On a Periodic Time-Dependent Model of Population Dynamics with Stage Structure and Impulsive Effects

Abstract

We consider a periodic time-dependent predator-prey system with stage structure and impulsive harvesting, in which the prey has a life history that takes them through two stages, immature and mature. A set of sufficient and necessary conditions which guarantee the permanence of the system are obtained. Finally, we give a brief discussion of our results.

1. Introduction

In the natural world, there are many species whose individual members have a life history that takes them through two stages, immature and mature. In particular, we have mammalian populations and some amphibious animals in mind, which exhibit these two stages. From the view point of mathematics, the description of the stage structure of the population in the life history is also an interesting problem in population dynamics. The permanence and extinction of species are significant concepts for those stage-structured population dynamical systems. Recently, stage structure models have been studied by many authors [1–3]. This is not only because they are much more simple than the models governed by partial differential equations but also because they can exhibit phenomena similar to those of partial differential models [4], and many important physiological parameters can be incorporated. Much research has been devoted to the models concerning single-species population growth with the stage structure of immature and mature [5, 6]. Two species models with stage structure were investigated by Wang and Chen (1997), Magnusson (1999), Xiao and Chen (2003), as well as Cui and Song (2004). Also, Zhang, Chen, and Neumann (2000) proposed the following autonomous stage structure predator-prey system:𝑥𝑥^1_2=𝛼𝑥₂−𝑟₁𝑥₁−𝛽𝑥₁−𝜂𝑥²¹−𝛽₁𝑥₁𝑥₃,𝑥_3=𝛽𝑥₁−𝑟₂𝑥₂,=𝑥₃−𝑟+𝑘𝛽¹𝑥₁−𝜂₁𝑥₃,(1.1)where 𝛼, 𝛽, 𝛽₁, 𝜂, 𝜂₁, 𝑟, 𝑟₁, 𝑟₂, and 𝑘 are all positive constants, 𝑘 is a digesting constant.

On the other hand, since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. So more realistic and interesting models should take into account the seasonality of the changing environment [7, 8]. This motivated Cui and Y. Takeuchi (2006) to consider the following periodically nonautonomous predator-prey model with stage structure for prey:𝑥𝑥̇̇₁₂=𝑎(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)𝑥²¹−𝑝(𝑡)𝜙𝑡,𝑥₁𝑥¹𝑦,𝑦̇==𝑐𝑦(𝑡)−𝑥₁𝑔−(𝑡)𝑓+(𝑡)ℎ𝑥(²²𝑡),𝜙𝑡,𝑥₁𝑥¹−𝑞(𝑡)𝑦,(1.2)in which 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑓(𝑡), 𝑔(𝑡), ℎ(𝑡), 𝑝(𝑡), 𝑞(𝑡), and 𝜙(𝑡,𝑥₁) are all continuous 𝜔-periodic functions; 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑓(𝑡), ℎ(𝑡), 𝑝(𝑡), 𝑞(𝑡) are all positive; 𝑔(𝑡), 𝜙(𝑡,𝑥₁) are nonnegative; 𝑥₁ and 𝑥₂ denote the density of immature and mature population (prey), respectively; and 𝑦 is the density of the predator that only prey on 𝑥₁ (immature prey). They provided a set of sufficient and necessary conditions to guarantee the permanence of the above system.

Systems with impulsive effects describing evolution processes are characterized by the fact that at certain moments of time, they experience a change of state abruptly. Processes of such type are studied in almost every domain of applied science. Impulsive equations [9, 10] have been recently used in population dynamics in relation to impulsive vaccination [11, 12], population ecology [13, 14], the chemotherapeutic treatment of disease [15], birth pulses [16], as well as the theory of the chemostat [17].

Let us assume that the predator population is affected by harvesting (e.g., fishing or hunting). Further, as we all know that the harvesting does not occur continuously, that is, the harvesting occurs in regular pulses, then let us assume that at some fixed moments, the predator population in system (1.2) is subject to a perturbation which incorporates the proportional decrease. After a perturbation at step 𝜏_𝑘>0(𝑘∈𝐍), the size of the population 𝑦(𝜏^+𝑘) becomes𝑦𝜏^+𝑘=1−𝑢^𝑘𝑦𝜏^𝑘,(1.3)where 𝑦(𝜏_𝑘) is the size of the predator population at step 𝜏_𝑘 before the impulsive perturbation, and 0<𝑢_𝑘<1 represents the rate at which the predator is harvested.

