# Dynamical Behaviors of a Stage-Structured Predator-Prey Model with Harvesting Effort and Impulsive Diffusion

Authors: Lingzhi Huang; Zhichun Yang
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We consider a delayed predator-prey model with harvesting effort and impulsive diffusion between two patches. By the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map,
we obtain some sufficient conditions on the existence and global attractiveness of predator-eradicated periodic solution for the system. Furthermore, the permanence of the system is derived. The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems.


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## Body

## 1. Introduction

The effect of spatial factors in population dynamics is an interesting topic since dispersal often occurs between patches in ecological environment [1, 2]. Some models focused on dispersal process in continuous time meaning [3, 4]. However, many species diffuse only during a single period, and the diffusion often occurs in regular pulses. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse during other seasons. Since the short-time diffusion is often assumed to be in the form of impulses in the modeling process, some mathematical models on impulsive diffusion have been studied by impulsive differential equations (see, e.g., [5–11]). In [5], a single species model with impulsive diffusion was initially formulated by(1)x1′t=x1ta1-b1x1t,x2′t=x2ta2-b2x2t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,t=nτ,n∈N,where, for the ith patch, xi is the density of species in the logistic growth; Δxi(t)=xi(t+)-xi(t-) and xi(t)=xi(t-), xi(t+), xi(t-) represent the right-hand limit and left-hand one at t; the parameters ai,ai/bi are the intrinsic growth rates and carrying capacities, respectively; the diffusion occurs at impulsive moments nτ, n=0,1,2,…,τ>0; di∈(0,1) is dispersal rate from the jth patch to ith, i,j=1,2, i≠j. We assume that the net exchange from the jth patch to ith patch is proportional to the difference xj-xi of population densities.On the other hand, almost all species go through the life stage from the immature to the mature stage. Since it is necessary to spend units of time that the immature becomes the mature, time delay plays important role in stage-structured model. Aiello and Freedman [6] introduced the following stage-structured single species model:(2)y′t=αzt-ωyt-αe-ωτzt-τ,z′t=αe-ωτzt-τ-rz2t,where y,z represent the immature and mature populations densities, respectively, α is the birth rate, ω,r are the immature and mature death rate, τ is the maturation time delay, and the term αe-ωτz(t-τ) is the number of immature populations who were born at time t-τ and survive at time t, and this term represents the transformation from the immature to the mature stages. Note that the death number of mature population is of a logistic nature, which is proportional to the square of the population with proportionality constant r.Recently, some stage-structured models with time delays and impulsive diffusion were investigated in [7–11]. Jiao et al. [7, 9] and Shao et al. [8, 10] investigated a class of predator-prey models with prey-impulsive diffusion between two patches in which the predator is subject to stage-structured effects (delayed effects) only in one of the patches. Dhar and Jatav [11] consider a delayed stage-structured predator-prey model with impulsive diffusion, and both of the predators in two patches are subject to stage-structured effects. Many interesting results in the mentioned publications mainly focused on the global attractiveness of the predator-eradicated periodic solution and the permanence of the systems. To manage effectively the species, we should know or estimate how many members the population has at large time in every patch when a species is uniformly permanent. Mathematically, this corresponds to the ultimately low and upper boundedness of solutions of the systems. Although the permanence was derived by the estimate for the ultimately upper boundedness of solutions, there are some errors or negligence on the estimate in [7–11] (see Remarks 4 and 6 below). In addition, according to Aiello and Freedman [6], it may be more reasonable to take the square function of the mature population as the death numbers of the mature than to take the linear one in [7–11].Motivated by the above discussion, we propose the following stage-structured predator-prey model with generalized functional response, the harvesting effort of the mature predator, and impulsive diffusion between two predators’ territories:(3)x1′t=x1ta1-b1x1t-p1x1tz1t,y1′t=α1z1t-α1e-ω1τ1z1t-τ1-ω1y1t,z1′t=α1e-ω1τ1z1t-τ1+β1p1x1tz1t-E1z1t-r1z12t,x2′t=x2ta2-b2x2t-p2x2tz2t,y2′t=α2z2t-α2e-ω2τ2z2t-τ2-ω2y2t,z2′t=α2e-ω2τ2z2t-τ2+β2p2x2tz2t-E2z2t-r2z22t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,Δy1t=0,Δy2t=0,Δz1t=0,Δz2t=0,t=nτ,n∈N,where for the ith patch i=1,2, xi,yi,zi are prey population density and predator populations density of the immature and the mature at time t, respectively; αi represents the growth rate from the immature predator to the mature one; ωi is the death rate of the immature predator; ri is the death rate of mature predator, which is of a logistic nature; Ei is the harvesting rate of the mature population; βi is the rate of conversion of nutrients into the reproduction rate of the mature predator; τi represents a constant time to maturity; τ>0 is the period of impulsive diffusion; and di∈(0,1) is dispersal rate; the other parameters have the same biological meaning in system (1). In addition, pi(xi(t)) represents the functional response including Holling-type and Ivlev-type ones, which satisfies(4)pi∈H≜f∈CR+,R+∣f0=0,f′s>0,f′′s≤0,s>0,R+=0,∞.Mathematically, the proposed system (3) generalizes some existing models. For example, (3) is the model in [7] when p1(·)=0, p2(x2)=βx2, ri=0, and y1,z1 are removed; (3) becomes the model in [11] when ri=0, pi(xi)=cixi. In this paper, we will mainly investigate the existence and global attractiveness of periodic solution and the permanence for system (3) by employing the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map. The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems. Some examples and their simulations are given to illustrate the effectiveness of our results.


