# Positive Periodic Solutions of Nicholson-Type Delay Systems  with Nonlinear Density-Dependent Mortality Terms

Authors: Wei Chen; Lijuan Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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.


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## Body

## 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 [1–4]. As a famous and common delay dynamic system, Nicholson’s blowflies model and its modifications have made remarkable progress that has been collected in [5] 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. [6], Wang et al. [7], and Liu [8] 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,τij∈C(R,(0+∞)), and i=1,2, j=1,2,…,m.In [5], 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. [5] 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 [5–8], Liu and Gong [9] 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 [10–13]. 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,φ∈C+=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=max1≤k≤l{supt∈R1τik(t)}, i,j=1,2, k=1,…,l.For convenience, we introduce some notations. Throughout this paper, given a bounded continuous functiong defined on R1, let g+ and g- be defined as
(1.5)g-=inft∈R1g(t),g+=supt∈R1g(t).
We also assume that aij,bij,cik,γik:R→(0,+∞) and τik:R→[0,+∞) are all ω-periodic functions, ri=max1≤k≤l{τik+}, and i,j=1,2, k=1,…,l.Set(1.6)Ai=2∫0ω‍aii(t)bii(t)dt,Bi=∑j=1l‍∫0ω‍cij(t)dt,γi+=max1≤j≤l{γij+},γi-=min1≤j≤l{γij-},D1=∫0ω‍a12(t)dt,D2=∫0ω‍a21(t)dt,Ci=∫0ω‍aii(t)dt,i=1,2.LetRn(R+n) be the set of all (nonnegative) real vectors; we will use x=(x1,x2,…,xn)T∈Rn 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||=max1≤i≤n|xi|. For matrix A=(aij)n×n, AT denotes the transpose of A. A matrix or vector A≥0 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, A≥B (resp. A>B) means that A-B≥0 (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 [14]).
LetX and Z be two Banach spaces. Suppose that L:D(L)⊂X→Z is a Fredholm operator with index zero and N~:X→Z is L -compact on Ω¯, where Ω is an open subset of X. Moreover, assume that all the following conditions are satisfied:(1)
Lx≠λN~x, for all x∈∂Ω∩D(L), λ∈(0,1);(2)
N~x∉ImL, for all x∈∂Ω∩KerL;(3)
the Brouwer degree(2.1)deg{QN~,Ω∩KerL,0}≠0.Then equation Lx=N~x has at least one solution in domL∩Ω¯.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.
SetN(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))T∈C(R,R2):x(t+ω)=x(t) for all t∈R} be Banach spaces equipped with the supremum norm ||·||. For any x∈X, because of periodicity, it is easy to see that Δ(x,·)=(Δ1(x,·),Δ2(x,·))T∈C(R,R2) is ω-periodic. Let
(2.5)L:D(L)={x∈X:x∈C1(R,R2)}∋x↦x′=(x1′,x2′)T∈Z,P:X∋x↦(1ω∫0ω‍x1(s)ds,1ω∫0ω‍x2(s)ds)T∈X,Q:Z∋z↦(1ω∫0ω‍z1(s)ds,1ω∫0ω‍z2(s)ds)T∈Z,N~:X∋x↦Δ(x,·)∈Z.
It is easy to see that
(2.6)ImL={x∣x∈Z,∫0ω‍x(s)ds=(0,0)T},KerL=R2,ImP=KerL,KerQ=ImL.
Thus, the operatorL is a Fredholm operator with index zero. Furthermore, denoting by LP-1:ImL→D(L)∩KerP the inverse of L|D(L)∩KerP, we have
(2.7)LP-1y(t)=-1ω∫0ω‍∫0t‍y(s)dsdt+∫0t‍y(s)ds=(-1ω∫0ω‍∫0t‍y1(s)dsdt+∫0t‍y1(s)ds,-1ω∫0ω‍∫0t‍y2(s)dsdt+∫0t‍y2(s)ds)T.
It follows that