In this article, we extend the model (1.2) to the case when the predator population is omnivorous and affected by impulsive effects, which is governed by the following system:𝑥𝑥̇̇₁₂=𝑎(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)𝑥²¹−𝑝₁(𝑡)𝜙₁𝑡,𝑥¹𝑥²¹𝑦,𝑡≠𝜏_𝑘,𝑦̇==𝑐𝑦(𝑡𝑔)𝑥(𝑡₁)−+𝑓(₂𝑡)𝑥²²−𝑝₂(𝑡)𝜙₂𝑡,𝑥²𝑥²²𝑦,𝑡≠𝜏_𝑘,_𝑖=1Δ𝑦(𝑡)=−𝑢_𝑘ℎ_𝑖(𝑡)𝜙_𝑖𝑡,𝑥^𝑖𝑥^2𝑖−𝑞(𝑡)𝑦(𝑡),𝑡≠𝜏_𝑘,𝑦(𝑡),𝑡=𝜏_𝑘,𝑘∈𝐍,(1.4)with the initial value conditions𝑥₁(0)=𝑥₁₀≥0,𝑥₂(0)=𝑥₂₀≥0,𝑦(0)=𝑦₀≥0,(1.5)in which 𝑥₁ and 𝑥₂ are the densities of immature and mature prey, respectively; and 𝑦 is the density of the predator that can prey on 𝑥₁and𝑥₂. When its favorite food is severely scarce, population 𝑦 can eat other resources: Δ𝑦(𝑡)=𝑦(𝑡⁺)−𝑦(𝑡). Also, there exists a positive integer 𝑞 such that0=𝜏₀𝑢_𝑘+𝑞=𝑢_𝑘,𝜏_𝑘+𝑞=𝜏_𝑘+𝜔,<𝜏₁<𝜏₂<⋯<𝜏_𝑘<𝜏_𝑘+1<⋯,𝑘∈𝐍.(1.6) Here, 𝑥^2𝑖𝜙_𝑖(𝑡,𝑥_𝑖), the number of the prey 𝑥_𝑖 consumed per predator in unit time, is called the predator functional response. We assume that there exists a positive constant 𝐿 such that0<𝜙_𝑖𝑡,𝑥^𝑖<𝐿;𝜕𝜕𝑥_𝑖𝑥^2𝑖𝜙_𝑖𝑡,𝑥^𝑖≥0for𝑥^𝑖>0,𝑖=1,2.(1.7)The last condition in (1.7) implies that as the prey population increases, the consumption rate of prey consumed per predator increases. The birth rate of the immature prey population is proportional to the existing mature prey population with a proportionality function 𝑎(𝑡). For the immature prey population, the death rate is proportional to the existing immature prey population with a proportionality function 𝑏(𝑡). The variable parameter 𝑑(𝑡) represents that the immature prey population is density restriction. The transition rate from immature individuals to the mature individuals is assumed to be proportional to the existing immature population with proportionality coefficient 𝑐(𝑡). The death rate of the mature population is of a logistic nature with proportionality coefficient 𝑓(𝑡). Also, 𝑝_𝑖(𝑡) and ℎ_𝑖(𝑡)(𝑖=1,2) are the coefficients that relate to conversion rates of the immature and, respectively, mature prey biomass into predator biomass. The coefficients in (1.4) are all continuous 𝜔-periodic for 𝑡≥0. In fact, 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑓(𝑡), 𝑝_𝑖(𝑡), ℎ_𝑖(𝑡), and 𝑞(𝑡) are all strictly positive, and 𝜙_𝑖(𝑡,𝑥_𝑖) is nonnegative (𝑖=1,2).

The organization of this paper is as follows. In Section 2, we provide some preliminary results which will be useful. In Section 3, we investigate the permanence and extinction of system (1.4) by using analysis technique. In the last section, we give a biological example and a brief discussion of our result.