## 2. Preliminaries

Letx(t)=(x1(t),y1(t),z1(t),x2(t),y2(t),z2(t))T be the solution of (3) with the initial conditions(5)ϕ1ζ,ϕ2ζ,ϕ3ζ,ϕ4ζ,ϕ5ζ,ϕ6ζ∈C-τ,0,R+6,ϕi0>0,i=1,2,3,4,5,6.From the theory of impulsive differential equations and the biological meanings, we can obtain the global existence and uniqueness of solution x1(t),y1(t),z1(t),x2(t),y2(t),z2(t)≥0.Note that the variablesy1,y2 do not appear in the first, third, fifth, and sixth equations in (3). We will simplify the model and need to restrict our attention to the following system:(6)x1′t=x1ta1-b1x1t-p1x1tz1t,z1′t=α1e-ω1τ1z1t-τ1+β1p1x1tz1t-E1z1t-r1z12t,x2′t=x2ta2-b2x2t-p2x2tz2t,z2′t=α2e-ω2τ2z2t-τ2+β2p2x2tz2t-E2z2t-r2z22t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,Δz1t=0,Δz2t=0,t=nτ,n∈N=1,2,3,….Suppose thatz1(t)=0, z2(t)=0; the above subsystem becomes model (1). Integrating and solving the first two equations of (1) between impulsive moments, we have(7)xit=biai+1xinτ+-biaie-ait-nτ-1,nτ<t≤n+1τ,i=1,2.Combining with the impulsive diffusion, we obtain the following stroboscopic map:(8)x1,n+1τ=x1,nτh1+c1x1,nτ+d1x2,nτh2+c2x2,nτ-x1,nτh1+c1x1,nτ,x2,n+1τ=x2,nτh2+c2x2,nτ+d2x1,nτh1+c1x1,nτ-x2,nτh2+c2x2,nτ.Here xi,nτ=xi(nτ+), ci=(bi/ai)(1-e-aiτ)>0, and hi=e-aiτ<1, i=1,2, n∈N.We define a mapF:R+2→R+2 such that(9)F1x=x1h1+c1x1+d1x2h2+c2x2-x1h1+c1x1,F2x=x2h2+c2x2+d2x1h1+c1x1-x2h2+c2x2.The set of all iterations of the map is equivalent to the set of all density sequences generated by system (8).By using the theory of monotone dynamical systems in [12], Hui and Chen [5] gave the following lemma.Lemma 1.
There exists a unique positive fixed pointq=(q1,q2) of map F in (9), Fn(x)→q as n→∞. Furthermore, all nontrivial trajectories of system (1) approach the positive periodic solution (x~1(t),x~2(t)) with period τ, where(10)x~it=biai+1qi-biaie-ait-nτ-1,nτ<t≤n+1τ,i=1,2.To obtain our results, we also need the following lemma.Lemma 2 (see [13]).
Consider the following delay differential equation:(11)x′t=axt-τ-bxt-cx2t,where a,b,c, and τ are positive constants and x(t)>0 for t∈[-τ,0]. One has the following:(i)
ifa>b, then limt→+∞x(t)=(a-b)/c;(ii)
ifa<b, then limt→+∞x(t)=0.


