# Positive Periodic Solutions of a Neutral Impulsive Predator-Prey Model

Authors: Guirong Liu; Xiaojuan Song
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We investigate a ratio-dependent predator-prey model with Holling type III functional response based on system of neutral impulsive differential equations. Sufficient conditions for existence of positive periodic solutions are obtained by applying continuation theorem. Our main results demonstrate that under the suitable periodic impulse perturbations, the neutral impulsive system preserves the periodicity of the corresponding neutral system without impulse. In addition, our results can be applied to the corresponding system without impulsive effect, and thus, extend previous results.


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## Body

## 1. Introduction

During the recent three decades, impulsive differential equations were intensively investigated. Some researchers devote themselves to the research on the theory of impulsive differential equations. The detailed introductions of impulsive differential equations were given in [1].In the real world, there are many man-made and natural factors that always lead to sudden changes of population number at certain moments of time. For example, the births of many species are not continuous but happen at some regular time. These changes can often be characterized mathematically in the form of impulses. If these impulsive factors are incorporated into population dynamical models, the corresponding models should be governed by impulsive differential equations. With the development of impulsive differential equations, many models of impulsive differential equations have been considered extensively. For example, many authors have studied some qualitative properties of the solutions for impulsive differential equations such as periodicity [2–6], stability [7–10], and permanence and extinction [2, 11].Recently, Du and Feng [12] studied the existence of periodic solutions for the following neutral impulsive predator-prey model with Beddington-DeAngelis functional response: (1)x′t=xtr(t)-a(t)x(t-σ1(t))-ρtx′(t-σ2(t))b(t)y(t)f(t)+n(t)x(t)+m(t)y(t)-btytft+ntxt+mtyt,y′t=yt-dtc(t)x(t-σ3(t))f(t)+n(t)x(t-σ3(t))+m(t)y(t-σ3(t))+ctxt-σ3tft+ntxt-σ3t+mtyt-σ3t,t≠tk,k=1,2,…,x(tk+)-x(tk)=θ1kx(tk),ytk+-ytk=θ2kytk,t=tk,k=1,2,….In 2011, Liu and Yan [13] studied the existence of periodic solutions for the following neutral ratio-dependent predator-prey model with Holling type III functional response: (2)x′t=x(t)[r(t)-a(t)x(t-σ1)-ρx′(t-σ2)]-c(t)x2(t)m2y2(t)+x2(t)y(t),y′(t)=y(t)-d(t)+h(t)x2(t-τ(t))m2y2(t-τ(t))+x2(t-τ(t)).Now, we will consider the following neutral impulsive ratio-dependent predator-prey system with Holling type III functional response:(3)x′t=x(t)a(t)-b(t)x(t-σ1(t))-ρtx′(t-σ2(t))-c(t)x2(t)m2(t)y2(t)+x2(t)y(t),y′t=yt-dt+etx2t-σ3tm2ty2t-σ3t+x2t-σ3t,t≠tk,k=1,2,…,x(tk+)-x(tk)=θ1kx(tk),ytk+-ytk=θ2kytk,t=tk,k=1,2,…,where x and y represent the prey and predator densities; a is the intrinsic growth rate of the prey; b is the density-dependent coefficient of the prey; d is the death rate of the predator; θ1k,θ2k(k∈N+={1,2,…}) are the regular pulses at time tk of the prey and predator, respectively. In addition, x(tk+),x(tk-) represent the right and left limits of x(t) at tk; y(tk+),y(tk-) represent the right and left limits of y(t) at tk.Throughout the paper, we give the hypothesis as follows:(H1)
b,c,e,m,ρ∈C(R+,(0,+∞)), a,d∈C(R+,R), σ1,σ3∈C1R+,R, σ2∈C2(R+,R) are ω-periodic functions; ∫0ω‍a(t)dt>0, ∫0ω‍d(t)dt>0; 0<t1<t2<⋯tk<⋯ are fixed points and limk→∞tk=+∞;(H2)
θik>-1, i=1,2; k∈N+; ∏0<tk<t(1+θik) is ω-periodic function for i=1,2.The rest of this paper is organized as follows. We first introduce a lemma and the continuation theorem. Then, in Section3, we obtain the existence of periodic solutions for system (3). In Section 4, a conclusion is given.


