# Coexistence for an Almost Periodic Predator-Prey Model with Intermittent Predation Driven by Discontinuous Prey Dispersal

Authors: Yantao Luo; Long Zhang; Zhidong Teng; Tingting Zheng
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

An almost periodic predator-prey model with intermittent predation and prey discontinuous dispersal is studied in this paper, which differs from the classical continuous and impulsive dispersal predator-prey models. The intermittent predation behavior of the predator species only happens in the channels between two patches where the discontinuous migration movement of the prey species occurs. Using analytic approaches and comparison theorems of the impulsive differential equations, sufficient criteria on the boundedness, permanence, and coexistence for this system are established. Finally, numerical simulations demonstrate that, for an intermittent predator-prey model, both the intermittent predation and intrinsic growth rates of the prey and predator species can greatly impact the permanence, extinction, and coexistence of the population.


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## Body

## 1. Introduction

In real ecosystems, since the spatial distribution and dynamics of a population are greatly affected by their spatial heterogeneity and population mobility, dispersal becomes one of the dominant themes in mathematical biology. In fact, animal dispersal movements between patches are extremely prevalent in ecological environments; for example, many types of birds and mammals will migrate from cold regions to warm regions in search of a better habitat or a breeding site [1]. Therefore, to take spatial heterogeneity into account, realistic population models should contain the dispersal process. During the past couple of decades, predator-prey models with diffusion in a patchy environment have attracted significant attention from ecologists, biologists, and biomathematicians. Many important works and monographs about the properties of population dynamics in a spatial idiosyncratic environment, for example, permanence, extinction, and global asymptotic stability of positive periodic solutions, have been written (see [2–14]). Teng and Chen [6] considered a nonautonomous predator-prey Lotka-Volterra type dispersal system with periodic coefficients and distributed delays:(1)x˙1t=x1ta1t-b1tx1t-ct∫-∞0k12syt+sds+∑j=1nd1jtxjt-x1t,x˙it=xitait-bitxit+∑j=1ndijtxjt-xit,i=2,3,…,n,y˙t=yt-et+ft∫-∞0k21sx1t+sds-gt∫-∞0k21sx1t+sds,where y is the population density of the predator species confined to the 1st patch. Criteria for the permanence, extinction, and existence of positive periodic solutions for system (1) were established. In this model, the prey dispersal behavior occurs at every point in time and simultaneously between any patches; that is, it is a continuous bidirectional dispersal.However in practice, it is often the case that diffusion occurs in the form of regular pulses. For example, when winter comes, birds will migrate between patches to seek a better habitat, whereas they do not diffuse in other seasons, and the dispersion of foliage seeds occurs at a fixed period of time every year. For another example, in the Pacific Northwest,Larimichthys polyactis cross over deep water during the winter and migrate to the coast during the spring; then, 3–6 months after spawning, they scatter offshore and return to the depths of the sea during late autumn [15]. All these types of migratory behaviors are appropriately assumed to be in the form of pulses in the modeling process. Thus, impulsive diffusion provides a more natural description. Currently, theories of impulsive differential equations [16] have been introduced into population dynamics. A large number of models have been described by impulsive diffusion (see [14, 17–24]) during the past couple of decades.Shao [23] considered the following predator-prey models with impulsive prey diffusion between two patches:(2)x˙1t=x1tr1-a1x1t-b1yt,x˙2t=x2tr2-a2x2t,y˙t=yt-r3+a3x1t-τ1-b2yt-τ2,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,Δyt=0,t=nτ,where the pulse diffusion of the species x occurs in every period T (a positive constant) and di is the dispersal rate in the ith patch satisfying 0<di<1 for i=1,2. The system evolves from its initial state without being further affected by diffusion until the next pulse appears. Δxi(nT)=xi(nT+)-xi(nT), where xi(nT+) represents the density of the population of the prey species x in the ith patch immediately after the nth diffusion pulse at time t=nT; xi(nT) represents the density of the population of the prey species in the ith patch before the nth diffusion pulse at time t=nT, n=0,1,…. Criteria for the global attractivity and permanence of system (2) were obtained.Furthermore, migration movements of the population will be influenced by many uncertain factors (such as the landscape and weather). Therefore, the dispersal movement of migratory species will have to be suspended when the environment becomes unavailable. In other words, patches would permit normal movement patterns between patches to occur only during certain time intervals instead of all the time. For instance, in the Canary Islands of Spain,Anas platyrhynchos undergo a spring migration from early March to the end of March and a fall migration from late September to the end of October, departing as late as early November, during which they are extensively killed by humans, and other carnivorous animals can prolong the journey [25]. As another example, the wild goose will fly to the south when winter comes; in this process, they will stop to rest in some places and at certain time periods. In other words, their diffusion behavior is neither continuous at all times nor impulsive at a fixed time, but it is intermittent within some time intervals. Therefore, it is more reasonable to model this kind of population dynamics with intermittent dispersals. Zhang et al. [26] considered the following nonautonomous almost periodic single species model with