## 3. Extinction of the Predator-Eradicated Periodic Solution

It is clear that system (6) has a predator-eradicated periodic solution (x~1(t),0,x~2(t),0), where the positive periodic functions x~1(t),x~2(t) with period τ are given in (10). In the following, we will show that it is globally attractive.Theorem 3.
Assume that(12)αie-ωiτi+βipiXi∗<Ei,Xi∗≔biai+1qi-biaie-aiΔiτ-1,i=1,2,where Δi=min0,sgnqi-ai/bi, (q1,q2) is a unique positive fixed point of map F defined in (9) with ci=(bi/ai)(1-e-aiτ) and hi=e-aiτ. Then the predator-eradicated periodic solution (x~1(t),0,x~2(t),0) of (6) is globally attractive.Proof.
From (12), we can choose ɛ0>0 sufficiently small such that(13)α1e-ω1τ1+β1p1X1∗+ɛ0<E1,α2e-ω2τ2+β2p2X2∗+ɛ0<E2.It follows from system (6) that(14)x1′t≤x1ta1-b1x1t,x2′t≤x2ta2-b2x2t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,t=nτ,n∈N.By Lemma 1 and the comparison theorem of impulsive differential equation, we have limsupt→∞xi(t)≤x~i(t), where x~i(t) are defined in (10). So, for given ɛ0>0, there exists an n1 such that for n≥n1(15)x1t≤x~1t+ɛ0≤X1∗+ɛ0,x2t≤x~2t+ɛ0≤X2∗+ɛ0,nτ<t≤n+1τ.From the second and fourth equations in system (6), we get(16)z1′t≤α1e-ω1τ1z1t-τ1-E1-β1p1X1∗+ɛ0z1t-r1z12t,t>nτ+τ1,n>n1,z2′t≤α2e-ω2τ2z2t-τ2-E2-β2p2X2∗+ɛ0z2t-r2z22t,t>nτ+τ2,n>n1.By using the comparison theorem, it follows from Lemma 2 and the inequalities (13) that limt→∞z1(t)=0 and limt→∞z2(t)=0. Therefore, for any sufficiently small positive number ɛ1, there is an n2 such that z1(t)<ɛ1, z2(t)<ɛ1 for all t>n2τ. Since pi∈H implies that pi(xi(t))≤pi′(0)xi(t), from system (6) we have(17)x1′t≥x1ta1-p1′0ɛ1-b1x1t,x2′t≥x2ta2-p2′0ɛ1-b2x2t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,t=nτ,n∈N.By using the comparison theorem and Lemma 1, we can obtain that xi(t)≥si(t), i=1,2, where(18)si=biai-pi′0ɛ1+1q¯i-biai-pi′0ɛ1e-ai-pi′0ɛ1τ-1,and (q¯1,q¯2) is a unique positive fixed point of map F defined in (9) with ci=(bi/(ai-pi′(0)ɛ1))(1-e-(ai-pi′(0)ɛ1)τ) and hi=e-(ai-pi′(0)ɛ1)τ. From (15) and (18), letting ɛ0,ɛ1→0 and t→∞, then(19)x~it≥xit≥sit⟶xi~t,i=1,2.The proof is complete.Remark 4.
X
i
∗ is the maximum of the periodical function x~i(t) defined in (10), which is dependent on the sign of the constant 1/qi-bi/ai, i=1,2. However, the authors neglected the details when they derived the global attractivity of predator-eradicated periodic solution in [7, 8, 10, 11].