## 2. Preliminaries

In view of the actual applications of (3), we will only consider the solutions of system (3) with the initial condition: (4)x(t)=φ(t),φ∈C1-σ,0,0,∞,φ(0)>0,y(t)=ψ(t),ψ∈C1([-σ,0],[0,∞)),ψ(0)>0,where σ=max⁡t∈[0,ω]{σ1(t),σ2(t),σ3(t)}.Definition 1.
Assume that(x,y)T∈C([-σ,∞),(0,+∞)×(0,+∞)) satisfies the initial condition (4). Moreover, (i)
x and y are absolutely continuous on each interval (0,t1] and (tk,tk+1],k∈N+;(ii)
fork∈N+,(x(tk+),y(tk+))T and (x(tk-),y(tk-))T exist and (x(tk-),y(tk-))T=(x(tk),y(tk))T;(iii)
fork∈N+, (x,y)T satisfies (3) for almost everywhere in [0,∞)∖{tk}.Then (x,y)T is a positive solution of the initial value problem (3) and (4) on [-σ,∞).Consider the following system:(5)p′t=p(t)a(t)-B(t)p(t-σ1(t))-δtp′(t-σ2(t))C(t)p(t)q(t)M12(t)q2(t)+N12(t)p2(t)-C(t)p(t)q(t)M12(t)q2(t)+N12(t)p2(t),q′t=q(t)·-d(t)+E(t)p2(t-σ3(t))M22(t)q2(t-σ3(t))+N22(t)p2(t-σ3(t)),where(6)B(t)=b(t)∏0<tk<t-σ1(t)‍(1+θ1k),δ(t)=ρ(t)∏0<tk<t-σ2(t)‍(1+θ1k),C(t)=c(t)∏0<tk<t‍(1+θ1k)(1+θ2k),M1(t)=m(t)∏0<tk<t‍(1+θ2k),N1(t)=∏0<tk<t‍(1+θ1k),E(t)=e(t)∏0<tk<t-σ3(t)‍1+θ1k2,M2(t)=m(t)∏0<tk<t-σ3(t)‍(1+θ2k),N2(t)=∏0<tk<t-σ3(t)‍(1+θ1k).By (H1) and (H2), the above functions are all positive ω-periodic functions.Lemma 2.
Assume that(H1) and (H2) hold. If (p,q)T is a positive solution of system (5), then (x,y)T is a positive solution of system (3), where (7)x(t)=p(t)∏0<tk<t‍(1+θ1k),y(t)=q(t)∏0<tk<t‍(1+θ2k).Proof.
It is easy to see thatx and y are positive and absolutely continuous on each interval (tk,tk+1]. For t≠tk, k∈N+,(8)x′t-xtat-btxt-σ1t-ρtx′t-σ2t-ctxtytm2ty2t+x2t=p′(t)∏0<tk<t‍(1+θ1k)-p(t)∏0<tk<t‍(1+θ1k)·at-btpt-σ1t∏0<tk<t-σ1t‍1+θ1k-ρtp′t-σ2t∏0<tk<t-σ2t‍1+θ1kctptqt∏0<tk<t‍1+θ1k1+θ2km2tq2t∏0<tk<t‍1+θ2k2+p2t∏0<tk<t‍1+θ1k2-ctptqt∏0<tk<t1+θ1k1+θ2km2tq2t∏0<tk<t1+θ2k2+p2t∏0<tk<t1+θ1k2=∏0<tk<t‍(1+θ1k)p′t-ptat-btpt-σ1t∏0<tk<t-σ1t‍1+θ1kctptqt∏0<tk<t1+θ1k1+θ2km2tq2t∏0<tk<t1+θ2k2+p2t∏0<tk<t1+θ1k2-ρtp′(t-σ2(t))∏0<tk<t-σ2t‍(1+θ1k)-ctptqt∏0<tk<t1+θ1k1+θ2km2tq2t∏0<tk<t1+θ2k2+p2t∏0<tk<t1+θ1k2=∏0<tk<t‍1+θ1kp′(t)-p(t)at-Btpt-σ1t-δtp′(t-σ2(t))C(t)p(t)q(t)M12(t)q2(t)+N12(t)p2(t)-C(t)p(t)q(t)M12(t)q2(t)+N12(t)p2(t)=0.Hence, the