# The Dynamics of an Impulsive Predator-Prey System with Stage Structure and Holling Type III Functional Response

Authors: Zhixiang Ju; Yuanfu Shao; Xiaolan Xie; Xiangmin Ma; Xianjia Fang
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

Based on the biological resource management of natural resources, a stage-structured  predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments is proposed in this paper. By applying comparison theorem and some analysis techniques, the global attractivity of predator-extinction periodic solution and the permanence of this system are studied. At last, examples and numerical simulations are given to verify the validity of the main results.


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## Body

## 1. Introduction

In recent decades, with the increase of population and the development of science and technology, human accelerated the exploitation of natural resources. Many harvesting predator-prey models have been studied. The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance; see [1–6] and the references cited therein. On the other hand, in the real world, many species usually go through two or even more life stages as they proceed from birth to death. Thus, it is practical to introduce the stage structure into predator-prey models; see [7–12]. For example, Wei and Wang [13] considered the following predator-prey system with stage structure:(1)x˙t=xtr1-a11xt-a12y2t,y˙1t=αxty2t-γy1t-αe-γτxt-τy2t-τ,y˙2t=αe-γτxt-τy2t-τ-r2y2t-a22y22t,where x(t),  y1(t), and y2(t) denote the densities of prey population and immature and mature individual predators at time t, respectively. The meanings of all parameters may refer to [13]. Authors obtained the sufficient conditions of the persistence for system (1).However, it is well known that many evolution processes are characterized by the fact that at certain moments their stage changes abruptly. For example, for IPM strategy on ecosystem, the predators are released periodically every timeT, and periodic catching or spraying pesticide is also applied. Hence, the predator and prey experience a change of state abruptly. Consequently, it is natural to assume that these processes act in the form of impulse. Impulsive methods have been applied in almost every field of applied sciences. For example, many population models assume that the populations are born throughout the year, whereas it is often the case that many species give birth only during a single period of the year; that is, births occur in regular pulses. Hence, the authors Z. Xiang, D. Long, and X. Y. Song gave a single population logistic model with birth pulse and impulsive harvesting at different moments as follows:(2)x˙(t)=-μx(t),t≠(n+l)T,t≠n+1T,Δxt=rxt1-xtK,t=n+lT,Δxt=-pxt,t=n+1T,n∈Z+,where x(t) represents the density of the resource population at time t. Parameter r is the intrinsic growth rate, the positive constant K is referred to as the environmental carrying capacity, and parameter μ is the death rate of resource population. Parameter p denotes the harvest rate of resource population. For more details of the biological meaning of system (2), we can refer to [14–17].In addition, in the description of dynamical interactions on predators and their preys, a crucial element of all models is the classic definition of a predator’s functional response. A functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes. For example, Zhang et al. [18] suggested a function(3)φx=x2c+x2,called Holling type III. It is monotonic in the first quadrant, that is, if the prey population increases, then the consumption rate of prey per predator will increase too. And c is the half-saturation constant. The field of research on the dynamics of impulsive predator-prey model with functional response seems to be a new increasingly interesting area, which draws many scholars’ attention.Moreover, by the picture of ocean’s food-chain, we know that small fish can prey on fish larvae as a predator; also it can be eaten by the higher predator as a prey (see [19]). Hence, according to the nature of biological resource management, it is interesting to investigate impulsive harvesting on prey and mature predator (e.g., the small fish and higher predator in the ocean’s food-chain) simultaneously at some fixed time.Based on the above discussion, we consider the stage-structured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments as follows:(4)x˙t=-βx2(t)c+x2(t)y2(t)-d1x(t),y˙1t=λβx2tc+x2ty2ty˙1t=-λβe-d2τx2t-τc+x2t-τy2t-τ-d2y1t,y˙2t=λβe-d2τx2t-τc+x2t-τy2t-τy˙2t=-d3y2t-d4y22t,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit≠n+l-1T,t≠nT,xt+=xt+rxt1-xtK,y1t+=y1t,y2t+=y2t,iiiiiiiiiiiiiiiiiiiiiiiit=n+l-1T,xt+=(1-p1)x(t),y1t+=y1t,y2t+=(1-p2)y2(t),iiiiiiiiiiiiiiiiiiiiiiit=nT,ϕ1s,ϕ2s,ϕ3s∈C-τ,0,R+3,ϕi0>0,i=1,2,3,where x(t) denotes the density of the prey and y1(t),  y2(t) represent the immature and mature predator densities, respectively. Parameters r, K, β, λ, d1, d2, d3, and d4 are positive constants, where r is the intrinsic growth rate of the prey, K denotes the capacity rate, concerned with the maintaining of the evolution of the population, β represents the predation rate of predator, λ is the conversion rate that translated into predator population increase, d1, d2, and d3 denote the death rate of prey, immature predator, and mature predator, respectively, and d4 is the intrinsic-specific competition rate of the mature predator. Parameter τ represents a constant time from immaturity to maturity. Parameters p1, p2 denote the harvesting rates of prey and mature predator at t=nT,  n∈Z+, respectively, Z+={1,2,…}, and T is the period of the impulsive effect.It is well known that, in the sustainable development of natural resources, it is very important to study the sustainable survival of species. So, in this paper, we aim to investigate the global attractivity of predator-extinction periodic solution and the permanence of system (4). From the biological point of view, we only consider (4) in the biological meaning region(5)D=xt,y1t,y2t:xt≥0,y1t≥0,y2t≥0.This paper is organized as follows. Firstly, some preliminaries are given in Section2. In Section 3, the sufficient conditions for the global attractivity of predator-extinction periodic solution are obtained. The permanence of system (4) is investigated in Section 4. In Section 5, we present some examples and simulations to illustrate our results. At last, a brief conclusion is given in Section 6.