2. Preliminary Results

Before stating and proving our main results, we give the following definitions, notations, and lemmas which will be useful in the following section.

Let 𝐑₊=[0,∞), 𝐑^𝑛+={(𝑥₁,…,𝑥_𝑛)∶𝑥_𝑖>0,𝑖=1,…,𝑛} and 𝛼(𝑡) be a continuous 𝜔-periodic function defined on [0,+∞), then we set𝐀_𝜔(𝛼)=𝜔⁻¹^𝜔0𝛼(𝑡)𝑑𝑡,𝛼^𝑈=max_{𝑡∈[0,𝜔]}𝛼(𝑡),𝛼^𝐿=min_{𝑡∈[0,𝜔]}𝛼(𝑡).(2.1)Lemma 2.1. System (1.4) is dissipative.Proof. Define a function 𝑉(𝑡,𝑥₁,𝑥₂,𝑦) such that𝑉𝑡,𝑥₁,𝑥₂,𝑦=ℎ𝑥₁𝑡+𝑥²(𝑡)+𝑝𝑦(𝑡),(2.2)in which ℎ=max_{𝑡∈[0,𝜔]}{ℎ₁(𝑡),ℎ₂(𝑡)}, 𝑝=min_{𝑡∈[0,𝜔]}{𝑝₁(𝑡),𝑝₂(𝑡)}. After a simple computations, we have𝐷⁺𝑉||₍₄₎+𝑤𝑉<ℎ𝑎^𝑈+𝑤𝑥₂−ℎ𝑓^𝐿𝑥²²+ℎ𝑐^𝑈+𝑤−ℎ𝑏^𝐿𝑥¹−ℎ𝑑^𝐿𝑥²¹+𝑤+𝑝𝑔^𝑈𝑦−𝑝𝑞^𝐿𝑦²,(2.3)in which 𝑤 is a positive constant. Obviously, the right-hand side of the above inequality is bounded above for all (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡))∈𝑅³⁺. Hence,𝐷⁺𝑉||₍₄₎+𝑤𝑉<𝜆,(2.4)where 𝜆 is a positive constant. When 𝑡=𝜏_𝑘, we get𝑉𝜏^+𝑘≤𝑉𝜏^𝑘.(2.5)According to Lemma 2.2 in [9, page 23 ], we derive that𝑉(𝑡)=𝑉(0)𝑒^−𝑤𝑡+^𝑡0𝜆𝑒^{−𝑤(𝑡−𝑠)}𝑑𝑠<2𝜆𝑤,as𝑡→∞,(2.6)which implies that system (1.4) is dissipative. This completes the proof.Next, we consider the following two subsystems of system (1.4):𝑥𝑥̇̇₁₂=𝑎(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)𝑥²¹,𝑦̇==𝑐𝑦(𝑡𝑔)𝑥(𝑡₁)−−𝑓𝑞((𝑡𝑡))𝑦𝑥(²²𝑡,),𝑡≠𝜏_𝑘(2.7)Δ𝑦(𝑡)=−𝑢_𝑘,𝑦(𝑡),𝑡=𝜏_𝑘,𝑘∈𝐍.(2.8)Lemma 2.2 ([18]). The system (2.7) has a positive 𝜔-periodic solution (𝑥^∗1(𝑡),𝑥^∗2(𝑡)) which is globally asymptotically stable with respect to 𝐑²⁺.Lemma 2.3. If the following conditions 𝐀_𝜔(𝑔)>0,_𝑞𝑘∏₌₁1−𝑢^𝑘>exp−𝜔𝐀^𝜔(𝑔)(2.9)hold, then