## 4. Permanence

Firstly, we can estimate the ultimately upper boundedness as follows.Lemma 5.
If(20)βipiXi∗<Ei<αie-ωiτi+βipiXi∗,then limsupt→∞xi(t)≤Xi∗, limsupt→∞zi(t)≤Zi∗, i=1,2, in system (6), where Xi∗ is given in (12) and(21)Zi∗=αie-ωiτi-Ei-βipiXi∗ri.Proof.
It follows from system (6) that(22)x1′t≤x1ta1-b1x1t,x2′t≤x2ta2-b2x2t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,t=nτ,n∈N.By Lemma 1 and the comparison theorem of impulsive differential equation, we have limsupt→∞xi(t)≤x~i(t)≤Xi∗. From the second and fourth equations in system (6),(23)z1′t≤α1e-ω1τ1z1t-τ1-E1-β1p1X1∗+ɛ0z1t-r1z12t,t>nτ+τ1,n>n2,z2′t≤α2e-ω2τ2z2t-τ2-E2-β2p2X2∗+ɛ0z2t-r2z22t,t>nτ+τ2,n>n2.It follows from Lemma 2 and (20) that(24)limsupt→∞z1t≤α1e-ω1τ1-E1-β1p1X1∗+ɛ0r1,limsupt→∞z2t≤α2e-ω2τ2-E2-β2p2X2∗+ɛ0r2.From the arbitrariness of ɛ0, we obtain the conclusion. The proof is complete.Remark 6.
In [7–11], the authors gave the estimate of the ultimately upper boundedness of solutions to derive the persistence of the system by constructing the V-functions. Nevertheless, there are errors on the estimate since the constructed V-functions do not satisfy V(nτ+)≤V(nτ) when t=nτ, n=1,2,…. In Lemma 5, we make a modification for the estimate.Theorem 7.
LetXi∗,Zi∗ be given in Lemma 5 and(25)xi∗≔biai-pi′0Zi∗+1qi~-biai-pi′0Zi∗e-ai-pi′0Zi∗Δi~τ-1,zi∗≔αie-ωiτi-Ei-βipixi∗ri,where Δi~=max0,sgnqi~-(ai-pi′0Zi∗)/bi; (q1~,q2~) is a unique positive fixed point of map F defined in (9) with ci=bi(1-e-(ai-pi′(0)Zi∗))/(ai-pi′0Zi∗) and hi=e-(ai-pi′(0)Zi∗)τ, i=1,2. If(26)ai>pi′0Zi∗,βipiXi∗<Ei<αie-ωiτi+βipixi∗,i=1,2,then system (6) is uniformly persistent in the following meaning:(27)xi∗≤liminft→∞xit≤limsupt→∞xit≤Xi∗,zi∗≤liminft→∞zit≤limsupt→∞zit≤Zi∗.Proof.
We can see that (26) implies (20). Therefore, limsupt→∞xi(t)≤Xi∗, limsupt→∞zi(t)≤Zi∗, i=1,2. Furthermore, we can choose ɛ>0 sufficiently small such that(28)α1e-ω1τ1>E1-β1p1x1∗-ɛ,α2e-ω2τ2>E2-β2p2x2∗-ɛ.Note that pi(xi(t))≤pi′(0)xi(t) since pi∈H. From system (6) and Theorem 7, there is an n3 such that(29)x1′t≥x1ta1-p1′0Z1∗-b1x1t,x2′t≥x2ta2-p2′0Z2∗-b2x2t,t≠nτ,Δx1t=d1x2t-x1t,Δx2t=d2x1t-x2t,t=nτ,n>n3,n∈N.By Lemma 1 and the comparison theorem, there exists an n4>n3 such that(30)x1t≥l1~t-ɛ≥x1∗-ɛ,x2t≥l2~t-ɛ≥x2∗-ɛ,nτ<t≤n+1τ,n>n4,where(31)li~=biai-pi′0Zi∗+1qi~-biai-pi′0Zi∗e-ai-pi′0Zi∗t-nτ-1and qi~ can be confirmed homoplastically to qi. Letting σ1=x1∗-ɛ, σ2=x2∗-ɛ, from the second and fourth equations in system (6), we have(32)z1′t≥α1e-ω1τ1z1t-τ1-E1-β1p1σ1z1t-r1z12t,t>nτ+τ1,n>n4,z2′t≥α2e-ω2τ2z2t-τ2-E2-β2p2σ2z2t-r2z22t,t>nτ+τ2,n>n4.It follows from (28), Lemma 2, and comparison differential system that(33)liminft→∞z1t≥α1e-ω1τ1-E1-β1p1σ1r1,liminft→∞z2t≥α2e-ω2τ2-E2-β2p2σ2r2.So there is an n5 sufficiently large such that z1(t)≥z1∗>0, z2(t)≥z2∗>0 for all t>n5τ. Hence, by Theorem 7 and the above discussion, we get that system (6) is permanent. The proof is complete.Remark 8.
Theorem7 derives the persistence of the system and estimates how many members the population has at large time in every patch. This is important to manage effectively the species, but the estimate may be conservative and further results need to be developed in this direction.


