12B1+x1(η1)+γ1+ex1(η1)+A1. It follows that (2.21)x1(η1)>ln(1γ1+(ln2B1A1-A1)). Again from (2.14), we have (2.22)x1(t)x1(η1)-0ω|x1(t)|dt>ln(1γ1+(ln2B1A1-A1))-A1:=H11,t[0,ω]. Similarly, we can obtain (2.23)x2(t)x2(η2)-0ω|x2(t)|dt>ln(1γ2+(ln2B2A2-A2))-A2:=H21,t[0,ω]. Since x1(ξ1)=0, from (2.10), we have (2.24)a11(ξ1)b11(ξ1)+ex1(ξ1)=j=1lc1j(ξ1)ex1(ξ1-τ1j(ξ1))-x1(ξ1)-γ1j(ξ1)ex1(ξ1-τ1j(ξ1))+a12(ξ1)ex2(ξ1)-x1(ξ1)b12(ξ1)+ex2(ξ1). Hence, from (2.24) and the fact that supu0ue-u=1/e, we have (2.25)ex1(ξ1)b11++ex1(ξ1)ex1(ξ1)b11(ξ1)+ex1(ξ1)=j=1lc1j(ξ1)a11(ξ1)γ1j(ξ1)γ1j(ξ1)ex1(ξ1-τ1j(ξ1))e-γ1j(ξ1)ex1(ξ1-τ1j(ξ1))+a12(ξ1)a11(ξ1)(1+b12(ξ1)e-x2(ξ1))<j=1lc1j+a11-γ1j-e+a12+a11-. Noting that u/(b11++u) is strictly monotone increasing on [0,+) and (2.26)supu0ub11++u=1>j=1lc1j+a11-γ1j-e+a12+a11-,it is clear that there exists a constant k1>0 such that (2.27)ub11++u>j=1lc1j+a11-γ1j-e+a12+a11-u[k1,+). In view of (2.25) and (2.27), we get (2.28)ex1(ξ1)k1,x1(ξ1)lnk1. In the same way, there exists a constant k2>0 such that (2.29)x2(ξ2)lnk2. Again from (2.14), (2.15), (2.28), and (2.29), we get (2.30)x1(t)x1(ξ1)+0ω|x1(t)|dt<lnk1+A1,x2(t)x2(ξ1)+0ω|x2(t)|dt<lnk1+A2. Then, we can choose two sufficiently large positive constants H12>lnk1+A1 and H22>lnk2+A2 such that (2.31)x1(t)<H12,x2(t)<H22,lnb11+<H12,lnb22+<H22.

Let H>max{|H11|,|H21|,H12,H22} be a fix constant such that (2.32)eH>1γ1-(H-lnC1-2D12B1),eH>1γ2-(H-lnC2-2D22B2). Then (2.22), (2.23), and (2.31) imply that ||x||<H, if xX is solution of (2.10). So we can define an open bounded set as Ω={xX:||x||<H} such that there is no λ(0,1) and xΩ such that Lx=λN~x. That is to say LxλN~x for all xΩD(L),λ(0,1).

Secondly, we prove that N~xImL for all xΩKerL. That is ((QN~(x))1,(QN~(x))2)T(0,0)T for all xΩKerL.

If x(t)=(x1(t),x2(t))TΩKerL, then x(t) is a constant vector in R2, and there exists some i{1,2} such that |xi|=H. Assume |x1|=H, so that x1=±H. Then, we claim (2.33)(QN~(x))1>0forx1=-H,(QN~(x))1<0forx1=H. If (QN~(x))10 for x1=-H, it follows from (2.2) and (2.8) that (2.34)0ωΔ1(x,t)dt0,forx1=-H. Hence, (2.35)A12=0ωa11(t)b11(t)dt>0ωa11(t)b11(t)+e-Hdt0ω[j=1lc1j(t)e-γ1j(t)e-H+a12(t)ex2+Hb12(t)+ex2]dt>0ωj=1lc1j(t)e-γ1j+e-Hdte-γ1+e-Hj=1l0ωc1j(t)dt=B1e-γ1+e-H, which implies (2.36)-H>ln(1γ1+ln2B1A1)>ln(1γ1+(ln2B1A1-A1))-A1=H11. This is a contradiction and implies that (QN~(x))1>0 for x1=-H.