intermittent dispersals and dispersal delays between two patches:(3)x˙1t=x1ta1t-b1tx1t,x˙2t=x2ta2t-b2tx2t,t∈τ2k,τ2k+1,x1τ2k+1=d1x1τ2k+1-,x2τ2k+1=d2x2τ2k+1-,t=τ2k+1,x˙1t=x1ta~1t-b~1tx1t+D12tx2t-τ1-x1t,x˙2t=x2ta~2t-b~2tx2t+D21tx1t-τ2-x2t,t∈τ2k+1,τ2k+2,x1τ2k+2=D1x1τ2k+2-,x2τ2k+2=D2x2τ2k+2-,t=τ2k+2,where all the parameters are almost periodic and the dispersal movement happens only in the time interval t∈τ2k+1,τ2k+2 but not in t∈τ2k,τ2k+1. Here, k∈N. Criteria for the existence, uniqueness, and global attractivity of positive almost periodic solution for system (3) were established.Moreover, in a real ecological system, there always exist natural enemies during the migratory process between patches. For example, annually, at the end of July, with the arrival of the dry season, millions of wildebeests, zebras, and other herbivorous wildlife form a migratory army, migrating from the Serengeti National Park, Tanzania, Africa, to Kenya’s Masai Mara National Nature Reserve to find enough water and food. Along the way, they will be preyed upon by lions, leopards, and so on. Additionally, crocodiles and hippopotamus will wait and ambush the migration species in the Mara River. As the seasons alternate, i.e., when the rainy season comes, the migration movement starts again, and these species will return to the Serengeti National Park, and vice versa [27]. Obviously, the predation behavior for the above situation is intermittent and only happens in the channels where the dispersals of migratory species occur.Motivated by the above consideration, in this paper, we introduce an almost periodic predator-prey model with intermittent predation and discontinuous prey dispersal between two patches:(4)x˙1t=x1ta1t-b1tx1t,x˙2t=x2ta2t-b2tx2t,y˙t=yta3t-b3tyt,t∈τ2k,τ2k+1,x1τ2k+1=d1x1τ2k+1-,x2τ2k+1=d2x2τ2k+1-,yτ2k+1=yτ2k+1-,t=τ2k+1,x˙1t=x1ta~1t-b~1tx1t-c1tyt+D12tx2t-τ1-x1t,x˙2t=x2ta~2t-b~2tx2t-c2tyt+D21tx1t-τ2-x2t,y˙t=yta~3t-b~3tyt+c~1tx1t+c~2tx2t,t∈τ2k+1,τ2k+2,x1τ2k+2=D1x1τ2k+2-,x2τ2k+2=D2x2τ2k+2-,yτ2k+2=yτ2k+2-,t=τ2k+2,where xi(t) denotes the prey population density in the ith patch (i=1,2) and y(t) represents the predator population density in channels between two patches. When t∈τ2k,τ2k+1 with k∈N, the prey species xi inhabits the ith patch and does not disperse. At the same time, the predator species y inhabits the channels between two patches with other food sources. When t→τ2k+1, the intrinsic discipline of the species x in each patch changes. The channels between the two patches will open, and the species x disperses bidirectionally from one patch to another; this dispersal movement will continue for the time interval t∈τ2k+1,τ2k+2. Meanwhile, the predator species y preys on the species x in the channels. When t→τ2k+2, the gate of the channels will close, the species xi will stop dispersing and inhabit patch i. At the same time, the predator species y will also stop preying on species x. Obviously, the predation behavior only happens in the time interval t∈τ2k+1,τ2k+2; that is, it is intermittent. Here, ai(t),a~i(t)(i=1,2) represent the intrinsic growth rates of the species x in the ith patch over the time intervals τ2k,τ2k+1 and τ2k+1,τ2k+2, respectively; a3(t),a~3(t) denote the intrinsic growth rates of the species y over the time intervals τ2k,τ2k+1 and τ2k+1,τ2k+2, respectively; bi(t),b~i(t)(i=1,2) represent the intercompetition rates of the species x in the ith patch over the time intervals τ2k,τ2k+1 and τ2k+1,τ2k+2, respectively; b3(t),b~3(t) denote the intercompetition rates of the species y over the time intervals τ2k,τ2k+1 and τ2k+1,τ2k+2, respectively; di represents the survival rates of switching from stage 1 (without dispersal) to stage 2 (dispersal movement); Di denotes the survival rates of switching from stage 2 to stage 1; Dij(t) is the dispersal rates from the ith patch to the jth patch during the time interval τ2k+1,τ2k+2; and τi is the time for the population to disperse from patch j to i(i≠j,i,j=1,2). During the whole process, the predator species y never disperses.In allusion to system (4) above, our main purpose in this paper is to establish a series of criteria on the ultimate boundedness, permanence, and coexistence of the two populations for system (4). The methods used in this paper are motivated by the works on the permanence and extinction for periodic predator-prey systems in patchy environments given by Teng and Chen in [6] and the works on the survival analysis for a periodic predator-prey model given by Zhang et al. in [22].This paper is organized as follows. In Section2, some definitions, assumptions, and useful lemmas are introduced. In Section 3, we state and prove the main results. Finally, special examples and numerical simulations are illustrated to demonstrate our theoretical results in Section 4.


## 2. Preliminaries

LetR and R2 denote the set of real numbers and the 2-dimensional Euclidean linear space, respectively, and τk be a time sequence, satisfying τk<τk+1, with τk→∞ as k→∞. Here, k∈N. Define τ^=max⁡{τ1,τ2}, PCt-τ^,t,R2={ψ:([t-τ^,t]→R2)∣φ=(φ1,φ2) is continuous everywhere except at t=τk∈t-τ^,t, and φ(τk+),φ(τk-) exist with φ(τk+)=φ(τk)}. Define BPC={φ∈PC-∞,0,R2:φisbounded}; the norm of φ is defined by φ=supθ∈-τ^,0φθ. Let BPC+={φ=φ1,φ2∈BPC:φi(θ)≥0 for all θ∈[-∞,0] and φi(0+)>0 for i=1,2}.In this paper, we assume that all solutions of system (4) satisfy the following initial conditions:(5)xis,ϕ=ϕis,s∈-τ^,0,xi0+,ϕ=ϕi0+,s∈-τ^,0,where ϕ=(ϕ1,ϕ2)∈BPC+. It is not hard to prove that the functional of right of system (4) is continuous and satisfies the local Lipschitz condition with respect to ϕ in the space R×BPC. Therefore, by the fundamental theory of the impulsive functional differential equations with finite delays [16, 28, 29], system (4) has a unique solution (x1(t,ϕ),x2(t,ϕ),y(t)) satisfying the initial conditions (5). Obviously, the solution (x1(t,ϕ),x2(t,ϕ),y(t)) is positive in its maximal interval of the existence.Before going into details, we first draw some very useful definitions and lemmas.Definition 1 (see [30]).