## 2. Preliminaries

In this section, some definitions and lemmas are introduced.LetR+=0,∞ and  R+3={x∈R3,x≥0}. Denotef=(f1,f2,f3)T to be the map defined by the right hand of system (4). Let V:R+×R+3→R+; then V is said to belong to class V0, if(i)
V is continuous in ((n-1)T,(n+l-1)T]×R+3 and ((n+l-1)T,nT]×R+3 for each x∈R+3, n∈N, lim(t,y)→((n+l-1)T+,x)V(t,y)=V((n+l-1)T+,x) and lim(t,y)→(nT+,x)V(t,y)=V(nT+,x) exists,(ii)
V is locally Lipschitzian in x.Definition 1.
LetV∈V0, and (t,v)∈((n-1)T,(n+l-1)T]×R+3 and ((n+l-1)T,nT]×R+3, the upper right derivative of V(t,v) with respect to the impulsive differential system (4) is defined as(6)D+V(t,v)=limh→0+sup⁡1hVt+h,v+hft,v-Vt,v.The solution of (4), denote by v=(x(t),y1(t),y2(t))T, is a piecewise continuous function v:R+→R+3, v(t) is continuous on ((n-1)T,(n+l-1)T]×R+3 and(7)((n+l-1)T,nT]×R+3n∈N,0≤l≤1.Obviously the existence and uniqueness of solutions of (4) is guaranteed by the smoothness properties of f, which denotes the map defined by the right hand of system (4).The following lemmas are useful for the proof of the main results.Lemma 2 (see [5, 7]).
Consider the following differential equation:(8)x˙t=axt-τ-bxt-cx2t,where a, b, c, and τ are positive constants and x(t)>0 for t∈[-τ,0]. We have the following:(i)
ifa<b, then limt→+∞⁡x(t)=0;(ii)
ifa>b, then limt→+∞x(t)=a-b/c.Lemma 3 (see [14]).
Consider the following system:(9)u˙t=-wut,t≠n+l-1T,t≠nT,ut+=ut+rut1-utK,t=n+l-1T,ut+=1-put,t=nT.Then, system (9) has a positive periodic solution u*(t) with period T, which is globally stable, where(10)u*(t)=ewlT1+rrK-ewT1-prKe-w(t-(n-1)T),iiiiiiiiiiiiiiiiiiiiit∈n-1T,n+l-1T,ewT1-p1+rrK-ewT1-prKe-w(t-(n+l-1)T),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit∈n+l-1T,nT,if p<1-(ewT/1+r).Lemma 4.
There is a positive constant M such thatx(t)≤M/λ,  y1(t)≤M, and  y2(t)≤M for every solution (x(t),y1(t),y2(t))of system (4) with all t sufficiently large, where λ is a positive constant defined in system (4).Proof.
DefineVt=λxt+y1t+y2t.
Ift≠(n+l-1)T, t≠nT, we let d=min⁡{d1,d2,d3}; then(11)D+Vt+dVt=-λd1-dxt-d2-dy1t-d3-dy2t-d4y22t≤0,If t=n+l-1T, we have(12)Vn+l-1T+=Vn+l-1T+λrxt1-xtK≤Vn+l-1T+λrK4,and if t=nT, it is clear that V(nT+)≤V(nT).
Then, by Lemma 2.2 of [20], for all t≥0, we have(13)Vt≤V0+e-d1t+λrK4e-dt-T1-e-dT+λrK4edTedT-1⟶λrK4edTedT-1,t⟶∞.Therefore, there exists a positive constant(14)M=λrK4edTedT-1such that x(t)≤M/λ, y1(t)≤M, and y2(t)≤M for t large enough. This completes the proof.


