# Global Analysis of a Delayed Impulsive Lotka-Volterra Model with Holling III Type Functional Response

Authors: Hui Wang; Xiaomin Hu; Zhixing Hu; Fucheng Liao
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A delayed impulsive Lotka-Volterra model with Holing III type functional response was established. With the help of Mawhin’s Continuation Theorem in coincidence degree theory, a sufficient condition is found for the existence of positive periodic solutions of the system under consideration. By applying the comparison theorem and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are proved. Two numerical simulations are also given to illustrate our main results.


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## Body

## 1. Introduction

Recently, many complicated but realistic predator-prey systems based on classical Holling type functional responses have been analyzed by ecologists and mathematicians; see papers [1–6] and so forth. For example, a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response was proposed by Lv and Du in [2]; Zhang and his coworkers studied positive periodic solutions in a predator-prey model with Hassell-Varley type functional response, nonselective harvesting, and multiple delays in paper [6]. The theoretical values of these studies not only have great significance in biological economics but also provide strong support for the management and development of renewable energy.Hassell [7] introduced the following predator-prey system with mutual interference m(0<m≤1):(1)x˙=xg(x)-φ(x)ym,y˙=y-d-kφ(x)ym-1-q(y).After that more scholars have further conducted research. For example, Du and Lv investigated a Lotka-Volterra model with mutual interference and time delays in [8]:(2)x˙=xtr1t-b1txt-τt-c1(t)x2(t)x2(t)+k2ym(t),y˙=yt-r2t-b2tyt+c2(t)x2(t)x2(t)+k2ym(t),(0<m<1,k>0).Some criteria on the permanence and global attractivity of the above system are found. As far as we know, delay models have been studied and applied extensively in biology, physics, population dynamics, and other fields. However, the assumption of these models with constant environment is rarely the case in real life. A system must be nonautonomous if the environmental fluctuation is taken into account, such as seasonal effects of weather, food supplies, and harvesting. Therefore, it is rational to consider the ecosystem with periodic or almost periodic coefficients. On the other hand, there exist a few discontinuous and impulsive phenomena; for instance, many species are given birth seasonally. If we introduce these impulsive factors into the systems, it is more realistic to analyze the ecology models. For example, Wang and Zhu [4] considered a delayed impulsive prey-predator system with mutual interference:(3)x˙=x(t)r1(t)-b1(t)x(t-τ)-c1(t)x(t)ym(t),y˙=y(t)-r2(t)-b2(t)y(t-σ)+c2(t)x(t)ym(t),t≠tk,k∈Z+,Δx(t)=dkx(t),Δy(t)=fky(t),t=tk,k∈Z+.But as far as we all know, there are few results on the existence and global attractivity of positive periodic solutions of model (1) with delays and impulses. Motivated by these facts, we formulate a delayed impulsive Lotka-Volterra model with Holing III type functional response(4)x˙=x(t)r1(t)-b1(t)x(t-τ1(t))-c1(t)x2(t)x2(t)+K2ym(t),y˙=y(t)-r2(t)-b2(t)y(t-τ2(t))+c2(t)x2(t)x2(t)+K2ym(t),t≠tk,k∈Z+,Δx(t)=dkx(t),Δy(t)=hky(t),t=tk,k∈Z+,with initial conditions(5)x(t)=φ(t),φ∈C[-τ,0],R+,φ(0)>0,y(t)=ϕ(t),ϕ∈C[-τ,0],R+,ϕ(0)>0,where dk and hk represent the regular harvest or death from spraying pesticide of the predator and prey at time tk, τ=maxt∈[0,ω]{τ1(t),τ2(t)}. In this paper, Z, Z+, and R denote integers, positive integers, and real numbers, respectively, and R+=0,∞; ri(t), bi(t), and ci(t)(i=1,2) are continuously nonnegative periodic functions with period ω>0; τi(t)(i=1,2) are nonnegative and continuously differentiable periodic functions with period ω on R, and min1-τ˙i(t)>0; m and K are positive constants and m<1; x and y satisfy Δx(t)=x(t+)-x(t)=lims→t+x(s)-lims→t-x(s).Furthermore, suppose that the following conditions hold.(H1)
There exists an integerl≥1, such that 0,ω∩tk=t1,t2,…tl, tk+l=tk+ω, and 0<t1<t2<⋯<tk<⋯ are fixed points with limk→∞tk=+∞.(H2)
-1<dk≤0, -1<hk≤0 are constants, and dk=dk+l, hk=hk+l.This paper is organized as follows. In Section2, we introduce some useful lemmas. In Section 3, we not only prove the existence of periodic solutions but also study the permanence and the global attractivity of system (4). In Section 4, two examples are given to illustrate the feasibility of our results by using simulation. The last section is a brief conclusion.


## 2. Preliminaries

Under the assumptions (H1) and (H2), we consider a new system as follows:(6)p˙t=p(t)r1(t)-B1tpt-τ1t-C1(t)p2(t)Ap2(t)+K2qm(t),q˙t=q(t)-r2(t)-B2tqt-τ2t+C2(t)p2(t)Ap2(t)+K2qm(t),with initial conditions(7)p(t)=φ(t),φ∈C[-τ,0],R+,φ(0)>0,q(t)=ϕ(t),ϕ∈C[-τ,0],R+,ϕ(0)>0,where(8)B1(t)=b1(t)∏0<tk<t-τ1(t)‍(1+dk),C1(t)=c1(t)∏0<tk<t‍(1+dk)1+hkm,B2(t)=b2(t)∏0<tk<t-τ2(t)‍(1+hk),C2(t)=c2(t)∏0<tk<t‍1+dk21+hkm-1,A=∏0<tk<t‍1+dk2,0<A<1.Lemma 1.