## 5. Examples and Their Simulations

In this section, we will give some examples and their simulations to illustrate the effectiveness of the obtained results.Example 1.
Consider system (6) with Holling II-type functional response pi(xi)=μixi/(1+νixi) and τ=1,(34)a1=1,b1=0.2,α1=0.5,β1=0.6,r1=0.5,ω1=0.5,τ1=1,μ1=2,ν1=3,d1=0.4,E1=0.8,a2=0.8,b2=0.1,α2=0.2,β2=0.5,r2=0.8,ω2=1,τ2=1,μ2=1,ν2=2,d2=0.3,E2=0.8.We can compute the unique positive fixed point (q1,q2)=(6.1952,6.8051) and(35)α1e-ω1τ1+β1p1X1∗+ɛ0-E1=-0.1201<0,α2e-ω2τ2+β2p2X2∗+ɛ0-E2=-0.4922<0.It follows from Theorem 3 that the predator-eradicated periodic solution (x~1(t),0,x~2(t),0) of (6) is globally attractive. Figure 1 shows the asymptotical behaviors when taking the initial values xi(s)=0.8, zi(s)=0.2, -1≤s≤0, i=1,2.Figure 1
Time response in Example1.
(a)
(b)
(c)
(d)Example 2.
Consider system (6) with Holling II-type functional response pi(xi)=μi(1-e-νixi) and τ=2,(36)a1=1,b1=0.2,α1=0.5,β1=0.6,r1=0.01,ω1=1,τ1=1,μ1=0.5,ν1=0.03,d1=0.4,E1=0.05,a2=1.2,b2=0.1,α2=0.2,β2=0.5,r2=0.01,ω2=1,τ2=1,μ2=0.4,ν2=0.02,d2=0.3,E2=0.1.We can compute the positive fixed points (q1,q2)=(7.8602,9.8111), (q~1,q~2)=(7.7286,9.6796), and(37)0.0252=β1p1X1∗<E1=0.05<α1e-ω1τ1+β1p1x1∗=0.0907,0.0417=β2p2X2∗<E2=0.1<α2e-ω2τ2+β2p2x2∗=0.1088.It follows from Theorem 7 that system (6) is permanent, and we obtained the estimate of the ultimately low and upper boundedness of solutions as follows:(38)3.1273=x1∗≤liminft→∞x1t≤limsupt→∞x1t≤X1∗=7.8602,4.6796=x2∗≤liminft→∞x2t≤limsupt→∞x2t≤X2∗=11.7619,3.0685=z1∗≤liminft→∞z1t≤limsupt→∞z1t≤Z1∗=4.8784,0.8777=z2∗≤liminft→∞z2t≤limsupt→∞z2t≤Z2∗=1.5499.Figure 2 shows the permanence of the system when taking the initial values x1(s)=z1(s)=0.8, x2(s)=z2(s)=0.2, -2≤s≤0.Figure 2
Time response in Example2.
(a)
(b)
(c)
(d)The following example and its simulation show that the conclusions in Theorems3 and 7 may not hold when their conditions are not satisfied and point out some errors in existing publications.Example 3.
Consider system (6) with Holling I-type functional response pi(xi)=μixi and τ=3,(39)a1=1,b1=1,α1=0,β1=0,r1=0,ω1=0,τ1=5,μ1=0,d1=0.2,E1=0.8,a2=1,b2=1.6,α2=0.2,β2=1,r2=0,ω2=2,τ2=10,μ2=4,d2=0.9,E2=2.6,which becomes system (9) given in [7] when z1 are removed. We can compute the unique positive fixed points (q1,q2)=(0.9239,0.9599), (X1∗,X2∗)=(0.9959,0.6360),(40)α1e-ω1τ1+β1p1X1∗-E1=-0.8<0,α2e-ω2τ2+β2p2X2∗-E2=1.2397>0.Clearly, those conditions in both Theorems 3 and 7 are not satisfied. So our results cannot verify the permanence and the global attractiveness of the predator-eradicated periodic solution for the system. In fact, Figure 3 shows that the system is not permanent and the predator-eradicated periodic solution is not globally attractive when taking the initial values x1(s)=0.8, x2(s)=0.5, z1(s)=z2(s)=0.1, -3≤s≤0. However, according to Theorem  4.1 in [7], the predator-eradicated periodic solution is globally attractive for the above system without the variable z1 because(41)α2e-ω2τ2+β2μ2biai+1qi-biaie-aiτ-1-E2=-0.0558<0.This leads to the false conclusion from the simulation and Remark 4, and some similar errors can be found in [8, Theorem  3.1], [10, Theorem  3.1], and [11, Theorem  4.1].Figure 3
Time response in Example3.
(a)
(b)
(c)
(d)



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