first equation of system (3) holds. Similarly(9)y′(t)-y(t)-d(t)+e(t)x2(t-σ3(t))m2(t)y2(t-σ3(t))+x2(t-σ3(t))=q′t∏0<tk<t‍1+θ2k-q(t)∏0<tk<t‍(1+θ2k)-dt+e(t)p2(t-σ3(t))∏0<tk<t-σ3t1+θ1k2m2(t)q2(t-σ3(t))∏0<tk<t-σ3t1+θ2k2+p2(t-σ3(t))∏0<tk<t-σ3t1+θ1k2=∏0<tk<t‍(1+θ2k)q′t-qt-d(t)+e(t)p2(t-σ3(t))∏0<tk<t-σ3(t)1+θ1k2m2(t)q2(t-σ3(t))∏0<tk<t-σ3(t)1+θ2k2+p2(t-σ3(t))∏0<tk<t-σ3(t)1+θ1k2·-d(t)+e(t)p2(t-σ3(t))∏0<tk<t-σ3(t)1+θ1k2m2(t)q2(t-σ3(t))∏0<tk<t-σ3(t)1+θ2k2+p2(t-σ3(t))∏0<tk<t-σ3(t)1+θ1k2=∏0<tk<t‍1+θ2kq′t-qt-dt+Etp2t-σ3tM22tq2t-σ3t+N22tp2t-σ3t.=0.That is, the second equation of system (3) holds. For t=tk, k∈N+,(10)xtk+=limt→tk+p(t)∏0<tj<t‍(1+θ1j)=p(tk)∏0<tj≤tk‍(1+θ1j),ytk+=limt→tk+q(t)∏0<tj<t‍(1+θ2j)=q(tk)∏0<tj≤tk‍(1+θ2j),x(tk)=p(tk)∏0<tj<tk‍(1+θ1j),y(tk)=q(tk)∏0<tj<tk‍(1+θ2j).Hence, for t=tk, k∈N+,(11)x(tk+)=(1+θ1k)x(tk),y(tk+)=(1+θ2k)y(tk).So (x,y)T is a positive solution of system (3).Next, we recall some concepts and results on coincidence degree.LetX,Z be normed vector spaces; let L:DomL⊂X→Z be a linear mapping. Assume that dimKerL=codimImL<+∞ and ImL is closed in Z; then the mapping L is said to be a Fredholm mapping of index zero.IfL is a Fredholm mapping of index zero, then there exist continuous projectors P:X→X and Q:Z→Z satisfying ImP=Ker⁡L,Ker⁡Q=ImL=Im(I-Q). It is obvious that the restriction LP of L to DomL∩Ker⁡P:(I-P)X→ImL has an inverse function. Denote KP:ImL→DomL by KP=LP-1.LetN:X→Z be a continuous mapping. N is L-compact on Ω¯; if Ω is an open bounded subset of X, QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is relatively compact.Lemma 3 (see [14]).
LetΩ⊂X be an open bounded set; L a Fredholm mapping of index zero; and NL-compact on Ω¯. If the following conditions hold: (i)
Lx≠λNx, for any λ∈(0,1), x∈∂Ω∩DomL;(ii)
QNx≠0, for any x∈∂Ω∩Ker⁡L;(iii)
deg⁡{JQN,Ω∩Ker⁡L,0}≠0, where J:ImQ→Ker⁡L is an isomorphism;then Lx=Nx has at least one solution in Ω¯∩DomL.


## 3. Main Results

Leth be a continuous ω-periodic function. Denote (12)h0=max⁡t∈0,ωht,h¯=1ω∫0ω‍h(t)dt,h^=1ω∫0ω‍htdt.In the following, we will present our main results.Theorem 4.