Suppose that (H1) and (H2) hold; then we have the following.(i)
Ifp(t),q(t)T is a solution of systems (6) and (7), then x(t),y(t)T is a solution of systems (4) and (5), where(9)xt=∏0<tk<t‍1+dkpt,yt=∏0<tk<t‍1+hkqt.(ii)
Ifx(t),y(t)T is a solution of systems (4) and (5), then p(t),q(t)T is a solution of systems (6) and (7), where(10)p(t)=∏0<tk<t‍1+dk-1xt,q(t)=∏0<tk<t‍1+hk-1y(t).Proof.
(i) For anyt≠tk, k∈Z+, we have(11)x˙t-xtr1t-b1txt-τ1t+c1(t)x2(t)x2(t)+K2ym(t)=∏0<tk<t‍(1+dk)×p˙(t)-p(t)∏0<tk<t‍1+hkmqm(t)c1(t)∏0<tk<t(1+dk)p2(t)∏0<tk<t1+dk2p2(t)+K2×r1(t)-b1(t)∏0<tk<t-τ1(t)‍(1+dk)p(t-τ1(t))-c1(t)∏0<tk<t(1+dk)p2(t)∏0<tk<t1+dk2p2(t)+K2∏0<tk<t‍1+hkmqm(t)=0.Similarly, we have(12)y˙t-yt-r2t-b2tyt-τ2t-c2(t)x2(t)x2(t)+K2ym(t)=∏0<tk<t‍1+hk×q˙(t)-q(t)c2(t)∏0<tk<t1+dk2p2(t)∏0<tk<t1+dk2p2(t)+K2×-r2t-b2t∏0<tk<t-τ2t‍1+hkqt-τ2t-c2(t)∏0<tk<t1+dk2p2(t)∏0<tk<t1+dk2p2(t)+K2∏0<tk<t‍1+hkm-1qm(t)=0.On the other hand, for any t=tk, by definition, we obtain that(13)x(tk+)=limt→tk+∏0<ti<t‍(1+di)p(t)=∏0<ti≤tk‍(1+di)p(tk),x(tk+)=limt→tk+∏0<ti<t‍(1+hi)q(t)=∏0<ti≤tk‍(1+hi)q(tk),and xtk=∏0<ti<tk1+diptk, y(tk)=∏0<ti<tk(1+hi)q(tk). Then we get xtk+=1+dkxtk, y(tk+)=(1+hk)y(tk). Hence, x(t),y(t)T is a solution of systems (4) and (5).(ii)
x(t)=∏0<tk<t(1+dk)p(t) and y(t)=∏0<tk<t(1+hk)q(t) are continuous on each interval tk,tk+1. Since x(tk+)=(1+dk)x(tk) and y(tk+)=(1+hk)y(tk), we have(14)ptk+=∏0<ti≤tk‍1+di-1x(tk+)=∏0<ti<tk‍1+di-1x(tk)=p(tk),ptk+=∏0<ti≤tk‍1+hi-1y(tk+)=∏0<ti<tk‍1+hi-1y(tk)=q(tk).And combining(15)ptk-=∏0<ti≤tk-1‍1+di-1x(tk-)=∏0<ti<tk‍1+di-1x(tk)=p(tk),qtk-=∏0<ti≤tk-1‍1+hi-1y(tk-)=∏0<ti<tk‍1+hi-1ytk=qtk,we know that p(t) and q(t) are continuous on interval -τ,+∞.
Similarly,(16)p˙t-ptr1t-B1tpt-τ1t+C1(t)p2(t)Ap2(t)+K2qm(t)=0,q˙t-qt-r2t-B2tqt-τ2t-C2(t)p2(t)Ap2(t)+K2qm(t)=0.Therefore, p(t),q(t)T is a solution of systems (6) and (7).From Lemma1, we notice that if we want to discuss the existence of an ω-periodic solution of systems (4) and (5), we only need to discuss the existence of an ω-periodic solution of systems (6) and (7).LetX and Y be two normed linear spaces, let L:DomL⊂X→Y be a linear map, and let N:X→Y be a continuous map. If dimKerL=codimImL<+∞ and ImL is closed in Y, then L is called Fredholm operator. If L is a Fredholm operator with index zero, there exist continuous projections P:X→X and Q:Y→Y such that KerL=ImP and KerQ=ImL=ImI-Q. Hence, LDom∩KerP:(I-P)X→ImL has an inverse mapping KP. The mapping N is called L-compact on Ω¯, if Ω∈X is an open bounded subset, and   QN(Ω¯) is bounded, Kp(I-Q)N:Ω¯→X is compact. Since KerL is isomorphic to ImQ, there exists an isomorphism J:ImQ→KerL.Lemma 2 (see [9], (Continuation Theorem)).
Let bothX and Y be Banach spaces; L:DomL⊂X→Y is a Fredholm operator with index zero; continuous projection N:Ω¯→Y is called L-compact on Ω¯, where Ω is an open bounded subset on X. If all the following conditions hold,(1)
for anyλ∈(0,1), each solution of Lx=λNx satisfies x∉∂Ω,(2)
anyx∈KerL∩∂Ω, QNx≠0,(3)
degJQN,Ω∩KerL,0≠0, where J:ImQ→KerL is an isomorphism, then the equation Lx=Nx has at least one solution in DomL∩Ω¯.Lemma 3 (see [10]).
Ifa>0, b>0, and x˙≥(≤)x(a-bxγ), where γ is a positive constant, then(17)limt→∞infxt≥ab1/γ,limt→∞supx(t)≤ab1/γ.Lemma 4 (see [11]).