system (2.8) has a unique 𝜔-periodic solution: 𝑦^∗(𝑡)=∏^{0<𝜏𝑘<𝑡}1/1−𝑢^𝑘exp−∫^𝑡0𝑔(𝜉)𝑑𝜉∫^𝜔0∏_{𝑠≤𝜏𝑘<𝜔}1/1−𝑢^𝑘exp−∫^𝜔𝑠𝑔(𝜉)𝑑𝜉𝑞(𝑠)𝑑𝑠+1−∏^𝑞𝑘=11/1−𝑢^𝑘exp−∫^𝜔0𝑔(𝜉)𝑑𝜉_𝑡0_𝑠≤_{𝜏𝑘<𝑡}11−𝑢_𝑘exp−^𝑡𝑠𝑔(𝜉)𝑑𝜉𝑞(𝑠)𝑑𝑠⁻¹(2.10)and for every solution 𝑦(𝑡) of system (2.8), |||𝑦(𝑡)−𝑦_∗(𝑡)|||→0as𝑡→∞.(2.11)Proof. Let 𝑦(𝑡)=1/𝑧(𝑡) and obtain the linear nonhomogeneous impulsive equation𝑧𝑧𝑡₊(̇𝑡)=−𝑔(𝑡)𝑧(𝑡)+𝑞(𝑡),𝑡≠𝜏_𝑘,=1−1𝑢_𝑘𝑧(𝑡),𝑡=𝜏_𝑘.(2.12)Let 𝑊(𝑡,𝑠)=∏_{𝑠≤𝜏𝑘<𝑡}(1/(1−𝑢_𝑘))exp(−∫^𝑡𝑠𝑔(𝜉)𝑑𝜉) be the Cauchy matrix for the relevant homogeneous equation, then the solution of (2.12) has the form𝑧(𝑡)=𝑊(𝑡,0)𝑧(0)+^𝑡0𝑊(𝑡,𝑠)𝑞(𝑠)𝑑𝑠.(2.13)The solution 𝑧(𝑡) will be 𝜔-periodic if 𝑧(𝜔)=𝑧(0), or if(1−𝑊(𝜔,0))𝑧(0)=^𝜔0𝑊(𝜔,𝑠)𝑞(𝑠)𝑑𝑠.(2.14)In view of conditions (2.9), (2.14) has a unique solution𝑧(0)=∫^𝜔0𝑊(𝜔,𝑠)𝑞(𝑠)𝑑𝑠1−𝑊(𝜔,0).(2.15)Then, (2.13) has a unique 𝜔-periodic solution𝑧^∗(𝑡)=𝑊(𝑡,0)∫^𝜔0𝑊(𝜔,𝑠)𝑞(𝑠)𝑑𝑠1−𝑊(𝜔,0)+^𝑡0𝑊(𝑡,𝑠)𝑞(𝑠)𝑑𝑠.(2.16)Hence, (2.8) has a unique 𝜔-periodic solution𝑦^∗(𝑡)=𝑊(𝑡,0)∫^𝜔0𝑊(𝜔,𝑠)𝑞(𝑠)𝑑𝑠1−𝑊(𝜔,0)+^𝑡0𝑊(𝑡,𝑠)𝑞(𝑠)𝑑𝑠⁻¹=∏^{0≤𝜏𝑘<𝑡}1/1−𝑢^𝑘exp−∫^𝑡0𝑔(𝜉)𝑑𝜉∫^𝜔0∏_{𝑠≤𝜏𝑘<𝜔}1/1−𝑢^𝑘exp−∫^𝜔𝑠𝑔(𝜉)𝑑𝜉𝑞(𝑠)𝑑𝑠+1−∏^𝑞𝑘=11/1−𝑢^𝑘exp−∫^𝜔0𝑔(𝜉)𝑑𝜉)_𝑡0𝑠≤_{𝜏𝑘<𝑡}11−𝑢_𝑘exp−^𝑡𝑠𝑔(𝜉)𝑑𝜉𝑞(𝑠)𝑑𝑠⁻¹.(2.17)From (2.12), we derive that𝑧(𝑡)−𝑧^∗(𝑡)^=−𝑔(𝑡)𝑧(𝑡)−𝑧^∗(𝑡),𝜏_𝑘<𝑡≤𝜏_𝑘+1.(2.18)Let𝜅≐^𝑞_𝑘=111−𝑢_𝑘exp−^𝜔0𝑔(𝑡)𝑑𝑡.(2.19)Since conditions (2.9) hold, we note that 0<𝜅<1. Then, from Lemma 2.1, the solution of (2.8) is governed by|||𝑦(𝑡)−𝑦_∗(𝑡)|||≤2𝜆𝑤𝑦^∗𝑈|||𝑧0⁺)−𝑧^∗0⁺|||_{0<𝜏𝑘<𝑡}11−𝑢_𝑘≤2𝜆exp−^𝑡0𝑔(𝑠)𝑑𝑠,𝑤𝑦^∗𝑈|||𝑧0⁺−𝑧^∗0⁺|||𝜅^{[𝑡/𝜔]+1}→0as𝑡→∞,(2.20)in which 𝜆 and 𝑤 are defined in Lemma 2.1. This completes the proof.