Suppose that(H1), (H2), and the following conditions hold. (H3)
Fort∈[0,∞), σ1′(t)<1, σ2′(t)<1; let t=ϕ1(p) be the inverse function of p=t-σ1(t) and G(t)>H′(t), where G(t)=B(ϕ1(t-σ2(t)))(1-σ2′(t))/(1-σ1′(ϕ1t-σ2t)) and H(t)=δ(t)/(1-σ2′t).(H4)
Consider that|δ|0eA1<1, where A1=ln⁡2a¯max⁡t∈[0,ω](1-σ2′t)/(Gt-H′t) + 2a¯|H|0max⁡t∈[0,ω]1/(Gt-H′t)+(a^+a¯)ω.(H5)
Consider that2a¯>C¯1/M1N10 and E¯>d¯|N22|0.Then system (3) has at least one positive ω-periodic solution.Proof.
Construct the system(13)u1′t=a(t)-B(t)eu1(t-σ1(t))-δ(t)eu1(t-σ2(t))u1′(t-σ2(t))-C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t),u2′(t)=-d(t)+E(t)e2u1(t-σ3(t))M22(t)e2u2(t-σ3(t))+N22(t)e2u1(t-σ3(t)).Apparently, if there exists one ω-periodic solution (u1*(t),u2*(t))T for system (13), then (p*(t),q*(t))T=(eu1*(t),eu2*(t))T is a positive ω-periodic solution of system (5). By (H2) and Lemma 2, Theorem 4 holds if we can show that there exists one ω-periodic solution of system (13).
LetZ be the real Banach space u=(u1,u2)T∈C(R,R2):ui(t+ω)=ui(t),t∈R,i=1,2 with the norm(14)u∞=max⁡t∈0,ωu1t+u2t.Let X be the real Banach space {u=(u1,u2)T∈C1(R,R2):ui(t+ω)=ui(t), t∈R, i=1,2} with the norm(15)u=u∞+u′∞.
DefineL:X→Z and N:X→Z by(16)Lu1t,u2tT=u1′t,u2′tT,Nu1(t)u2(t)=a(t)-B(t)eu1(t-σ1(t))-δ(t)eu1(t-σ2(t))u1′(t-σ2(t))-C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)-d(t)+E(t)e2u1(t-σ3(t))M22(t)e2u2(t-σ3(t))+N22(t)e2u1(t-σ3(t)).Therefore, system (13) can be written as(17)Lu=Nu,u∈X.Evidently, Ker⁡L=R2,ImL={(u1,u2)T∈Z:∫0ω‍ui(t)dt=0, i=1,2} is closed in Z, and dimKerL=codimImL=2. Hence L is a Fredholm mapping of index zero.
LetP:X→X and Q:Z→Z be (18)Pu1(t)u2(t)=u¯1u¯2,u1u2∈X,Qu1(t)u2(t)=u¯1u¯2,u1u2∈Z.The generalized inverse (to L) KP:ImL→Ker⁡P∩DomL is (19)KP(u)=∫0t‍u(s)ds-1ω∫0ω‍∫0t‍usdsdt.Clearly, P and Q are continuous projectors satisfying (20)ImP=Ker⁡L,Ker⁡Q=ImL=Im(I-Q).So QN:X→Z and KP(I-Q)N:X→X are(21)QNu=1ω∫0ω‍at-Bteu1t-σ1t+H′teu1t-σ2t-Cteu1teu2tM12te2u2t+N12te2u1tdt1ω∫0ω‍-dt+Ete2u1t-σ3tM22te2u2t-σ3t+N22te2u1t-σ3tdt,(KP(I-Q)N)u=∫0t‍a(s)-B(s)eu1s-σ1s+H′(s)eu1(s-σ2(s))-C(s)eu1(s)eu2(s)M12(s)e2u2(s)+N12(s)e2u1(s)ds-H(t)eu1(t-σ2(t))+H(0)eu1(-σ2(0))∫0t‍-d(s)+E(s)e2u1(s-σ3(s))M22(s)e2u2(s-σ3(s))+N22(s)e2u1(s-σ3(s))ds-1ω∫0ω‍∫0t‍a(s)-B(s)eu1s-σ1s+H′(s)eu1(s-σ2(s))-C(s)eu1(s)eu2(s)M12(s)e2u2(s)+N12(s)e2u1(s)dsdt-1ω∫0ω‍H(t)eu1(t-σ2(t))-H(0)eu1(-σ2(0))dt1ω∫0ω‍∫0t‍-d(s)+E(s)e2u1(s-σ3(s))M22(s)e2u2(s-σ3(s))+N22(s)e2u1(s-σ3(s))dsdt-tω-12∫0ω‍a(s)-B(s)eu1s-σ1s+H′(s)eu1(s-σ2(s))-C(s)eu1(s)eu2(s)M12(s)e2u2(s)+N12(s)e2u1(s)dstω-12∫0ω‍-d(s)+E(s)e2u1(s-σ3(s))M22(s)e2u2(s-σ3(s))+N22(s)e2u1(s-σ3(s))ds.Obviously, it is easy to check that QN and KP(I-Q)N are continuous. Furthermore, QN(Ω¯) is bounded; KP(I-Q)N(Ω¯) are relatively compact for any open bounded set Ω⊂X. Thus N is L-compact on Ω¯.