If (QN~(x))10 for x1=H, it follows from (2.2) and (2.8) that (2.37)0ωΔ1(x,t)dt0,forx1=H,C12e-H=0ωa11(t)2eHdt<0ωa11(t)b11(t)+eHdt0ωj=1lc1j(t)e-γ1j(t)eHdt+0ωa12(t)ex2-Hb12(t)+ex2dt0ωj=1lc1j(t)e-γ1-eHdt+0ωa12(t)b12(t)eH-x2+eHdt<e-γ1-eHj=1l0ωc1j(t)dt+0ωa12(t)eHdt=B1e-γ1-eH+D1e-H. Consequently, (2.38)eH<1γ1-(H-lnC1-2D12B1), a contradiction to the choice of H. Thus, (QN~(x))1<0 for x1=H.

Similarly, if |x2|=H, we obtain (2.39)(QN~(x))2>0forx2=-H,(QN~(x))2<0forx2=H. Consequently, (2.33) and (2.39) imply that N~xImL for all xΩKerL.

Furthermore, let 0μ1 and define continuous function H(x,μ) by setting (2.40)H(x,μ)=-(1-μ)x+μQN~x.

For all x(t)=(x1(t),x2(t))TΩKerL, then there exists some i{1,2} such that |xi|=H. There are two cases: x1=±H or x2=±H. When x1=H or x2=H, from (2.33) and (2.39), it is obvious that (H(x,μ))1<0 or (H(x,μ))2<0. Similarly, if x1=-H or x2=-H, it results that (H(x,μ))1>0 or (H(x,μ))2>0. Hence H(x,μ)(0,0)T for all xΩkerL.

Finally, using the homotopy invariance theorem, we obtain (2.41)deg{QN~,ΩkerL,(0,0)T}=  deg{-x,ΩkerL,(0,0)T}0. It then follows from the continuation theorem that Lx=N~x has a solution (2.42)x*(t)=(x1*(t),x2*(t))TDomLΩ¯, which is an ω-periodic solution to (2.4). Therefore N*(t)=(N1*(t),N2*(t))T=(ex1*(t),ex2*(t))T is a positive ω-periodic solution of (1.3) and the proof is complete.

3. An Example

In this section, we give an example to demonstrate the results obtained in the previous section.

Example 3.1.

Consider the following Nicholson-type delay system with nonlinear density-dependent mortality terms: (3.1)N1(t)=-(5+sint)N1(t)5+sint+N1(t)+(2+cost)N2(t)2+cost+N2(t)+e4π(1+cost4)N1(t-|2+cost|)e-e4π+|sint|N1(t-|2+cost|)+e4π(1+sint4)N1(t-|2+sint|)e-e4π+|cost|N1(t-|2+sint|),N2(t)=-(5+cost)N2(t)5+cost+N2(t)+(2+sint)N1(t)2+sint+N1(t)+e4π(1+sint4)N2(t-|2+sint|)e-e4π+|cost|N2(t-|2+sint|)+e4π(1+cost4)N2(t-|2+cost|)e-e4π+|sint|N2(t-|2+cost|). Obviously, Ai=4π, Bi=4πe4π, Ci=10π, Di=4π  (i=1,2), cij+=(5/4)e4π, γij-=e4π(i,j=1,2), a12+=a21+=3, a11-=a22-=4, then (3.2)ln2BiAi=ln2+4π>4π=Ai,Ci=10π>8π=2Di,i=1,2,j=1lc1j+a11-γ1j-e+a12+a11-=58e+340.9799<1,j=1lc2j+a22-γ2j-e+a21+a22-=58e+340.9799<1, which means the conditions in Theorem 2.2 hold. Hence, the model (3.1) has a positive 2π-periodic solution in Ω¯, where Ω={xX:||x||<10000}. The fact is verified by the numerical simulation in Figure 1.

Numerical solution N(t)=(N1(t),N2(t))T of systems (3.1) for initial value φ(t)(1,2)T.

Remark 3.2.

Equation (3.1) is a form of Nicholson’s blowflies delayed systems with nonlinear density-dependent mortality terms, but as far as we know there are no that results can be applicable to (3.1) to obtain the existence of positive 2π-periodic solutions. This implies the results of this paper are essentially new.

Acknowledgments

The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. In particular, the authors expresses the sincere gratitude to Prof. Bingwen Liu for the helpful discussion when this work is carried out. This work was supported by National Natural Science Foundation of China (Grant nos. 11201184 and 11101283), Innovation Program of Shanghai Municipal Education Commission (Grant no. 13YZ127), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant nos. Y6110436, LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).

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