(2.8)QN~x=1ω∫0ω‍N~x(t)dt=(1ω∫0ω‍Δ1(x(t),t)dt,1ω∫0ω‍Δ2(x(t),t)dt)T,(2.9)LP-1(I-Q)N~x=∫0t‍N~x(s)ds-tω∫0ω‍N~x(s)ds-1ω∫0ω‍∫0t‍N~x(s)dsdt+1ω∫0ω‍∫0t‍QN~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 Ω⊂X by using the Arzela-Ascoli theorem. Moreover, QN~(Ω¯) is clearly bounded. Thus N~ is L-compact on Ω¯ with any open bounded set Ω⊂X.
Considering the operator equationLx=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))T∈X is a solution of (2.10) for some λ∈(0,1).
Firstly, we claim that there exists a positive numberH such that ||x||<H. Integrating the first equation of (2.10) and in view of x∈X, it results that
(2.11)0=∫0ω‍x1′(t)dt=λ∫0ω‍Δ1(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)|dt≤λ∫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+λ∫0ω‍|a11(t)b11(t)+ex1(t)|dt<2∫0ω‍a11(t)b11(t)dt=A1,(2.15)∫0ω‍|x2′(t)|dt≤λ∫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+λ∫0ω‍|a22(t)b22(t)+ex2(t)|dt<2∫0ω‍a22(t)b22(t)dt=A2.
Sincex∈X, 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=∫0ω‍∑j=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=1l‍∫0ω‍c1j(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 supu≥0ue-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)supu≥0ub11++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.
LetH>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 x∈X is solution of (2.10). So we can define an open bounded set as Ω={x∈X:||x||<H} such that there is no λ∈(0,1) and x∈∂Ω such that Lx=λN~x. That is to say Lx≠λN~x for all x∈∂Ω∩D(L),λ∈(0,1).
Secondly, we prove thatN~x∉ImL for all x∈∂Ω∩KerL. That is ((QN~(x))1,(QN~(x))2)T≠(0,0)T for all x∈∂Ω∩KerL.
Ifx(t)=(x1(t),x2(t))T∈∂Ω∩KerL, 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))1≤0 for x1=-H, it follows from (2.2) and (2.8) that
(2.34)∫0ω‍Δ1(x,t)dt≤0,forx1=-H.
Hence,
(2.35)A12=∫0ω‍a11(t)b11(t)dt>∫0ω‍a11(t)b11(t)+e-Hdt≥∫0ω‍[∑j=1l‍c1j(t)e-γ1j(t)e-H+a12(t)ex2+Hb12(t)+ex2]dt>∫0ω‍∑j=1l‍c1j(t)e-γ1j+e-Hdt≥e-γ1+e-H∑j=1l‍∫0ω‍c1j(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))1≥0 for x1=H, it follows from (2.2) and (2.8) that
(2.37)∫0ω‍Δ1(x,t)dt≥0,forx1=H,C12e-H=∫0ω‍a11(t)2eHdt<∫0ω‍a11(t)b11(t)+eHdt≤∫0ω‍∑j=1l‍c1j(t)e-γ1j(t)eHdt+∫0ω‍a12(t)ex2-Hb12(t)+ex2dt≤∫0ω‍∑j=1l‍c1j(t)e-γ1-eHdt+∫0ω‍a12(t)b12(t)eH-x2+eHdt<e-γ1-eH∑j=1l‍∫0ω‍c1j(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~x∉ImL for all x∈∂Ω∩KerL.
Furthermore, let0≤μ≤1 and define continuous function H(x,μ) by setting
(2.40)H(x,μ)=-(1-μ)x+μQN~x.
For allx(t)=(x1(t),x2(t))T∈∂Ω∩KerL, 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 x∈∂Ω∩kerL.
Finally, using the homotopy invariance theorem, we obtain(2.41)deg{QN~,Ω∩kerL,(0,0)T}=deg{-x,Ω∩kerL,(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))T∈DomL⋂‍Ω¯,
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+34≈0.9799<1,∑j=1l‍c2j+a22-γ2j-e+a21+a22-=58e+34≈0.9799<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={x∈X:||x||<10000}. The fact is verified by the numerical simulation in Figure 1.Numerical solutionN(t)=(N1(t),N2(t))T of systems (3.1) for initial value φ(t)≡(1,2)T.
(a)
(b)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.



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