For anyλ∈(0,1), consider Lu=λNu; that is, (22)u1′t=λa(t)-B(t)eu1(t-σ1(t))-δ(t)eu1(t-σ2(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)·u1′(t-σ2(t))-C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t),u2′(t)=λ-d(t)+E(t)e2u1(t-σ3(t))M22(t)e2u2(t-σ3(t))+N22(t)e2u1(t-σ3(t)).Let (u1,u2)T∈X be a solution of (22). Thus (23)∫0ω‍a(t)-B(t)eu1(t-σ1(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)-δ(t)eu1(t-σ2(t))u1′(t-σ2(t))-C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)dt=0,(24)∫0ω‍-d(t)+E(t)e2u1(t-σ3(t))M22(t)e2u2(t-σ3(t))+N22(t)e2u1(t-σ3(t))dt=0.By (H3),(25)∫0ω‍δ(t)eu1(t-σ2(t))u1′(t-σ2(t))dt=∫0ω‍Hteu1(t-σ2(t)′dt=-∫0ω‍H′(t)eu1(t-σ2(t))dt.By (23),(26)∫0ω‍B(t)eu1t-σ1t-H′(t)eu1(t-σ2(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)+C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)dt=a¯ω.By (24),(27)∫0ω‍E(t)e2u1(t-σ3(t))M22(t)e2u2(t-σ3(t))+N22(t)e2u1(t-σ3(t))dt=d¯ω.Set t=ϕ2(p) be the inverse function of p=t-σ2(t). Clearly, B(ϕ1(p)),σ1′(ϕ1(p)),B(ϕ1(p-σ2(p))), σ1′(ϕ1(p-σ2(p))) are all ω-periodic functions. In addition, ϕ2(p+ω)=ϕ2(p)+ω. By (H3),(28)∫0ω‍Bteu1t-σ1tdt=∫-σ1(0)ω-σ1(ω)‍B(ϕ1(p))1-σ1′(ϕ1(p))eu1(p)dp=∫0ω‍B(ϕ1(s))1-σ1′(ϕ1(s))eu1(s)ds=∫ϕ2(0)ϕ2(ω)‍B(ϕ1(t-σ2(t)))(1-σ2′(t))1-σ1′(ϕ1(t-σ2(t)))eu1(t-σ2(t))dt=∫ϕ2(0)ϕ2(0)+ω‍G(t)eu1(t-σ2(t))dt=∫0ω‍G(t)eu1(t-σ2(t))dt.Thus(29)∫0ω‍(G(t)-H′(t))eu1(t-σ2(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)+C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)dt=a¯ω.By (22), (29), and (H3),(30)∫0ω‍ddtu1(t)+λH(t)eu1(t-σ2(t))dt=λ∫0ω‍a(t)-[G(t)-H′(t)]eu1(t-σ2(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)-Cteu1teu2tM12te2u2t+N12te2u1tdt≤∫0ω‍atdt+∫0ω‍(G(t)-H′(t))eu1(t-σ2(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)+Cteu1teu2tM12te2u2t+N12te2u1tdt=(a^+a¯)ω.Note that