The set of sequences{τki∣τki=τk+1-τk,k,i∈N,τk∈B} is said to be uniformly almost periodic if, for arbitrary ε>0, there exists a relatively dense set in R of ε-almost periodic common for all of the sequences; here B={τk∣τk∈R,τk<τk+1,k∈N,limk→±∞τk=±∞}.Definition 2 (see [30]).
Assume that the following conditions hold:(1)
The set of sequences{τki} is almost periodic, k,i∈N.(2)
For anyε>0 there exists a real number δ>0 such that if the points t′,t′′ belong to the same interval of continuity of φ(t) and satisfy the inequality t′-t′′<δ, then φt′-φt′′<ε.(3)
For anyε>0 there exists a relatively dense set C such that if σ∈C, then φt+σ-φt<ε for all t∈R satisfying the condition t-σk>ε,k∈N. The elements of C are called ϵ-almost periods.
We claim that functionφ∈PC([t-τ^,t],R2) is almost periodic, and we denote φ∈APC.Definition 3 (see [31]).
System (4) is said to be permanent if there exist positive constants m and M, such that for any positive solutions x(t)=(x1(t),x2(t),y(t)) of system (4) with initial value that satisfy condition (5)(6)m≤liminft→∞xit≤limsupt→∞xit≤M,i=1,2,m≤liminft→∞yt≤limsupt→∞yt≤M.Definition 4 (see [24]).
LetV:R×R2→R2, then V is said to belong to class L if(i)
V is continuous in (τk,τk+1)×R2 for each x∈R2,k∈N, (7)limt,y→τk+1-,xVt,y=Vτk+1,x.(ii)
V is locally Lipschitzian in x.
In this paper, there are some notations and assumptions that shall be used:(N1)
Iff(t),t∈R, is an almost periodic function, we define(8)Af=limt→∞1T∫tt+Tfsds,fu=supt∈Rft,fl=inft∈Rft,where T is a positive constant.(N2)
We letλi(t)=t-τi,λi-1(t) be an inverse function of the function λi(t)(i=1,2).(H1)
Functionsai(t),a~i(t),bi(t),b~i(t), and Dij(t) are τ-almost periodic and bounded continuous functions for all t∈R and bil≥0,b~il≥0,0<di,Di≤1(i=1,2,3,j=1,2).(H2)
A(ai(t))>0,A(a~i(t))>0,A(bi(t))>0,A(b~i(t))>0(i=1,2).(H3)
There exists a constantτ>0, such that, for any t≥0, (9)∫tt+τ-1k+12ais+-1k+1+12a~isds+∑t<τk+1≤t+τln⁡hik>0i=1,2,k∈N.(H4)
There exists a constantξi>0(i=1,2), such that, for any t≥0,(10)∑t<τk+1≤t+sln⁡hik≤ξi,s∈0,τ.(H5)
A(aik(t)-dijk(t))>0(i,j=1,2,k∈N).(H6)
There exist constantsθ>0,ei>0(i=1,2), such that limt→∞infBi(t)=limt→∞[ei(bik(t))-dijk(λj-1(t))/m~]≥θ, where m~=max⁡1,m and [m] is the integer part of m(k∈N).(H7)
The set of sequences{τki},k∈N, is uniformly almost periodic, and τk+2=τk+τ,infk∈Nτk+1-τk>0, where τ belongs to the relatively dense set T(i=1,2,3).(H8)
There exists a constantτ~>0, such that, for any t≥0,(11)∫tt+τ~-1k+12a3s+-1k+1+12a~3sds>0.
Now, we give some useful lemmas which will be used in the proofs of the main results.
If there is no predatory in system (4), we have the following predator-free system:(12)x˙it=xitaikt-biktxit+dijktxjt-τi-xit,t∈τk,τk+1,xiτk+1=hikxiτk+1-,t=τk+1,where (13)aikt=-1k+12ait+-1k+1+12a~it,bikt=-1k+12bit+-1k+1+12b~it,dijkt=-1k+1+12Dijt,hik=-1k+12di+-1k+1+12Di,i≠j,i,j=1,2.For system (12), we have the following result.Lemma 5 (see [26] Theorem 3.4).
Suppose that assumptions(H1)–(H7) hold; then system (12) has a unique globally attractive positive τ-almost periodic solution x∗(t)=(x1∗(t),x2∗(t)).If there is no preyx in system (4), we have the following prey-free system:(14)y˙t=yta3kt-b3ktyt,t∈τk,τk+1,yτk+1=yτk+1-,t=τk+1,where(15)a3kt=-1k+12a3t+-1k+1+12a~3t,b3kt=-1k+12b3t+-1k+1+12b~3t.For system (14), we have the following result.Lemma 6 (see [26]).