## 3. Global Attractivity of a Predator-Extinction Periodic Solution

In this section, we will demonstrate the existence and global attractivity of a predator-extinction periodic solution, in which the predator individuals are entirely absent from the population permanently; that is,y1(t)=0 and  y2(t)=0 for all t≥0.Firstly, by Lemma3, we can easily obtain the existence of predator-extinction periodic solution for system (4).Theorem 5.
System (4) has a predator-extinction periodic solution (x*(t),0,0) which is globally stable; that is, for t∈((n-1)T,nT] and any solution (x(t),y1(t),y2(t)) of system (4), we have x(t)→x*(t) as t→∞, where(15)x*t=ed1lT1+rrK-ed1T1-p1rKe-d1(t-(n-1)T),iiiiiiiiiiiiiiiiiiiiiiiit∈n-1T,n+l-1T,ed1T1-p11+rrK-ed1T1-p1rKe-d1(t-(n+l-1)T),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit∈n+l-1T,nT,and x*0+=ed1lT(1+r/rK-(ed1T/1-p1r)K).
Next, we give the conditions on the global attractivity of the predator-extinction periodic solution(x*(t),0,0) of the system (4).Theorem 6.
The predator-extinction periodic solution(x*(t),0,0) of system (4) is globally attractive, if(A1)
1-p1-(ed1T/1+r)>0,(A2)
λβe-d2τρ2/c+ρ2<d3.Proof.
Let(x(t),y1(t),y2(t)) be any solution of system (4); from the first, the fourth, and the seventh equations of system (4), we have(16)x˙(t)≤-d1x(t),t≠(n+l-1)T,t≠nT,xt+=xt+rxt1-xtK,t=n+l-1T,x(t+)=(1-p1)x(t),t=nT.Consider the auxiliary system of (16) as follows:(17)z˙1t=-d1z1t,t≠n+l-1T,t≠nT,z1t+=z1t+rz1t1-z1tK,t=n+l-1T,z1t+=1-p1z1t,t=nT.By Lemma 3 and condition (A1), we have(18)z1*t=ed1lT1+rrK-ed1T1-p1rKe-d1t-n-1T,iiiiiiiiiiiiiiiiiiiiiiit∈n-1T,n+l-1T,ed1T1-p11+rrK-ed1T1-p1rKe-d1t-n+l-1T,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit∈n+l-1T,nT,which is unique and positive periodic solution of system (17) and is globally attractive. Applying comparison theorem of impulsive differential equation [21], there exist n1∈Z+ and a sufficiently small constant ε1>0 such that(19)xt≤z1t≤z1*t+ε1≤ed1T1-p11+rrK-ed1T1-p1rK+ε1=Δρ,for t≥(n1-1)T.
From inequality (19), we can obtain(20)y˙2t≤λβe-d2τρ2c+ρ2y2t-τ-d3y2t-d4y22t,t≠nT,y2(t+)=(1-p2)y2(t)≤y2(t),t=nT.Consider the auxiliary system of (20):(21)z˙2(t)=λβe-d2τρ2c+ρ2z2t-τ-d3z2t-d4z22t.Applying Lemma 2, (A2), and comparison theorem, we have z2(t)→0 as t→∞. Then, for any small constant ε2>0, there exists t1>0, such that(22)y2t≤z2t≤ε2,t>t1.By the second equation of system (4) and (22), we have(23)y˙1t≤λβρ2c+ρ2ε2-d2y1t.Consider the auxiliary system of (23) as follows:(24)z˙3t=λβρ2c+ρ2ε2-d2z3t.Integrating (24), one can easily getz3(t)=λβρ2/d2(c+ρ2)ε2 as t→∞. Then, for any small constant ε3>0, there exists t2>0 such that(25)y1(t)≤z3(t)≤λβρ2d2(c+ρ2)ε2+ε3,t>t2.According to the first, the fourth, and the seventh equations of (4), we have(26)x˙t≥-βξε2+d1xt,t≠n+l-1T,t≠nT,x(t+)=x(t)+rx(t)(1-x(t)K),t=(n+l-1)T,xt+=1-p1xt,t=nT,where ξ=max⁡{c/2c,ρ/c+ρ2}. Again, consider the auxiliary system of (26):(27)z˙4(t)=-d1z4(t),t≠(n+l-1)T,t≠nT,z4t+=z4t+rz4t1-z4tK,t=n+l-1T,z4t+=1-p1z4t,t=nT.In view of Lemma 3, system (27) has a unique positive periodic solution as follows:(28)z4*t=eβξε2+d1lT1+rrK-eβξε2+d1T1-p1rK×e-βξε2+d1t-n-1T,iiiiiiiiiiit∈n-1T,n+l-1T,eβξε2+d1T1-p11+rrK-eβξε2+d1T1-p1rK×e-βξε2+d1t-n+l-1T,iiiiiiiiiiiiiiiiiiiit∈n+l-1T,nT.Similarly, for any arbitrary small constant ε1>0, there exists t3>0 such that x(t)≥z4*(t)-ε1=Δδ. Let ε2→0; then, z4*(t)→x*(t), and(29)x(t)≥x*(t)-ε1.It follows from (29) and (19) that x(t)→x*(t) as t→∞. Let ε2,ε3→0; then, we have y1(t)→0, y2(t)→0 as t sufficiently large enough. The proof is complete.