G(ϕ2(p)), H′(ϕ2(p)),σ2′(ϕ2(p)) are all ω-periodic functions. By (29) and (H3),(31)a¯ω≥∫0ω‍[G(t)-H′(t)]eu1(t-σ2(t))dt=∫-σ2(0)ω-σ2(ω)‍G(ϕ2(p))-H′(ϕ2(p))1-σ2′(ϕ2(p))eu1(p)dp=∫0ω‍G(ϕ2(t))-H′(ϕ2(t))1-σ2′(ϕ2(t))eu1(t)dt,which yields (32)∫0ω‍(G(t)-H′(t))eu1(t-σ2(t))G(ϕ2(t))-H′(ϕ2(t))1-σ2′(ϕ2(t))+G(ϕ2(t))-H′(ϕ2(t))1-σ2′(ϕ2(t))eu1(t)dt≤2a¯ω.So there exists ξ∈[0,ω] satisfying (33)(G(ξ)-H′(ξ))eu1(ξ-σ2(ξ))+G(ϕ2(ξ))-H′(ϕ2(ξ))1-σ2′(ϕ2(ξ))eu1(ξ)≤2a¯.By (H3),(34)u1(ξ)≤ln⁡2a¯max⁡t∈[0,ω]1-σ2′(t)G(t)-H′(t),eu1(ξ-σ2(ξ))≤2a¯max⁡t∈[0,ω]1G(t)-H′(t).By (30) and (H4), for t∈[0,ω],(35)u1(t)+λH(t)eu1(t-σ2(t))≤u1ξ+λHξeu1ξ-σ2ξ+∫0ω‍ddtu1(t)+λH(t)eu1(t-σ2(t))dt≤ln⁡2a¯max⁡t∈[0,ω]1-σ2′(t)G(t)-H′(t)+2a¯H0max⁡t∈[0,ω]1G(t)-H′(t)+(a^+a¯)ω=A1.As λH(t)eu1(t-σ2(t))≥0,(36)u1t≤A1,t∈[0,ω].By (22) and (36),(37)u1′t=λa(t)-B(t)eu1(t-σ1(t))C(t)eu1(t)eu2(t)M12(t)e2u2(t)+N12(t)e2u1(t)-δteu1t-σ2tu1′t-σ2t-Cteu1teu2tM12te2u2t+N12te2u1t≤a0+B0eA1+δ0eA1u1′0+C2M1N10.By (H4),(38)u1′0≤11-δ0eA1a0+B0eA1+C2M1N10=:D1.In addition, there exist ξi,ηi∈[0,ω] satisfying (39)ui(ξi)=min⁡t∈[0,ω]{ui(t)},uiηi=max⁡t∈0,ωuit,i=1,2.By (29) and (H3),(40)a¯ω-(G¯-H′¯)eu1(η1)ω≤a¯ω-∫0ω‍Gt-H′teu1t-σ2tdt=∫0ω‍Cteu1teu2tM12te2u2t+N12te2u1tdt≤C2M1N1¯ω≤12M1N10C¯ω.By (H5) and (40),(41)u1(η1)≥ln⁡2a¯-1/M1N10C¯2(G¯-H′¯).By (38),(42)u1t≥u1(η1)-∫0ω‍u1′tdt≥ln⁡2a¯-1/M1N10C¯2(G¯-H′¯)-D1ω=:A2,t∈[0,ω].By (36) and (42),(43)u10≤max⁡{A1,A2}=:A.By (27), there exists ζ1∈[0,ω] satisfying (44)d¯ω=e2u1(ζ1-σ3(ζ1))M22(ζ1)e2u2(ζ1-σ3(ζ1))+N22(ζ1)e2u1(ζ1-σ3(ζ1))E¯ω.Set ζ1-σ3(ζ1)=ζ2+kω, where k is an integer, ζ2∈[0,ω]. So (45)d¯ω=e2u1(ζ2)M22(ζ1)e2u2(ζ2)+N22(ζ1)e2u1(ζ2)E¯ω.Denote (46)M2(ξ3)=min⁡t∈[0,ω]{M2(t)},M2(η3)=max⁡t∈[0,ω]{M2(t)},N2(ξ4)=min⁡t∈[0,ω]{N2(t)},N2(η4)=max⁡t∈[0,ω]{N2(t)}.Clearly, the function x2/(kx2+l) is increasing with respect to x, where k>0, l>0. By (39), (43), (45), and (46),(47)d¯E¯≤e2u1(ζ2)M22(ξ3)e2u2(ξ2)+N22(ξ4)e2u1(ζ2)≤e2AM22(ξ3)e2u2(ξ2)+N22(ξ4)e2A,d¯E¯≥e2u1(ζ2)M22(η3)e2u2(η2)+N22(η4)e2u1(ζ2)≥e-2AM22(η3)e2u2(η2)+N22(η4)e-2A.By (H5),(48)u2ξ2≤A+12ln⁡E¯-d¯N22ξ4d¯M22ξ3,u2(η2)≥-A+12ln⁡E¯-d¯N22(η4)d¯M22(η3).By (22) and (27),(49)∫0ω‍u2′tdt=λ∫0ω‍-dt+Ete2u1t-σ3tM22te2u2t-σ3t+N22te2u1t-σ3tdt≤∫0ω‍dtdt+∫0ω‍Ete2u1t-σ3tM22te2u2t-σ3t+N22te2u1t-σ3tdt=(d^+d¯)ω.Hence(50)u2t≤u2(ξ2)+∫0ω‍u2′tdt≤A+12ln⁡E¯-d¯N22ξ4d¯M22ξ3+d^+d¯ω=:B1,t∈[0,ω],u2t≥u2(η2)-∫0ω‍u2′tdt≥-A+12ln⁡E¯-d¯N22η4d¯M22η3-d^+d¯ω=:B2,t∈[0,ω].So (51)u20≤max⁡B1,B2=:B.By (22),(52)u2′0=λ-d(t)+E(t)e2u1(t-σ3(t))M22(t)e2u2(t-σ3(t))+N22(t)e2u1(t-σ3(t))≤d0+EN220=:D2.By (38), (43), (51), and (52),(53)u=u∞+u′∞≤A+B+D1+D2.Consider the algebraic system (54)a¯-B¯eu1-1ω∫0ω‍C(t)eu1eu2M12(t)e2u2+N12(t)e2u1dt=0,-d¯+1ω∫0ω‍E(t)e2u1M22(t)e2u2+N22(t)e2u1dt=0.If (u1*,u2*)T is a solution of (54), then there exist ξ5,ξ6∈[0,ω] such that (55)a¯-B¯eu1*-C¯eu1*eu2*M12(ξ5)e2u2*+N12(ξ5)e2u1*=0,-d¯+E¯e2u1*M22(ξ6)e2u2*+N22(ξ6)e2u1*=0,which yields (56)u1*=ln⁡a¯B¯-lC¯B¯[M12(ξ5)l2+N12(ξ5)],u2*=u1*+ln⁡l,where l=(E¯-d¯N22ξ6)/M22(ξ6)d¯. Further, (57)u1*≤max⁡t∈[0,ω]ln⁡a¯B¯,ln⁡1B¯a¯-12C¯1M1N10=:D3,u2*≤D3+max⁡t∈[0,ω]ln⁡E¯-d¯N220d¯M220,ln⁡1M20E¯d¯=:D4.In addition, it is not difficult to see that the algebraic system (58)a¯-B¯eu1=0,-d¯+1ω∫0ω‍E(t)e2u1M22(t)e2u2+N22(t)e2u1dt=0has a unique solution (u~1,u~2)T. Let F be a constant satisfying (59)u~1+u~2<F.Set K=A+B+D1+D2+D3+D4+F. Obviously, K is independent of λ. Set (60)Ω=u1,u2T∈X:u1t,u2tT<K.Clearly, the condition (i) in Lemma 3 holds. If (u1,u2)T∈∂Ω∩Ker⁡L=∂Ω∩R2, then (u1(t),u2(t))T is a constant vector in R2 with |u1|+|u2|=K. Thus (61)QNu1u2=a¯-B¯eu1-1ω∫0ω‍C(t)eu1eu2M12(t)e2u2+N12(t)e2u1dt-d¯+1ω∫0ω‍E(t)e2u1M22(t)e2u2+N22(t)e2u1dt≠00.So the condition (ii) in Lemma 3 holds.