Ifτ∈C1(R,R) with τ(t+ω)=τ(t) and 1-τ˙(t)>0 for any t∈[0,ω], then function μ(t)=t-τ(t) has a unique inverse μ-1(t) satisfying μ∈C(R,R) with μ-1(s+ω)=μ-1(s)+ω, for s∈[0,ω].


## 3. Main Results

In order to express the formulas conveniently, we introduce a few concepts(18)f¯=1ω∫0ω‍ftdt,fL=minft,fU=maxf(t),t∈0,ω,where f is a periodic function with period ω.Letp(t),q(t)T∈R2 be an arbitrary positive solution of systems (6) and (7), for all t∈R. Set u(t)=lnp(t) and v(t)=lnq(t). Consider the following model:(19)u˙(t)=r1(t)-B1(t)eu(t-τ1(t))-C1(t)eu(t)+mv(t)Ae2u(t)+K2,v˙(t)=-r2(t)-B2(t)ev(t-τ2(t))+C2(t)e2u(t)+(m-1)v(t)Ae2u(t)+K2.Apparently, if system (19) has an ω-periodic solution u(t),v(t)T, then (p(t),q(t))T=eu(t),ev(t)T is an ω-periodic solution of systems (6) and (7). Hence, we only need to show that system (19) has an ω-periodic solution.Set(20)X=Y=z(t)=u(t),v(t)T∈C(R,R2)∣z(t+ω)=z(t),where C(R,R2)=z:R→R2∣ziscontinuousatt≠tk,z(tk+),z(tk-)=z(tk)exist and the norm z=u(t),v(t)T=maxu(t)+v(t), t∈[0,ω]. Then both X and Y are Banach spaces.Define operatorsL, P, and Q as follows, respectively:(21)L:DomL⊂X⟶Y,Lz=z˙;P(z)=1ω∫0ω‍z(t)dt;Q(z)=1ω∫0ω‍z(t)dt.Define N:X→Y, satisfying(22)Nz=r1(t)-B1(t)eu(t-τ1(t))-C1(t)eu(t)+mv(t)Ae2u(t)+K2-r2(t)-B2(t)ev(t-τ2(t))+C2(t)e2u(t)+(m-1)v(t)Ae2u(t)+K2.Note that KerL=R2, dimKerL=codimImL=2, and ImL=z∈Y∣∫0ω‍ztdt=0 is closed in Y and P and Q are continuous maps satisfying KerL=ImP and KerQ=ImL=ImI-Q. Hence, L is a Fredholm operator with index zero. It implies that L has a unique inverse KP:ImP→DomL∩KerP. So we have(23)KP(z)=∫0t‍z(s)ds-1ω∫0ω‍∫0t‍zsdsdt.By a straightforward calculation, we get(24)QNz=1ω∫0ω‍r1(t)-B1(t)eu(t-τ1(t))-C1(t)eu(t)+mv(t)Ae2u(t)+K2dt1ω∫0ω‍-r2(t)-B2(t)ev(t-τ2(t))+C2(t)e2u(t)+(m-1)v(t)Ae2u(t)+K2dt,KPI-QNz=∫0t‍r1(s)-B1(s)eu(s-τ1(s))-C1(s)eu(s)+mv(s)Ae2u(s)+K2ds∫0t‍-r2(s)-B2(s)ev(s-τ2(s))+C2(s)e2u(s)+(m-1)v(s)Ae2u(s)+K2ds-E+F(t),where(25)E=1ω∫0ω‍∫0t‍r1(s)-B1(s)eu(s-τ1(s))-C1(s)eu(s)+mv(s)Ae2u(s)+K2dsdt1ω∫0ω‍∫0t‍-r2(s)-B2(s)ev(s-τ2(s))+C2(s)e2u(s)+(m-1)v(s)Ae2u(s)+K2dsdt,F(t)=(12-tω)∫0ω‍r1(s)-B1(s)eu(s-τ1(s))-C1(s)eu(s)+mv(s)Ae2u(s)+K2ds(12-tω)∫0ω‍-r2(s)-B2(s)ev(s-τ2(s))+C2(s)e2u(s)+(m-1)v(s)Ae2u(s)+K2ds.By the Lebesgue Convergence Theorem, it is not difficult to notice that QN and KP(I-Q)N are continuous. By applying the Arzela-Ascoli Theorem, we know that the operator QN(Ω¯) is bounded and KP(I-Q)N(Ω¯) is compact, for any open set Ω∈X. Therefore, N∈Ω is L-compact on Ω¯.In order to use Lemma2, we need to find an appropriate open and bounded set Ω.Theorem 5.
Assume the following.
(i)C¯2>Ar¯2+Δ2U, where Δ2=B2μ2-1(t)/1-τ˙2μ2-1t, min1-τ˙2t>0.
(ii) The following algebraic equation set(26)r¯1-B¯1eu(t)-C¯1eu(t)+mv(t)Ae2u(t)+K2=0,-r¯2-B¯2ev(t)+C¯2e2u(t)+(m-1)v(t)Ae2u(t)+K2=0,has finite solutions ui∗(t),vi∗(t), i=1,2,…,n, and then systems (6) and (7) have at least one ω-periodic solution.Proof.