## 4. Permanence of System (4)

In the real world, from the principle of ecosystem balance and saving resources, we only need to control the predator under the economic threshold level and not to eradicate the predator totally. Thus, we focus on the permanence of system (4).First, we give the definition of permanence.Definition 7.
System (4) is said to be persistent if there exist positive constants m and M such that every positive solution (x(t),y1(t),y2(t)) of system (4) satisfies m≤x(t),y1(t),y2(t)≤M for t sufficiently large enough.Theorem 8.
System (4) is permanent, if the following conditions hold:(A3)
λβe-d2τq2/c+q2-d3-Md4>0,(A4)
1-p1>eβM/λcm2*+d1T/1+r,where M, m2*, and q are defined in (14), (34), and (39), respectively.Proof.
It is obvious that the third equation of system (4) can be rewritten as follows:(30)y˙2t=λβe-d2τx2tc+x2t-d3-d4y2ty2t-λβe-d2τddt∫t-τtx2θc+x2θy2θdθ.Define V(t)=y2(t)+λβe-d2τ∫t-τtx2(θ)/c+x2(θ)y2(θ)dθ.
By computation, we have(31)V˙t=λβe-d2τx2tc+x2t-d3-d4y2ty2t.Applying Lemma 4, we have(32)V˙(t)>λβe-d2τx2tc+x2t-d3-Md4y2(t).By hypothesis (A3), for the arbitrary small constant ε4>0, we have(33)λβe-d2τq+ε42c+q+ε42>d3+Md4.Let m2* be determined by the following equation:(34)1+rrK-e(βM/cλm2*+d1)T(1-p1)rK=cd3+Md4λβe-d2τ-d3-Md4.Then, for any t4>0, it is impossible that y2(t)<m2* for all t>t4. Suppose that the claim is invalid; then, there exists t4>0 such that y2(t)<m2* for all t>t4. It follows from the first equation of system (4) that(35)x˙(t)≥-βMcλm2*+d1xt,t≠n+l-1T,t≠nT,xt+=xt+rxt1-xtK,t=n+l-1T,xt+=1-p1xt,t=nT,for all t>t4. Consider the auxiliary system of (35) as follows:(36)z˙5t=-βMλcm2*+d1z5t,t≠n+l-1T,t≠nT,z5t+=z5(t)+rz5t1-z5tK,t=(n+l-1)T,z5t+=1-p1z5t,t=nT.By hypothesis (A4), we can obtain from Lemma 3 that(37)z5*t=eβM/cλm2*+d1lT1+rrK-eβM/cλm2*+d1T1-p1rK×e-βM/cλm2*+d1t-n-1T,iiiiiiiiiiiiiiiiiiiiiiiiiiit∈n-1T,n+l-1T,eβM/cλm2*+d1T1-p11+rrK-eβM/cλm2*+d1T1-p1rK×e-βM/cλm2*+d1t-n+l-1T,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit∈n+l-1T,nT,which is the unique positive periodic solution of system (36) and is globally asymptotically stable. By the comparison theorem of impulsive differential equation, we know there exists t5(>t4+τ) such that x(t)≥z5*(t)-ε4.