3. Permanence and Extinction of System

Theorem 3.1. System (1.4) is permanent if and only if 𝐀_𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖>0,_𝑞𝑘∏₌₁1−𝑢^𝑘>exp−𝜔𝐀^𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖,(3.1)in which (𝑥^∗1(𝑡),𝑥^∗2(𝑡)) is the positive 𝜔-periodic solution of system (2.7).We need the following lemmas to prove Theorem 3.1.Lemma 3.2. There exist positive constants 𝑀_𝑥 and 𝑀_𝑦 such that 𝑙𝑖𝑚𝑠𝑢𝑝_𝑡→∞𝑥_𝑖(𝑡)≤𝑀_𝑥(𝑖=1,2),𝑙𝑖𝑚𝑠𝑢𝑝_𝑡→∞𝑦(𝑡)≤𝑀_𝑦.(3.2)Proof. In fact, from Lemmas 2.1 and 2.3, the proof of the lemma is obvious. This completes the proof.Lemma 3.3. There exists a positive constant 𝜌_𝑥(𝜌_𝑥<𝑀_𝑥) such that 𝑙𝑖𝑚𝑖𝑛𝑓_𝑡→∞𝑥_𝑖(𝑡)≥𝜌_𝑥(𝑖=1,2).(3.3)Proof. From Lemma 3.2, one notes that there exists a positive constant 𝑇₀ such that0<𝑦(𝑡)≤𝑀_𝑦for𝑡≥𝑇₀,(3.4)we then have𝑥̇₁≥𝑥𝑎̇₂(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)+𝑝₁(𝑡)𝐿𝑀_𝑦𝑥²¹,≥𝑐(𝑡)𝑥₁−𝑓(𝑡)+𝑝₂(𝑡)𝐿𝑀_𝑦𝑥²²(3.5)for 𝑡≥𝑇₀. According to Lemma 2.2, the following auxiliary equations:𝑢𝑢̇̇₁₂=𝑎(𝑡)𝑢₂−𝑏(𝑡)𝑢₁−𝑑(𝑡)+𝑝₁(𝑡)𝐿𝑀_𝑦𝑢²¹,=𝑐(𝑡)𝑢₁−𝑓(𝑡)+𝑝₂(𝑡)𝐿𝑀_𝑦𝑢²²(3.6)has a globally asymptotically stable 𝜔-periodic solution (𝑢^∗1(𝑡),𝑢^∗2(𝑡)). Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.6) with (𝑢₁(0),𝑢₂(0))=(𝑥₁(0),𝑥₂(0)). By the vector comparison theorem [19], we obtain𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡),𝑖=1,2∀𝑡≥0.(3.7)According to the global asymptotic stability of (𝑢^∗1(𝑡),𝑢^∗2(𝑡)), for any positive constant 𝜀(≤min_{𝑡∈[0,𝜔]}{𝑢^∗𝑖(𝑡)/3,𝑖=1,2}), there exists a 𝑇₁(>𝑇₀) such that for all 𝑡≥𝑇₁,|||𝑢_𝑖(𝑡)−𝑢^∗𝑖(𝑡)|||<𝜀,𝑖=1,2.(3.8)Hence, for all 𝑡≥𝑇₁, we derive that𝑢_𝑖(𝑡)≥𝑢^∗𝑖(𝑡)−𝜀,𝑖=1,2.