Suppose assumptions(H1)-(H2) and (H7)-(H8) hold; then system (14) has a unique globally attractive positive τ-almost periodic solution y∗(t)∈APC.For(t,x)∈(τk,τk+1)×R2, we define(16)D+Vt,x=limsuph→0+1hVt+h,x+hft,x-Vt,x,where f=(f1,f2) is the right-hand side of system (12). We give the following vector comparison results of the impulsive differential equations.Lemma 7 (see [1]).
LetV:R+×R+2 and V∈L. Assume that(17)D+Vt,x≤gt,Vt,xt∈τk,τk+1,Vt,xt≤ψkVt,x,t=τk+1,where g(t,u):R×R2→R2 is continuous in (τk,τk+1)×R2 and quasi-monotone nondecreasing in u, for v∈R2,lim(t,u)→(τk+1-)g(t,u)=g(τk+1-,v) exists, and ψk:R2→R2 is nondecreasing. Let r(t) be the maximal solution of the following vector impulsive differential system:(18)u˙=gt,Vt,xt∈τk,τk+1,ut=ψkut-,ut0=ut0-=u0t=τk+1,existing on t0,∞. Then V(t0,x0)≤u0 implies that(19)Vt,xt≤rt,t≥t0,where x(t)=x(t,t0,x0) is any solution of system (12) existing on [t0,∞). Here, we state that function g(t,u) is quasi-monotone nondecreasing in u; if u,v∈R2,u≤v, and xi=yi for some 1≤i≤2, then gi(t,x)≤gi(t,y).


## 3. Main Results

First, in terms of the ultimate boundedness for system (4), we obtain the following result.Theorem 8.
Suppose that assumptions(H1)–(H9) hold. Then there is a constant M>0 such that (20)limt→∞supxit,ϕ≤Mi=1,2,limt→∞supyt≤Mfor any positive solutions (x1(t,ϕ),x2(t,ϕ),y(t,ϕ)) of system (4).Proof.
Let(x1(t,ϕ),x2(t,ϕ),y(t)) be any positive solution of system (4) satisfying the initial condition (5). From system (4), we have(21)x˙it≤xitaikt-biktxit+dijktxjt-τi-xit,t∈τk,τk+1,xiτk+1=hikxiτk+1-,t=τk+1,for all t≥0,i,j=1,2(i≠j). Then by the Lemmas 5 and 7, we obtain(22)xit≤x-it,∀t≥0,i=1,2,where (x-1(t),x-2(t)) is the solution of system (12) with the initial condition (5). Under assumptions (H1)–(H7), from Lemma 5 we obtain x-i(t)→x∗(t)(i=1,2,t→∞), where x∗(t)=(x1∗(t),x2∗(t)) is the globally asymptotically stable positive τ-almost periodic solution of system (12). Hence, x-i(t)(i=1,2) is bounded on R+. Next, we choose a constant M1=supt∈Rx∗t, where x∗t=∑i=12‍xi∗(t). Since x-i(t)→x∗(t)(i=1,2,t→∞), there is a pair of T1>0 and ρ>0 such that(23)x-it≤xi∗t+ρ≜M1∀t≥T1,i=1,2.Hence, by (22) we have(24)xit≤M1∀t≥T1,i=1,2.Consequently,(25)limt→∞supxit,ϕ≤M1,i=1,2.Further, we prove that there is a constant M2>0 such that(26)limt→∞supyt≤M2.From systems (4) and (24) we obtain(27)y˙t≤yta3kt+c~1ktM1+c~2ktM2-b3ktyt,t∈τk,τk+1,yτk+1=yτk+1-,t=τk+1,for all t≥T1, where(28)c~1kt=-1k+1+12c1t,c~2kt=-1k+1+12c2t,k∈N.From the comparison theorem of the impulsive differential equations [16, 28, 29], we have y(t)≤z(t) for all t≥T1, where z(t) is the solution of the following auxiliary equation:(29)z˙t=zta3kt+c~1ktM1+c~2ktM2-b3ktzt,t∈τk,τk+1,zτk+1=zτk+1-,t=τk+1,with initial value z(T1)=y(T1). According to assumptions (H1)-(H2) and (H7)-(H8), we also have(30)∫tt+τ-1k+12a3s+-1k+1+12a~3s+c~1ktM1+c~2ktM1ds>0.Hence from Lemma 6, system (29) has a unique globally attractive positive τ-almost periodic solution z∗(t). For any constant λ>0, there is a T2≥T1 such that z(t)<z∗(t)+λ for all t≥T2. Therefore, we have(31)yt<z∗t+λ≤12maxt∈0,τz∗t+λ≜M2,for all t≥T2. Consequently,(32)limt→∞supyt≤M2.Therefore, (26) holds. Choose a constant M=max⁡{M1,M2}; then we can see(33)limt→∞supxit,ϕ≤Mi=1,2,limt→∞supyt≤M;this completes the proof of Theorem 8.Next, on the permanence of system (4), we have the following results.Theorem 9.
Suppose that assumptions(H1)-(H2) and (H7)-(H8) hold. Then the predator species y of system (4) is permanent.Proof.
Let(x1(t,ϕ),x2(t,ϕ),y(t)) be any positive solutions of system (4) satisfying the initial condition (5). By system (4), we easily obtain(34)y˙t≥yta3kt-b3ktyt,t∈τk,τk+1,yτk+1=yτk+1-,t=τk+1.Using comparison theorem of the impulsive differential equations [16, 28, 29] and Lemma 6, we can easily obtain y(t)≥y-(t), for all t∈R, where y-(t) is the solution of system (14) with condition (5). Under the assumptions (H1)-(H2) and (H7)-(H8), by Lemma 6, we can obtain y-(t)→y∗(t) as t→∞, where y∗(t) is the globally asymptotically stable positive τ-almost periodic solution of system (14). Hence, we can easily obtain that, for ε0=1/2mint∈[0,τ]y∗(t), there exists a T1>0 such that(35)yt>y∗t-ε0≥12mint∈0,τy∗t≜m2,for all t≥T1. Together with Theorem 8, we have that the predator species y of system (4) is permanent. This completes the proof of Theorem 9.Remark 10.