Considering the operator equationLz=λNz, λ∈(0,1), we have(27)u˙(t)=λr1(t)-B1(t)eu(t-τ1(t))-C1(t)eu(t)+mv(t)Ae2u(t)+K2,v˙(t)=λ-r2(t)-B2(t)ev(t-τ2(t))+C2(t)e2u(t)+(m-1)v(t)Ae2u(t)+K2.Integrating (27) on the interval [0,ω], we have(28)r¯1ω=∫0ω‍B1(t)eu(t-τ1(t))+C1(t)eu(t)+mv(t)Ae2u(t)+K2dt,r¯2ω+∫0ω‍B2(t)ev(t-τ2(t))dt=∫0ω‍C2(t)e2u(t)+(m-1)v(t)Ae2u(t)+K2dt,yielding(29)∫0ω‍u˙(t)dt≤2r¯1ω,∫0ω‍v˙(t)dt≤2r¯2ω+∫0ω‍B2(t)ev(t-τ2(t))dt.In view of Lemma 4, we obtain that(30)∫0ω‍B1(t)eu(t-τ1(t))dt=∫0ω‍B1μ1-1(t)eu(t)1-τ˙1μ1-1(t)dt,∫0ω‍B2(t)ev(t-τ2(t))dt=∫0ω‍B2μ2-1(t)ev(t)1-τ˙2μ2-1(t)dt,which together with (28) give(31)r¯1ω=∫0ω‍B1μ1-1(t)eu(t)1-τ˙1μ1-1(t)+C1(t)eu(t)+mv(t)Ae2u(t)+K2dt,(32)r¯2ω+∫0ω‍B2μ2-1(t)ev(t)1-τ˙2μ2-1(t)dt=∫0ω‍C2(t)e2u(t)+(m-1)v(t)Ae2u(t)+K2dt.From (31), we have(33)∫0ω‍eu(t)dt≤r¯1ωΔ1L≡ωN1,where Δ1=B1μ1-1(t)eu(t)/1-τ˙1μ1-1t.
Multiplying the first equation of system (27) by eu(t) and the second one by ev(t) and integrating them on [0,ω], we get(34)∫0ω‍r1(t)eu(t)dt=∫0ω‍B1μ1-1(t)e2u(t)1-τ˙1μ1-1(t)dt+∫0ω‍C1(t)e2u(t)+mv(t)Ae2u(t)+K2dt,(35)∫0ω‍r2(t)ev(t)dt+∫0ω‍B2μ2-1(t)e2v(t)1-τ˙2μ2-1(t)dt=∫0ω‍C2(t)e2u(t)+mv(t)Ae2u(t)+K2dt,which imply that(36)∫0ω‍e2u(t)+mv(t)Ae2u(t)+K2dt≤r1UC1L∫0ω‍eu(t)dt,(37)∫0ω‍ev(t)dt≤C2Ur2L∫0ω‍e2u(t)+mv(t)Ae2u(t)+K2dt≤r1UC2UωN1r2LC1L≡ωN2.
Ifv(t)≥0, then ev(t)≥1 and N2≥(1/ω)∫0ω‍ev(t)dt≥1, which implies that there must be a constant η1∈[0,ω] such that v(η1)≤lnN2. From (33) and (37),(38)N1≥1ω∫0ω‍eutdt≥r2LC1Lr1UC2Uω∫0ω‍evtdt≥r2LC1Lr1UC2U;then there exists a constant ξ1∈[0,ω] such that u(ξ1)≤maxlnN1,ln⁡(r2LC1L/r1UC2U).
On the other hand, it follows from (35) and the Hölder inequality that(39)r2Lω≤C2UA∫0ω‍emvtdt≤C2UAω1-m∫0ω‍evtdtm;that is(40)∫0ω‍ev(t)dt≥r2LωAC2Uω1-m1/m≡ωN3.
Sincev(t)≥0 and m<1, then 0<e(m-1)v(t)<1 and ev(t)≥1.
From (32), r¯2ω+Δ2Lω≤(C2U/K2)∫0ω‍e2u(t)dt, that is,(41)∫0ω‍e2u(t)dt≥r¯2+Δ2LωK2C2U≡ωN4.
Ifv(t)<0, then 0<emv(t)<1, 0<ev(t)<1, and e(m-1)v(t)>1.
From (31), we obtain that(42)r¯1ω≤Δ1U∫0ω‍eu(t)dt+C1UK2∫0ω‍eu(t)+mv(t)dt≤Δ1U∫0ω‍eu(t)dt+C1UK2∫0ω‍eu(t)dt,that is,(43)∫0ω‍eu(t)dt≥r¯1ωK2Δ1UK2+C1U≡ωN5.Together with (33) and (41), we notice that there exists a constant ξ2∈[0,ω] such that(44)u(ξ2)≤maxlnN1,lnr2LC1Lr1UC2U,12lnN4,lnN5≡A1.Furthermore, according to the condition C¯2>Ar¯2+Δ2U and (32), we have(45)∫0ω‍C2(t)e(m-1)v(t)dt-Ar¯2ω+∫0ω‍B2μ2-1(t)ev(t)1-τ˙2μ2-1(t)dt=∫0ω‍K2C2tem-1vtAe2ut+K2dt;then(46)0<ωC¯2-Ar¯2-AΔ2U≤K2C2UA∫0ω‍e(m-1)v(t)e2u(t)dt≤K2C2Uω3/2A∫0ω‍e2(m-1)v(t)dt1/2∫0ω‍eu(t)dt2≤K2C2Uω3/2Aω2N52∫0ω‍e2(m-1)v(t)dt1/2,that is(47)∫0ω‍e2(m-1)v(t)dt≥ω3/2N52AC¯2-Ar¯2-AΔ2UK2C2U2≡ωN6.We can find a constant η2∈[0,ω] such that(48)v(η2)≤maxlnN2,lnN3,lnN62(m-1)≡A2.Hence, (19), (44), and (48) combine together to make us know that(49)u(t)≤u(ξ2)+12∫0ω‍u˙(t)dt≤A1+r¯1ω≡D1,vt≤vη2+12∫0ω‍v˙tdt≤A2+r¯2ω+Δ2UωeA2≡D2.