On the other hand, for allt≥t5, we have(38)z5*t≥1+rrK-eβM/cλm2*+d1T1-p1rK1+rrK-eβM/cλm2*+d1T1-p1rKeβM/cλm2*+d1lT1-p1≥1+rrK-eβM/cλm2*+d1T1-p1rK=Δσ.Thus, (39)x(t)≥σ-ε4=Δq,∀t≥t5.By (33), we have (40)λβe-d2τσ2c+σ2>d3+Md4.Hence, by (32) and (40), we have(41)V˙(t)≥λβe-d2τσ2c+σ2-d3-Md4y2t,t≥t5.Let y2m=min⁡t∈[t1,t1+τ]⁡y2(t). We will show that y2(t)≥y2m for all t≥t5. Otherwise, there exists a constant α>0 such that y2(t)≥y2m for t5≤t≤t5+τ+α, y2(t5+τ+α)≥y2m, and y˙2(t5+τ+α)<0. However, by the third equation of system (4) and (40), we have(42)y˙2t5+τ+α≥λβe-d2τσ2c+σ2-d3-Md4y2m>0,t≥t5,which is a contradiction. Thus, we have y2(t)≥y2m for all t≥t5.
By (40) and (41), therefore,(43)V˙t>λβe-d2τσ2c+σ2-d3-Md4y2m>0,t≥t5,which means that V(t)→∞ as t→∞. It is a contradiction with V(t)≤M/λ(1+rτe-d1τ). Therefore, for any t4>0, the inequality y2(t)<m2* cannot hold for all t≥t4. So, there exist the following two cases: (i)
ify2(t)≥m2* holds for all t large enough, then our goal is obtained;(ii)
ify2(t) is oscillatory about m2*, let(44)m2=min⁡m2*2,m2*e-d3+d4m2*τ.We prove that y2(t)≥m2 for all t sufficiently large. Suppose that there exist positive constants μ,η such that y2(μ)=y2(μ+η)=m2* and y2(t)<m2* for all μ<t<μ+η, and inequality (40) holds true for μ<t<μ+η, where μ is sufficiently large enough. Since y2(t) is continuous and bounded and is not affected by impulses, we conclude that y2(t) is uniformly continuous. Hence, there exists a constant T0  (0<T0<τ and T0 is independent of the choice of μ) such that y2(μ)>m2*/2 for μ≤t≤μ+T0. If η≤T0, our aim is obtained. If T0<η≤τ, from the third equation of (4), we have that, for μ<t<μ+η,  y˙2(t)≥-d3y2(t)-d4y22(t). According to the assumptions y2(μ)=m2* and y2(μ)<m2* for μ<t<μ+η, we have y˙2(t)≥-(d3+d4m2*)y2(t) for μ<t≤μ+η≤μ+τ. Then, we derive that y2(t)≥m2*e-(d3+d4m2*)τ. It is clear that y2(t)≥m2 for μ≤t≤μ+η. If η≥τ, then we have y2(t)≥m2 for μ<t≤μ+τ. The same arguments can be continued. We can obtain y2(t)≥m2 for μ+τ≤t≤μ+η. Since the interval [μ,μ+η] is arbitrarily chosen, we get that y2(t)≥m2 for t sufficiently large. In view of our arguments above, the choice of m2 is independent of the positive solution of (4) which satisfies that y2(t)≥m2 for t large enough.
Next, from the second equation of system (4), we have(45)y˙1t≥λβm21-e-d2τ-d2y1t.Integrating (45) and using comparison theorem, we can easily get(46)y1(t)≥min⁡y10+,λβm21-e-d2τd2=Δm1.On the other hand, by (29), we have x(t)≥x*(t)-ε1=Δm0. Let m=min⁡{m0,m1,m2}; then, we have x(t)≥m, y1(t)≥m, y2(t)≥m. By Lemma 4 and above discussion, system (4) is permanent. This completes the proof.