Set(36)θ=∫tt+τ-1k+12a3s+-1k+1+12a~3sds.In system (4), based on the assumptions and the actual biological meanings of the parameters a3(t),b3(t) and a~3(t),b~3(t), we can see that the conditions in Theorem 9 are easily satisfied. Additionally, the constant θ represents the minimal total growth rate of the predator species y. If θ>0, which means that the predator species y has another food resource. As a result, Theorem 9 implies that if θ>0, then the predator species y will be permanent regardless of whether the prey species x exists or not.Next, on the permanence of the prey speciesx of system (4), we have the following result.Theorem 11.
Suppose that assumptions(H1)–(H7) and the following inequality (37)∫tt+τ-1k+12ais+-1k+12a~is-cisy∗sds+∑t≤τk+1≤t+τln⁡hik>0for (i=1,2,k∈N) hold. Then the prey species x of system (4) is permanent.Proof.
Owing to condition (37), we have that there is a small enough constant ε1>0, such that(38)∫tt+τ-1k+12ais-bisε1+-1k+12a~is-b~isε1-cisy∗s+ε1ds+∑t<τk+1≤t+τln⁡hik>0.Consider the following auxiliary system:(39)n˙it=nitaikt-biktnit-cikty∗s+ε1+dijktnjt-τi-nit,t∈τk,τk+1,niτk+1=hikniτk+1-,t=τk+1,where(40)c1kt=-1k+1+12c1t,c2kt=-1k+1+12c2t.By (38), we have(41)∫tt+τ-1k+12ais+-1k+12a~is-cisy∗s+ε1ds+∑t<τk+1≤t+τln⁡hik>0.Then, from (41) and Lemma 5, we know system (39) has a unique globally attractive positive τ-almost periodic solution nε1∗(t)=(n1ε1∗(t),n2ε1∗(t)).
And then, based on assumptionH8 and Lemma 5, for arbitrary constant γ>0 the following system(42)u˙t=-1k+12uta3t-b3tut+-1k+1+12uta~3t+c~1tγ+c~2tγ-b~3tut,t∈τk,τk+1,uτk+1=uτk+1-,t=τk+1,has a unique globally attractive positive τ-almost periodic solution uγ∗(t). For above ε1>0 and M^=max{M2,m2-1}>0, there is a t3=t(ε1,M^)>0, such that, for any t0≥0 and u0∈[M^-1,M^], (43)uγt,t0,u0-uγ∗t<ε12for all t≥t0+t3, where uγ(t,t0,u0) is the solution of system (42) with the initial condition uγ(t0)=u0.
According to the theorem of the continuity of solutions with respect to parameters of the impulsive differential equations [17], there exist γ0∈(0,ε1) and γ0<nε1∗(t)-γ0 such that(44)uγ0∗t-y∗t<ε12,for all t∈[t,t+τ]. Also owing to the almost periodic property of uγ0∗(t) and y∗(t), we further have(45)uγ0∗t-y∗t<ε12,for all t∈R.
Suppose(x1(t),x2(t),y(t)) is arbitrary positive solution of system (4); from above inequalities (24), (31), and (35), we have(46)xt≤M1,M-1≤yt≤Mfor all t≥T^=max⁡{T1,T2}. If there exists a t2≥T^, such that xi(t)≤γ0 for all t≥t2, then from system (4), we can obtain(47)y˙t≤-1k+12yta3t-b3tyt+-1k+1+12yta~3t+c~1tγ0+c~2tγ0-b~3tyt,t∈τk,τk+1,yτk+1=yτk+1-,t=τk+1,for all t≥t2. By comparison theorem of impulsive differential equations [16, 28, 29], we also have y(t)≤uγ0(t) for all t≥t2, where uγ0(t) is the solution of system (42) satisfying γ=γ0 and with the initial value uγ0(t2)=y(t2). For (43), let t0=t2 and u(t0)=y(t2). From M-1≤y(t)≤M, we have for γ=γ0(48)uγ0t=uγ0t,t2,yt2<uγ0∗t+12ε1,for all t≥t2+t3. Then y(t)<uγ0∗(t)+1/2ε1 for all t≥t2+t3. Thus, by condition (45), we further get(49)yt<y∗t+ε1.
Taking into account system (4) again, from above inequality (49) and system (4), we have(50)x˙it≥xitaikt-biktxit-cikty∗t+ε1+dijktxjt-τi-xit,t∈τk,τk+1,xiτk+1=hikxiτk+1-,t=τk+1,for all t≥t2+t3. From Lemma 7, we also have xi(t)≥niε1(t)(i=1,2) for all t≥t2+t3, where nε1(t)=(n1ε1(t),n2ε1(t)) is the solution of system (39) satisfying the initial value niε1(t2+t3)=xi(t2+t3). Because of the globally attractivity of positive τ-almost periodic solution nε1∗(t) of system (39), we further have that, for previous constant γ0, there exists a t4≥t2+t3 such that(51)nε1t≥nε1∗t-γ0>γ0,for all t≥t4. Hence, we have xi(t)>γ0 for all t≥t4, which leads to a contradiction.