Consider the algebraic equation set(50)r¯1-B¯1eu(t)-C¯1eu(t)+mv(t)Ae2u(t)+K2=0,-r¯2-B¯2ev(t)+C¯2e2u(t)+(m-1)v(t)Ae2u(t)+K2=0.From the assumption, there exist finite solutions ui∗(t),vi∗(t), i=1,2,…,n, of the above system. By simple computing, we can get its Jacobian matrix(51)Gu,v=-B¯1eu-C¯1eu+mvAe2u+K2+2Ae3u+mvAe2u+K22-mC¯1eu+mvAe2u+K22C¯2e2u+m-1vAe2u+K2-2AC¯2e4u+m-1vAe2u+K22-B¯2ev+m-1C¯2e2u+m-1vAe2u+K2,obviously det(G(u,v))>0.
SetW=A1+A2+δ and Ω=z∈X∣z<W, z(tk+)∈Ω, where δ is a large enough number satisfying ui∗+vi∗<δ. If z∈∂Ω∩KerL=∂Ω∩R2, then z satisfies(52)QNz=r¯1-B¯1eu(t)-C¯1eu(t)+mv(t)Ae2u(t)+K2-r¯2-B¯2ev(t)+C¯2e2u(t)+(m-1)v(t)Ae2u(t)+K2≠0,and degJQNz,Ω∩KerL,0=∑i=1nsigndetG(ui∗,vi∗)=n≠0, where J:ImQ→KerL is an isomorphism. Hence, Ω is a bounded open set. System (19) has at least one ω-periodic solution u∗(t),v∗(t)T in DomL∩Ω¯; that is, system (6) has at least one ω-periodic solution eu∗(t),ev∗(t)T.Corollary 6.
Suppose that Theorem5 holds; system (4) also has at least one ω-periodic solution x∗(t),y∗(t)=∏0<tk<t(1+dk)eu∗(t),∏0<tk<t(1+hk)ev∗(t).Now, we discuss the permanence of models (6) and (7). Before the main results, we give the definition of permanence.Definition 7.
System (6) is permanent, if there exist positive constants mi, Mi0<mi<Mi,i=1,2 and time t0>0 such that any solution of system (6) with initial condition (7) satisfies mi≤xit≤Mi for all t≥t0.
Denote(53)M1=r1Uer1UτB1L,m1=r1LB1U-C1UM2mM1B1UK2×expr1L-B1UM1-C1UM2mM1K2τ,M2=C2UAr2L+c1/(1-m),m2=C2Lm12r2U+B2UM2-1AM12+K2-c1/(1-m).Theorem 8.
Ifr1L-C1UM1M2m/K2>0, then system (6) is permanent.Proof.
From the first equation of model (6), we have p˙(t)≤r1(t)p(t), integrating it on interval t-τ1(t),t,(54)pt≤pt-τ1(t)e∫t-τ1(t)t‍r1(s)ds≤pt-τ1(t)er1Uτ1(t)≤pt-τ1ter1Uτ,t>τ;then we get(55)p˙t≤ptr1U-B1Le-r1Uτpt,t>τ.According to Lemma 3, we have limt→∞supp(t)≤r1Uer1Uτ/B1L≡M1, t>τ, and then there exists a sufficiently large time t1>τ such that p(t)≤M1 for all t>t1.
At the same time, we can find timet2=t1+τ such that pt-τ1(t)≤M1 for t>t2.
From the second equation of model (6), we have(56)q˙t≤-r2Lqt+C2UAqmt=qmtC2UA-r2Lq1-mt;that is,(57)dq1-m(t)dt≤(1-m)C2UA-r2Lq1-m(t).By solving equation, we obtain that(58)q1-mt≤C2UAr2L+q1-m0-C2UAr2Le-r2L1-mt;then there must be t3>t2 such that q(t)≤C2U/Ar2L+c1/(1-m)≡M2 for all t>t3, where c is a constant.
Further, there exists timet4=t3+τ such that qt-τ2(t)≤M2 for all t>t4. In view of the first equation of model (6), we have(59)p˙t≥ptr1L-B1Upt-τ1t-C1UM2mM1K2,t>t4.Assume that pt^ is local minimal value of system (59); then p˙t^=0. Thus, we have that(60)r1L-B1Upt^-τ1t^-C1UM2mM1K2≤0,t>t4,that is,(61)pt^-τ1(t^)≥r1LB1U-C1UM2mM1B1UK2.Integrating (59) on t^-τ1(t^),t^ and noticing that(62)lnpt^pt^-τ1t^≥∫t^-τ1(t^)t^‍r1L-B1Upt-τ1(t)-C1UM2mM1K2dt≥r1L-B1UM1-C1UM2mM1K2τ,we can easily get that(63)pt^≥r1LB1U-C1UM2mM1B1UK2×expr1L-B1UM1-C1UM2mM1K2τ≡m1.Thus, there must be a large enough time t5>t4 such that p(t)≥p(t^)≥m1 for t>t5.
In view of the second equation of system (6), we have(64)q˙t≥qmtC2Lm12AM12+K2-r2U+B2UM2q1-mt,t>t5.
Similarly, by solving the equation, we obtain that(65)q1-m(t)≥C2Lm12r2U+B2UM2-1AM12+K2+q1-m(0)-C2Lm12r2U+B2UM2-1AM12+K2e(m-1)(r2U+B2UM2)t.For the above constant c, we can seek time t6>t5 such that, for t>t6,(66)q(t)≥C2Lm12r2U+B2UM2-1AM12+K2-c1/(1-m)≡m2.
In summary,m1≤p(t)≤M1 and m2≤q(t)≤M2 for t>t6. Therefore, system (6) is permanent.Theorem 9.