## 5. Examples and Numerical Simulations

In this section, we give some examples and numerical simulations to show the effectiveness of the main results.In system (4), we let r=2, K=5, β=1, c=10, d1=0.5, λ=1, d2=0.4, d3=0.3, d4=0.1, p1=0.4, p2=0.1, τ=0, T=1. By computation, one can obtain that the conditions of Theorem 6 are satisfied, so the predator-extinction periodic solution is globally attractive, which can be shown by Figure 1. If r=1, K=3, β=2, c=1, d1=0.5, λ=1, d2=0.3, d3=0.2, d4=0.1, p1=0.3, p2=0.1, τ=1, T=0.1, then by computation, the conditions of Theorem 8 are also satisfied; hence, by Theorem 8, system (4) is permanent; see Figure 2.Figure 1
Dynamical behavior of system (4) with r=2, K=5, β=1, c=10, d1=0.5, λ=1, d2=0.4, d3=0.3, d4=0.1, p1=0.4, p2=0.1, τ=0, T=1. (a) Time series of the prey population; (b) time series of the immature predator population; (c) time series of the mature predator population.
(a)
(b)
(c)Figure 2
Dynamical behavior of system (4) with r=1, K=3, β=2, c=1, d1=0.5, λ=1, d2=0.3, d3=0.2, d4=0.1, p1=0.3, p2=0.1, τ=1, T=0.1. (a) Time series of the prey population; (b) time series of the immature predator population; (c) time series of the mature predator population.
(a)
(b)
(c)


## 6. Conclusion

In this paper, a stage-structured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments is proposed. By using comparison theorem of impulsive differential equations and some analysis techniques, the sufficient conditions ensuring the predator-extinction periodic solution and the permanence of system (4) are obtained. Theorem 6 implies that reducing p1 and T appropriately and (A2) can be propitious to the global attractivity of the predator-extinction periodic solution (x*(t),0,0). That is, if the mature predator is caught excessively or the immature predator is stocked too few, it will lose the merits of exploitative mature predator population. Similarly, we can see from Theorem 8 that reducing p1 and T appropriately and (A4) are in favor of the permanence of system (4). It implies that reasonable harvesting on mature predator and appropriate protecting on immature predator play an important role in the permanence of system (4). Moreover, by Theorems 6 and 8, we believe that there exists a sharp threshold, which is beneficial for people to make the best use of biological resources but will not break the biological balance. It will be interesting for us to continue to study the optimal harvesting policy of system (4) in the near future.



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