Hence, there exists a constantt5≥t2 such that xi(t5)>γ0. We now prove that(52)xit≥γ0exp⁡-M0+φ0t3+2ξi∀t≥t5,where(53)φ0=maxt∈0,τaikt+biktγ0+cikty∗t+ε1+dijkt,M0=mint∈0,τaikt-biktM1-ciktM-dijkt.If inequality (52) is not true, then from xi(t5)>γ0>γ0exp⁡-M0+φ0t3+2ξi, there are constants t6 and t7 satisfying t7≥t6>t5, such that xi(t7)<γ0exp⁡-M0+φ0t3+2ξi,xi(t6-)≥γ0,xi(t6)≤γ0. There are two cases for t7.
Case  1. t6=t7.
Case  2. t6<t7.
For Case  1, we see thatt6 is an impulsive time. Hence, there is an integer k>0 such that t6=tk. We have(54)γ0exp⁡-M0+φ0t3+2ξi>xit6=hikxit6-≥γ0exp⁡ln⁡hik≥γ0exp⁡-ξi≥γ0exp⁡-M0+φ0t3+2ξi,which leads to a contradiction.
For Case  2, we havexi(t)<γ0, for all t∈t6,t7. Assume t7-t6≤t3, then for any t∈t6,t7 we have(55)x˙it≥xitaikt-biktM1-ciktM-dijktxit≥xitaikt-biktM1-ciktM-dijkt≥xitM0,t∈τk,τk+1,xiτk+1=hikxiτk+1-,t=τk+1,where(56)M0=mint∈0,τaikt-biktM1-ciktM-dijkt.Integrating (55) from t6 to t, for any t∈[t6,t7], we obtain(57)xit≥xit6exp⁡∫t6taiks-biksM1-ciksM-dijksds+∑t6<τk+1≤tln⁡hik≥γ0exp⁡-M0t3+ξi.Assume t7-t6>t3; then when t∈[t6,t6+t3], according to above discussion we directly have(58)xit≥γ0exp⁡-M0t3+ξi.Particularly, (59)xit6+t3≥γ0exp⁡-M0t3+ξi.Since xi(t)≤γ0 for all t∈(t6,t7]. Then we have(60)y˙t≤-1k+12yta3t-b3tyt+-1k+1+12yta~3t+c~1tγ0+c~2tγ0-b~3tyt,t∈τk,τk+1,yτk+1=yτk+1-,t=τk+1,for t∈(t6,t7]. From comparison theorem of impulsive differential equations [16, 28, 29], we also have yt⩽uγ0(t) for all t∈(t6,t7), where uγ0(t) is the solution of system (42) with γ=γ0 and satisfies uγ0(t6)=y(t6). In (43), choose a t0=t6 and u0=y(t6), since M-1≤y(t6)≤M, we have for γ=γ0(61)uγ0t=uγ0t,t6,yt6≤uγ0∗t+12ε1∀t≥t6+t3.Hence, (62)yt<uγ0∗+ε12∀t≥t6+t3.Thus, from (45) we further have (63)yt<y∗t+ε1∀t≥t6+t3.From (63), we have(64)x˙it≥xitaikt-biktγ0-cikty∗t+ε1-dijktxitt∈τk,τk+1,xiτk+1=hikxiτk+1-,t=τk+1,For arbitrary t∈t6+t3,t7. We choose an integer p≥0 such that t7∈t6+t3+pτ,t6+t3+p+1τ; then integrating (64) from t6+t3 to t7, we have(65)γ0exp-M0+φ0t3+2ξi>xit7=xit6+t3exp∫t6+t3t7aiks-biksγ0-ciksy∗s+ε1-dijksds+∑t6+t3<τk+1≤t7ln⁡hik≥γ0exp⁡-M0t3+ξiexp⁡∫t6+t3t6+t3+pω+∫t6+t3+pωt7aiks-biksγ0-ciksy∗s+ε1-dijksds+∑t6+t3<τk+1≤t7ln⁡hik≥γ0exp⁡-M0t3+ξi-φ0t3+ξi≥γ0exp⁡-M0+φ0t3+2ξi,which leads to a contradiction. According to the discussion above, we finally have(66)limt→∞infxit≥m1≜γ0exp⁡-M0+φ0t3+2ξi.Obviously, m1 is independent of any positive solutions of system (4). This completes the proof of Theorem 11.Remark 12.
Set(67)λ=∫tt+τ-1k+12ais+-1k+12a~is-ciksy∗tds+∑t≤τk+1≤t+τln⁡hik.Since y∗(t) is the globally asymptotically stable τ- periodic solution of the prey-free subsystem (14) of system (4), we obtain that the integral ∫tt+τ‍(-1)k+1/2ai(s)+(-1)k+1/2(a~i(s)-cik(s)y∗(t))ds is the net total growth rate of species x within a whole almost period, while ∑t≤τk+1≤t+τ‍ln⁡hik is the maximal total survivability of species x within the process in which species x disperses between patches. Therefore, λ is the net total growth rate of species x during a whole period of transference. Theorem 11 tells us that if λ>0, then species x in system (4) is permanent.


## 4. Numerical Simulation and Discussion

In this paper, an almost periodic predator-prey model with intermittent prey dispersals and intermittent predation has been investigated. By using the methods of analysis and comparison theorems of the impulsive differential equations, we obtain sufficient conditions of the boundedness and permanence for this system.To illustrate our results for system (4), we give some numerical simulations by using the following values of parameters in Table 1.Table 1
Parameter values used in the simulations of system (4).