Define the homologyΨ:DomL×[0,1]→Z by(62)Ψu1,u2T,μ=a¯-B¯eu1-d¯+1ω∫0ω‍Ete2u1M22te2u2+N22te2u1dt+μ-1ω∫0ω‍C(t)eu1eu2M12(t)e2u2+N12(t)e2u1dt0.Similar to the derivations of (57), for any (u1,u2)T∈∂Ω∩Ker⁡L and μ∈(0,1), Ψ(u1,u2)T,μ≠0. By (57) and (59), for any (u1,u2)T∈∂Ω∩Ker⁡L and μ∈[0,1],(63)Ψu1,u2T,μ≠0.
Define the isomorphismJ=I:ImQ→Ker⁡L by (u1,u2)T→(u1,u2)T. By homotopy invariance of topological degree,(64)deg⁡JQN,Ω∩Ker⁡L,0=deg⁡QN,Ω∩Ker⁡L,0=deg⁡Ψu1,u2T,1,Ω∩Ker⁡L,0=deg⁡⁡Ψu1,u2T,0,Ω∩Ker⁡L,0=sgndet⁡-B¯eu~101ω∫0ω‍2EtM22te2u~1e2u~2M22te2u~2+N22te2u~12dt-1ω∫0ω‍2EtM22te2u~1e2u~2M22te2u~2+N22te2u~12dt=sgn2B¯e3u~1e2u~2ω∫0ω‍E(t)M22(t)M22te2u~2+N22te2u~12dt≠0.So, the condition (iii) of Lemma 3 holds. By Lemma 3, there exists one ω-periodic solution of system (13). The proof of Theorem 4 is complete.Remark 5.
From the proof of Theorem4, we can find that Theorem 4 is also true if θ1k=θ2k=0, k∈N+. Therefore, we can obtain the following result.Corollary 6.
Suppose that(H1) and the following conditions hold. (H6)
Fort∈[0,∞),σ1′(t)<1,σ2′(t)<1, let t=ϕ1(p) be the inverse function of p=t-σ1(t) and W(t)>V′(t), where W(t)=b(ϕ1(t-σ2(t)))(1-σ2′(t))/(1-σ1′(ϕ1(t-σ2(t)))) and V(t)=ρ(t)/(1-σ2′t).(H7)
Consider|ρ|0eA3<1, where A3=ln⁡2a¯max⁡t∈[0,ω](1-σ2′t)/(Wt-V′t) + 2a¯|V|0max⁡t∈[0,ω]1/(Wt-V′t)+(a^+a¯)ω.(H8)
Consider2a¯>c¯1/m0 and e¯>d¯.Then the following neutral ratio-dependent predator-prey system with Holling type III functional response (65)x′(t)=xtat-btxt-σ1t-ρtx′t-σ2t-c(t)x2(t)m2(t)y2(t)+x2(t)y(t),y′(t)=y(t)-d(t)+e(t)x2(t-σ3(t))m2(t)y2(t-σ3(t))+x2(t-σ3(t))has at least one positive ω-periodic solution.Remark 7.
Ifσ1(t),σ2(t),ρ(t),m(t) are all constants, then Corollary 6 matches Theorem  2.1 in [13]. Therefore, our main results extend Theorem  2.1 in [13].


## 4. Conclusion

In this paper, we construct a ratio-dependent predator-prey model with Holling type III functional response based on system of neutral impulsive differential equations. By constructing suitable homology and applying some analysis techniques and homotopy invariance of topological degree, we give the estimate on a priori bounds of periodic solutions for the corresponding system. Then the existence of positive periodic solutions for this model is obtained by applying coincidence degree theory.Our main results demonstrate that under the suitable periodic impulse perturbations, the neutral impulsive system preserves the periodicity of the corresponding neutral system without impulse. In addition, our results can be applied to the corresponding neutral ratio-dependent predator-prey model without impulsive effect. Further, our results extend some results in the literature.



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*Source: 818506-2015-05-14.xml*