Ifr1L-C1UM1M2m/K2>0, then system (4) is permanent and enters eventually into the region D, where(67)D=x(t),y(t)∣x(t)∈∏0<tk<t‍(1+dk)m1,∏0<tk<t‍1+dkM1,y(t)∈∏0<tk<t‍1+hkm2,∏0<tk<t‍(1+hk)M2.Corollary 10.
Ifr1L-C1UM1M2m/K2>0, then systems (6) and (7) have a positive ω-periodic solution.Definition 11.
System (6) is globally attractive, if(68)limt→+∞p(t)-p∗(t)+q(t)-q∗(t)=0,for any two positive solutions p(t),q(t) and p∗(t),q∗(t) of systems (6) and (7).Theorem 12.
If system (6) satisfies r1L-C1UM1M2m/K2>0, liminft→+∞Φ(t)>0, and liminft→+∞Ψ(t)>0, then system (6) is globally attractive, where(69)Φt=αB1(t)-αAC1(t)M12M2mAm12+K22-2K2C2(t)M1M2m-1Am12+K22-∫tμ1-1(t)‍B1(ξ)dξ·αr1t+B1tM1+C1tM1M2mAm12+K2+αC1(t)M1M2mAM12+K2Am12+K22+2K2C2(t)M1M2mAm12+K22+αM1B1μ1-1(t)1-τ˙1μ1-1(t),Ψt=B2(t)-αC1(t)M1mm2m-1Am12+K2-∫tμ1-1(t)‍B1(ξ)dξ·αC1(t)M12mm2m-1Am12+K2+r2(t)+B2(t)M2+C2(t)M12M2m-1Am12+K2+C2(t)M12M2(1-m)m2m-2Am12+K2-M2B2μ2-1(t)1-τ˙2μ2-1(t)∫μ2-1(t)μ2-1μ2-1(t)‍B2(ξ)dξ.Proof.
Assumep(t),q(t) and p∗(t),q∗(t) are two positive solutions of systems (6) and (7). According to Theorem 8, there exist two positive constants mi, Mi(0<mi<Mi,i=1,2) and t6 such that, for t>t6,(70)m1≤p(t),p∗(t)≤M1,m2≤q(t),q∗(t)≤M2.Define a function(71)V1t=αlnpt-lnp∗t+lnqt-lnq∗t,t>t6.By calculating its upper right derivative along the solution of system (6), we get that(72)D+V1t=αsignp(t)-p∗(t)p˙(t)p(t)-p˙∗(t)p∗(t)+signq(t)-q∗(t)q˙(t)q(t)-q˙∗(t)q∗(t)=αsignpt-p∗t×B1tp∗t-τ1t-pt-τ1t+C1(t)p∗(t)q∗m(t)Ap∗2(t)+K2-p(t)qm(t)Ap2(t)+K2+C1(t)p∗(t)q∗m(t)Ap∗2(t)+K2-p(t)qm(t)Ap2(t)+K2+signqt-q∗t×B2tq∗t-τ2t-qt-τ2t+C2(t)p2(t)qm-1(t)Ap2(t)+K2-p∗2(t)q∗m-1(t)Ap∗2(t)+K2+C2(t)p2(t)qm-1(t)Ap2(t)+K2-p∗2(t)q∗m-1(t)Ap∗2(t)+K2≤-αB1(t)pt-p∗t+αB1(t)∫t-τ1tt‍p˙s-p˙∗sds+αC1(t)p∗(t)Ap∗2(t)+K2qm(t)-q∗m(t)+αAC1(t)p(t)p∗(t)qm(t)Ap∗2(t)+K2Ap2(t)+K2p(t)-p∗(t)-B2(t)q(t)-q∗(t)+B2(t)∫t-τ2(t)t‍q˙(s)-q˙∗(s)ds+K2C2(t)q∗m-1(t)p(t)+p∗(t)Ap∗2(t)+K2Ap2(t)+K2p(t)-p∗(t),where(73)∫t-τ1(t)t‍p˙(s)-p˙∗(s)ds=∫t-τ1(t)t‍p∗(s)q∗m(s)Ap∗2(s)+K2-p(s)qm(s)Ap2(s)+K2r1(s)-B1(s)ps-τ1(s)-C1(s)p(s)Ap2(s)+K2qm(s)×p(s)-p∗(s)+p∗(s)B1(s)p∗s-τ1(s)-ps-τ1(s)+p∗(s)C1(s)p∗(s)q∗m(s)Ap∗2(s)+K2-p(s)qm(s)Ap2(s)+K2ds≤∫t-τ1tt‍r1s+B1sps-τ1s+C1spsqmsAp2s+K2×ps-p∗s+B1(s)p∗(s)p∗s-τ1s-ps-τ1s+C1sp∗2sAp∗2s+K2qms-q∗ms+C1sp∗sqmsAp∗sps+K2Ap∗2s+K2Ap2s+K2×r1s+B1sps-τ1s+C1spsqmsAp2s+K2ps-p∗sds,∫t-τ2(t)t‍q˙(s)-q˙∗(s)ds=∫t-τ2(t)t‍-r2(s)-B2(s)qs-τ2(s)+C2(s)p2(s)Ap2(s)+K2qm-1(s)+C2(s)p2(s)Ap2(s)+K2qm-1(s)×q(s)-q∗(s)+q∗(s)B2(s)q∗s-τ2(s)-qs-τ2(s)+q∗(s)C2(s)×p2(s)qm-1(s)Ap2(s)+K2-p∗2(s)q∗m-1(s)Ap∗2(s)+K2ds≤∫t-τ2tt‍r2(s)+B2(s)qs-τ2(s)+C2(s)p2(s)qm-1(s)Ap2(s)+K2+C2(s)p2(s)qm-1(s)Ap2(s)+K2×q(s)-q∗(s)+q∗(s)B2(s)q∗s-τ2(s)-qs-τ2(s)+C2(s)p2(s)q∗(s)Ap2(s)+K2qm-1(s)-q∗m-1(s)+K2C2(s)q∗m(s)p∗(s)+p(s)Ap∗2(s)+K2Ap2(s)+K2×p(s)-p∗(s)r2(s)+B2(s)qs-τ2(s)+C2(s)p2(s)qm-1(s)Ap2(s)+K2ds.Thus, for