Parameter
Value
a1(t)
0.5+0.2sin⁡(1.2πt)+0.2sin⁡(1.5πt)
a~1(t)
0.6
+
0.14
sin
⁡
(
1.2
π
t
)
+
0.14
sin
⁡
(
1.5
π
t
)
a2(t)
0.55+0.1sin⁡(1.2πt)+0.1sin⁡(1.5πt)
a~2(t)
0.6
+
0.09
sin
⁡
(
1.2
π
t
)
+
0.09
sin
⁡
(
1.5
π
t
)
a3(t)
0.5+0.3sin⁡(1.2πt)+0.2sin⁡(1.5πt)
a~3(t)
0.1
+
0.12
sin
⁡
(
1.2
π
t
)
+
0.12
sin
⁡
(
1.5
π
t
)
b1(t)
0.25
+
0.1
cos
⁡
(
1.2
π
t
)
+
0.1
cos
⁡
(
1.5
π
t
)
b~1(t)
0.4+0.04cos⁡(1.2πt)+0.04cos⁡(1.5πt)
b2(t)
0.3+0.04cos⁡(1.2πt)+0.04cos⁡(1.5πt)
b~2(t)
0.4
+
0.05
cos
⁡
(
1.2
π
t
)
+
0.05
cos
⁡
(
1.5
π
t
)
b3(t)
0.2
+
0.1
cos
⁡
1.2
π
t
+
0.2
cos
⁡
1.5
π
t
b~3(t)
0.5+0.10cos⁡1.2πt+0.10cos⁡1.5πt
c1(t)
0.2+0.2sin⁡1.2πt+0.2sin⁡1.5πt
c~1(t)
0.10+0.1sin⁡1.2πt+0.1sin⁡(1.5πt)
c2(t)
0.15+0.2sin⁡1.2πt+0.2sin⁡1.5πt
c~2(t)
0.15+0.2sin⁡1.2πt+0.2sin⁡1.5πt
D12(t)
0.5+0.05cos⁡1.2πt+0.05cos⁡1.5πt
D21(t)
0.5+0.05cos⁡1.2πt+0.05cos⁡1.5πt
d1
0.5
d2
0.6
D1
0.55
D2
0.65
τ1
1.3
τ2
1.6Due to system (4) being an almost periodic system, we will show the numerical simulation on the following eight intervals in Table 2.Table 2
Parameter values used in the simulations of system (4).
Number
Interval
τk+1-τk
Interval length
Process
1
[0,5)
τ1-τ0
5
1
2
[5,9)
τ2-τ1
4
2
3
[14,20)
τ
3
-
τ
2
6
1
4
[20,25)
τ
4
-
τ
3
5
2
5
[25,28)
τ
5
-
τ
4
3
1
6
[28,35)
τ
6
-
τ
5
7
2
7
[40,46)
τ
7
-
τ
6
6
1
8
[46,50)
τ
8
-
τ
7
4
2Therefore, we can consider almost periodic system as periodic approximately and theτ- period is 10.First, for system (12), when we take each parameter as in Table 1, we have (68)∫010a1sds+∑0<τk+1≤10ln⁡d1≈4.3918>0,∫010a~1sds+∑0<τk+1≤10ln⁡D1≈5.4615>0,∫010a2sds+∑0<τk+1≤10ln⁡d2≈5.0316>0,∫010a~2sds+∑0<τk+1≤10ln⁡D2≈5.6074>0.We can see that all the assumptions (H1)–(H7) of Lemma 5 hold. Therefore, system (12) has a unique positive globally attractive almost periodic solution x∗(t)=(x1∗(t),x2∗(t)). Numerical simulations of these results can be observed in Figures 1(a) and 1(b). From numerical simulations, we obtain the minimum values of (x1(t),x2(t)), which are (0.6988, 0.8355).Figure 1
(a) The globally stable positive almost periodic solution of the speciesx of system (12). (b) The portrait of the globally stable positive almost periodic solution of the species x of system (12). Here, we take the initial values ϕi(s)=(ϕ1(s),ϕ2(s))=(1.5,1.8),(1.6,1.7),(1.7,1.6) for all s∈[-5,0].
(a)
(b)Then, for system (14), when we also take the above parameters a3k(t),b3k(t), we have (69)∫010a3sds+∑0<τk+1≤10ln⁡1≈5.0509>0,∫010a~3sds+∑0<τk+1≤10ln⁡1≈1.0509>0.We can easily demonstrate that the assumptions (H1)-(H2) and (H7)-(H8) of Lemma 6 and Theorem 9 hold. Obviously, system (14) has a unique almost periodic solution y∗(t), and the predator species y of system (4) always has the minimum value of 0.3528 and the maximum value of 1.9017 when taking different initial values; that is, it is permanent. All of these results can be observed in Figure 2(a). However, for system (4), when we keep the other parameters unchanged and only just adjust the value of a~3(t)=-0.65+0.12sin⁡1.2πt+0.12sin⁡(1.5πt), then we have (70)∫010a~3sds+∑0<τk+1≤10ln⁡1≈-6.4491<0.This is contrary to the assumptions (H1)-(H2) and (H7)-(H8) of Theorem 9, that is, in the whole period, if θ<0 without considering the prey species x, then the predator species y of system (4) will be extinct (see Figure 2(b)). Nevertheless, if we take the prey species x into account, using the above parameters c~1(t),c~2(t) and the minimum values of (x1(t),x2(t)) with 0.6988, 0.8355 to make (71)∫05a3sds+∫510a~3s+0.6988×c~1s+0.8355×c~2sds+∑0<τk+1≤10ln⁡1≈0.6275>0,then we find an interesting phenomenon where the predator species y of system (4) goes from extinct to permanent (see Figure 3(a)). It turns out that if the predator species y in the time intervals τ2k+1,τ2k+2 has no other food resource but can only rely on the prey species x, then the more the predation behavior happens, the more likely the predator species will be permanent.Figure 2
(a) The globally stable positive almost periodic solution of the speciesy of system (14), and we take the initial condition ϕi(s)=(1.2,0.8,0.4) for all s∈[-5,0]. (b) Extinction of the species y of system (14), and we take the initial values ϕi(s)=(0.8,0.4) for all s∈[-5,0].
(a)
(b)Figure 3
(a) The globally stable positive almost periodic solution of the speciesy of system (14), and the predator species y is permanent just relying on the predation; here, we take the initial condition ϕi(s)=(1.2,0.8,0.4) for all s∈-5,0. (b) The time series of almost periodic solution of the species x,y of system (72). (c) The 3-dimension time series of almost periodic solution of the species x,y of system (72).
(a)
(b)
(c)Moreover, to illustrate the difference between continuous and intermittent predation of the predator speciesy, we consider the following system [25]:(72)x˙1t=x1ta1t-b1tx1t-c1tyt+D12tx2t-x1t,x˙2t=x2ta2t-b2tx2t-c2tyt+D21tx1t-x2t,y˙t=yta3t-b3t+c~1tx1t+c~2tx2t.We keep the other parameters unchanged including the intrinsic growth rates of the species x, the interspecies competition rates of all species, the survival rates of the species x, and the dispersal rates of the species x in system (72). We take another set of parameters in Table 3, and we let τi=0,di=Di=1(i=1,2) for the purpose of contrast.Table 3
Parameter values used in the simulations of system (4).
Parameter
Value
a3(t)
0.1+0.3sin⁡1.2πt+0.2sin⁡1.5πt
a~3(t)
-
0.6
+
0.12
sin
⁡
1.2
π
t
+
0.12
sin
⁡
1.5
π
t
c1(t)
0.3+0.1sin⁡1.2πt+0.1sin⁡1.5πt
c~1(t)
0.3
+
0.14
sin
⁡
1.2
π
t
+
0.1
sin
⁡
1.5
π
t
c2(t)
0.3+0.1sin⁡1.2πt+0.1sin⁡1.5πt
c~2(t)
0.3
+
0.14
sin
⁡
1.2
π
t
+
0.1
sin
⁡
1.5
π
tBy simple calculation, we have(73)∫010a3s+0.6988×c~1s+0.8355×c~2sds+∑0<τk+1≤10ln⁡1≈5.7530>0,∫010a1s-1.9017×c1sds+∑0<τk+1≤10ln⁡1≈-0.7009<0,∫010a2s-1.9017×c2sds+∑0<τk+1≤10ln⁡1≈-0.2434<0.Then, we find that continuous predation makes the predator permanent and the prey extinct (see Figures 3(b) and 3(c)). However, if we keep all others parameters unchanged and just change system (72) to system (4), that is, changing the model from continuous predation to intermittent predation, we have(74)∫05a3sds+∫510a~3s+0.6988×c~1s+0.8355×c~2sds+∑0<τk+1≤10ln⁡1≈-0.0981<0,∫05a1sds+∫510a~1s-1.9017×c1sds+∑0<τk+1≤10ln⁡1≈2.6762>0,∫05a2sds+∫510a~1s-1.9017×c2sds+∑0<τk+1≤10ln⁡1≈2.8974>0.Then, there are some interesting phenomena that the predator goes from permanent to extinct, while the opposite occurs for the prey. All of these results can be observed (Figures 4(a) and 4(b)) by numerical simulations. By the above analysis, we can conclude that if the predator continues preying on prey and the predation intensity is higher than the intrinsic growth of the prey, then the prey will be extinct, while the predator will be permanent. We also obtain that if the predation behavior is intermittent and the predation intensity is small, then the predator will be extinct, while the prey will be permanent. So, for an intermittent predator-prey model, both the intermittent predation and the intrinsic growth rates of the prey and predator species can greatly impact the permanence or extinction of the system.Figure 4
(a) The time series of almost periodic solution of the speciesx of the predator-prey model with continuous predation. (b) The time series of almost periodic solution of the species x of the predator-prey model with intermittent predation. Here, we take the initial condition ϕi(s)=(ϕ1(s),ϕ2(s))=(1.5,1.8),(1.6,1.7),(1.7,1.6) for all s∈[-5,0].
(a)
(b)Finally, for system (4), to test our main results, we use the above parameters in Table 1. We can simply calculate that the assumptions (H1)–(H7) in Theorem 11 hold, and for (37), we let y∗(t)=1.9017 then(75)∫010a1sds+∑0<τk+1≤10ln⁡d1≈4.3918>0,∫010a~1s-1.9017×c~1sds+∑0<τk+1≤10ln⁡D1≈1.4967>0,∫010a2sds+∑0<τk+1≤10ln⁡d2≈5.0316>0,∫010a~2s-1.9017×c~1sds+∑0<τk+1≤10ln⁡D2≈2.5934>0.From numerical simulations, we obtain that the minimum value of x of system (3) is always 0.8085 when using different initial values; that is, the prey x is permanent (see Figures 5(a) and 5(b)).Figure 5
(a) The time series of the globally stable positive almost periodic solution of the speciesx. (b) The portrait of the globally stable positive almost periodic solution of the species x, and we take the initial condition ϕi(s)=(ϕ1(s),ϕ2(s))=(1.5,1.8),(1.6,1.7),(1.7,1.6) for all s∈[-5,0].
(a)
(b)



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