t>t6,(74)D+V1t≤-αB1tpt-p∗t+αB1(t)×∫t-τ1(t)t‍r1(s)+B1(s)M1+C1(s)M1M2mAm12+K2×p(s)-p∗(s)+M1B1(s)p∗s-τ1(s)-ps-τ1(s)+C1sM12Am12+K2qms-q∗ms+C1(s)M1M2mAM12+K2Am12+K22×p(s)-p∗(s)r1(s)+B1(s)M1+C1(s)M1M2mAm12+K2ds+αC1(t)M1Am12+K2qm(t)-q∗m(t)+αAC1(t)M12M2mAm12+K22p(t)-p∗(t)-B2(t)q(t)-q∗(t)+B2t×∫t-τ2tt‍+2K2C2(s)M1M2mAm12+K22p(s)-p∗(s)r2s+B2sM2+C2sM12M2m-1Am12+K2×qs-q∗s+M2B2(s)q∗s-τ2(s)-qs-τ2(s)+C2sM12M2Am12+K2qm-1s-q∗m-1s+2K2C2(s)M1M2mAm12+K22p(s)-p∗(s)ds+2K2C2(t)M1M2m-1Am12+K22p(t)-p∗(t)=-αB1(t)p(t)-p∗(t)+αB1(t)∫t-τ1tt‍W1(s)ds+αC1tM1Am12+K2qmt-q∗mt+αAC1(t)M12M2mAm12+K22p(t)-p∗(t)-B2(t)q(t)-q∗(t)+B2(t)∫t-τ2tt‍W2(s)ds+2K2C2tM1M2m-1Am12+K22pt-p∗t,where(75)W1s=r1(s)+B1(s)M1+C1(s)M1M2mAm12+K2p(s)-p∗(s)+M1B1(s)p∗s-τ1(s)-ps-τ1(s)+C1(s)M12Am12+K2qm(s)-q∗m(s)+C1(s)M1M2mAM12+K2Am12+K22p(s)-p∗(s),W2s=r2(s)+B2(s)M2+C2(s)M12M2m-1Am12+K2×q(s)-q∗(s)+M2B2(s)q∗s-τ2(s)-qs-τ2(s)+C2(s)M12M2Am12+K2qm-1(s)-q∗m-1(s)+2K2C2(s)M1M2mAm12+K22p(s)-p∗(s).
Define further(76)V2(t)=α∫tμ1-1t‍∫τ1ξt‍B1ξW1sdsdξ,V3(t)=∫tμ2-1(t)‍∫τ2(ξ)t‍B2(ξ)W2(s)dsdξ.For t>t6, yielding that(77)D+V1(t)+V˙2(t)+V˙3(t)≤-αB1(t)p(t)-p∗(t)+αC1(t)M1Am12+K2qm(t)-q∗m(t)+αAC1(t)M12M2mAm12+K22p(t)-p∗(t)+2K2C2(t)M1M2m-1Am12+K22p(t)-p∗(t)-B2tqt-q∗t+αW1t∫tμ1-1t‍B1ξdξ+W2(t)∫tμ2-1(t)‍B2(ξ)dξ.Further define(78)V4t=αM1∫t-τ1(t)t‍∫μ1-1(η)μ1-1μ1-1(η)‍B1(ξ)B1μ1-1(η)1-τ˙1μ1-1(η)×p(η)-p∗(η)dξdη,V5t=M2∫t-τ2(t)t‍∫μ2-1(η)μ2-1μ2-1(η)‍B2(ξ)B2μ2-1(η)1-τ˙2μ2-1(η)×q(η)-q∗(η)dξdη.Then we choose the Lyapunov functional as follows:(79)Vt=V1t+V2t+V3t+V4t+V5t;then(80)D+V(t)≤-αB1(t)p(t)-p∗(t)+αC1(t)M1Am12+K2qm(t)-q∗m(t)-B2(t)q(t)-q∗(t)+αAC1tM12M2mAm12+K22pt-p∗t+2K2C2(t)M1M2m-1Am12+K22p(t)-p∗(t)+∫tμ1-1t‍B1(ξ)dξ·αr1t+B1tM1+C1tM1M2mAm12+K2×p(t)-p∗(t)+∫tμ1-1t‍B1(ξ)dξ·αC1tM1M2mAM12+K2Am12+K22×p(t)-p∗(t)+∫tμ1-1(t)‍B1(ξ)dξ·αC1(t)M12Am12+K2qm(t)-q∗m(t)+∫tμ2-1t‍B2(ξ)dξ·r2t+B2tM2+C2tM12M2m-1Am12+K2×q(t)-q∗(t)+∫tμ2-1(t)‍B2(ξ)dξ·C2(t)M12M2Am12+K2qm-1(t)-q∗m-1(t)+∫tμ2-1(t)‍B2(ξ)dξ·2K2C2(t)M1M2mAm12+K22p(t)-p∗(t)+∫μ1-1(t)μ1-1μ1-1(t)‍B1(ξ)dξ·αM1B1μ1-1(t)1-τ˙1μ1-1(t)p(t)-p∗(t)+∫μ2-1(t)μ2-1μ2-1(t)‍B2(ξ)dξ·M2B2μ2-1(t)1-τ˙2μ2-1(t)q(t)-q∗(t).
Since0<m<1, then(81)qm(t)-q∗m(t)≤mm2m-1q(t)-q∗(t),qm-1(t)-q∗m-1(t)≤1-mm2m-2qt-q∗t;hence, we have(82)D+V(t)≤-Φ(t)p(t)-p∗(t)-Ψ(t)q(t)-q∗(t).It follows from ΦL>0 and ΨL>0 that(83)D+V(t)≤-ΦLp(t)-p∗(t)-ΨLq(t)-q∗(t),for t>t6. Integrating both sides of the above inequality on t6,t, we have(84)Vt+ΦLpt-p∗t+ΨLqt-q∗t≤V(t6)<+∞.
Sincep(t)-p∗(t) and q(t)-q∗(t) are uniformly continuous on [t6,+∞), by Barbalat’s Lemma [12], we obtain that(85)limt→+∞p(t)-p∗(t)=0,limt→+∞q(t)-q∗(t)=0,that is, limt→+∞p(t)-p∗(t)+q(t)-q∗(t)=0. Therefore, system (6) is globally attractive. Together with(86)limt→+∞x(t)-x∗(t)=limt→+∞∏0<tk<t‍(1+dk)p(t)-p∗(t)=0,limt→+∞yt-y∗t=limt→+∞∏0<tk<t‍(1+hk)qt-q∗t=0,that is, limt→+∞x(t)-x∗(t)+y(t)-y∗(t)=0, system (4) is globally attractive.Corollary 13.
Letτ(t)≡τ, where τ is a nonnegative constant. If system (6) satisfies r1L-C1UM1M2m/K2>0, liminft→+∞Φ(t)>0, and liminft→+∞Ψ(t)>0, then system (6) has a unique positive almost ω-periodic solution which is globally attractive.


## 4. Numerical Simulation

Example 14.
Assumeτ(t)≡τ. Considering the following model as application:(87)p˙t=p(t)8+0.1sint-8p(t-0.05)-(4+0.3sint)p2(t)0.5p2(t)+1q0.5(t),q˙t=q(t)-(2.5-0.5sint)-2q(t-0.1)+(2.5+0.5sint)p2(t)0.5p2(t)+1q0.5(t),corresponding to model (87), we take(88)r1t=8+0.1sint,B1(t)=8,C1(t)=4+0.3sint,r2t=2.5-0.5sint,B2(t)=2,C2(t)=2.5+0.5sint,A=0.5,m=0.5,K=1,τ1(t)=0.05,τ2(t)=0.1.By direct computation, we have C¯2-Ar¯2+Δ2U=0.25>0, which satisfies the conditions of Theorem 5 and Corollary 6. We can see that system (87) has a positive periodic solution (see Figures 1 and 2).Figure 1
The integral curves of prey-time and predator-time.
(a)
(b)Figure 2
The orbits of prey-predator and time-prey-predator.
(a)
(b)Example 15.
Assumeτ(t)≡τ. Considering the following model as application(89)p˙t=p(t)8+0.1sint-8p(t-0.04)-(3+0.3sint)p2(t)0.5p2(t)+1q0.5(t),q˙t=q(t)-(5-0.5sint)-3.5q(t-0.2)+(1.5+0.5sint)p2(t)0.5p2(t)+1q0.5(t),corresponding to model (89), we take(90)r1t=8+0.1sint,B1(t)=8,C1(t)=3+0.3sint,r2t=5-0.5sint,B2(t)=3.5,C2(t)=1.5+0.5sint,A=0.5,m=0.5,K=1,τ1(t)=0.04,τ2(t)=0.2.By direct computation, we have r1L-C1UM1M2m/K2≈1.2237>0, liminft→+∞Φ(t)>0, and liminft→+∞Ψ(t)>0. We can see that system (89) is permanent and has a unique positive 2π-periodic solution, which is globally attractive (see Figures 3 and 4). It is easy to verify the accuracy of Theorems 8, 9, and 12 and Corollaries 10 and 13.Figure 3
The integral curves of prey-time and predator-time.
(a)
(b)Figure 4
The orbits of prey-predator and time-prey-predator.
(a)
(b)


## 5. Conclusion

In the study of population dynamics, we focus on two aspects: (1) the time-varying evolution of the population and (2) how to implement manual intervention to protect, develop, and utilize the population. Precisely, these two issues are reflected in our model. Regarding the first aspect, we take into account the impact of limited resources on population size; that is, its density has a restriction on the growth of the population size. Hence, we not only use the interspecific growth termsb1(t)x(t-τ1(t)) and b2(t)x(t-τ2(t)) to reflect the model, where τ1(t) and τ2(t) are delays, but also consider the interference constant m and Holling III type functional response. Taking into account the second aspect, we propose a regular pulse harvest in the model. The corresponding ecological system of the model we considered is more complex and has practical significance.In this paper, we analyze the existence and global attractivity of positive periodic solutions of a delayed impulsive Lotka-Volterra model with Holing III type functional response. We propose two delays and impulses to describe the model. From Theorem5, we can conclude that the positive periodic solutions of system (4) are delay dependent. This is different from these results that the positive periodic solutions are delay independent, and our conclusion is more general. Furthermore, we have shown the permanence and global attractivity of system (4) under certain conditions. We have found that system (4) has a unique and globally attractive periodic solution, but how can we prove it? We leave it as our work in the future. In addition to delayed and impulsive biological systems, we hope that our analysis can provide valuable design insights and supports to future biological works.



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*Source: 473539-2015-06-23.xml*