(3.9)Let𝜌_𝑥=max_{𝑡∈[0,𝜔]}𝑢^∗𝑖(𝑡)2,𝑖=1,2,(3.10)then𝑥_𝑖(𝑡)≥𝜌_𝑥,𝑖=1,2.(3.11)Consequently,𝑙𝑖𝑚𝑖𝑛𝑓_𝑡→∞𝑥_𝑖(𝑡)≥𝜌_𝑥,𝑖=1,2.(3.12)This completes the proof.Lemma 3.4. Suppose that (3.1) holds, then there exists a positive constant 𝜚_𝑦(𝜚_𝑦<𝑀_𝑦) such that 𝑙𝑖𝑚𝑠𝑢𝑝_𝑡→∞𝑦(𝑡)≥𝜚_𝑦.(3.13)Proof. In view of (3.1), we can choose a positive constant 𝜀₀ such that𝐀_𝜔𝜓^𝜀₀(𝑡)>0,_𝑞𝑘∏₌₁1−𝑢^𝑘>exp(−𝜔𝐀^𝜔𝜓^𝜀₀(𝑡)),(3.14)in which𝜓_𝜀0(𝑡)=𝑔(𝑡)+²_𝑖=1ℎ_𝑖(𝑡)𝜙_𝑖𝑡,𝑥^∗𝑖(𝑡)−𝜀₀𝑥^∗𝑖(𝑡)−𝜀₀²−𝑞(𝑡)𝜀₀.(3.15)Consider the following system with a positive parameter 𝜇: 𝑥𝑥̇̇₁₂=𝑎(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)+𝐿𝜇𝑝₁(𝑡)𝑥²¹,=𝑐(𝑡)𝑥₁−𝑓(𝑡)+𝐿𝜇𝑝₂(𝑡)𝑥²².(3.16)By Lemma 2.2, system (3.16) has a positive 𝜔-periodic solution (𝑥^∗1𝜇(𝑡),𝑥^∗2𝜇(𝑡)), which is globally asymptotically stable. Let (𝑥_1𝜇(𝑡),𝑥_2𝜇(𝑡)) be the solution of (3.16) with initial condition 𝑥_𝑖𝜇(0)=𝑥^∗𝑖(0), 𝑖=1,2, where (𝑥^∗1(𝑡),𝑥^∗2(𝑡)) is the positive periodic solution of (2.7). Hence, for the above 𝜀₀, there exists 𝑇₂>𝑇₁ such that|||𝑥_𝑖𝜇(𝑡)−𝑥^∗𝑖𝜇(𝑡)|||<𝜀⁰4(3.17)for 𝑡≥𝑇₂, 𝑖=1,2. According to the continuity of the solution in the parameter 𝜇, we have 𝑥_𝑖𝜇(𝑡)→𝑥^∗𝑖(𝑡)(𝑖=1,2) uniformly in [𝑇₂,𝑇₂+𝜔] as 𝜇→0. Hence, for 𝜀₀>0, there exists 𝜇₀=𝜇₀(𝜀₀)(0<𝜇₀<𝜀₀) such that|||𝑥_𝑖𝜇(𝑡)−𝑥^∗𝑖(𝑡)|||<𝜀⁰4,0≤𝜇≤𝜇⁰,(3.18)𝑡∈[𝑇₂,𝑇₂+𝜔], 𝑖=1,2. Thus, from (3.17) and (3.18), we get|||𝑥_∗𝑖𝜇(𝑡)−𝑥^∗𝑖(𝑡)|||<𝜀⁰2,0≤𝜇≤𝜇⁰,(3.19)𝑡∈[𝑇₂,𝑇₂+𝜔], 𝑖=1,2. Since 𝑥^∗𝑖𝜇(𝑡) and 𝑥^∗𝑖(𝑡) are all 𝜔-periodic, we have|||𝑥_∗𝑖𝜇(𝑡)−𝑥^∗𝑖(𝑡)|||<𝜀⁰2,0≤𝜇≤𝜇⁰,(3.20)𝑡≥0,𝑖=1,2.
Choose a constant 𝜇₁(0<𝜇₁<𝜇₀,𝜇₁<𝜀₀), from (3.20), we derive𝑥^∗𝑖𝜇₁(𝑡)≥𝑥^∗𝑖(𝑡)−𝜀⁰2,𝑡≥0,𝑖=1,2.(3.21)Suppose that (3.13) is not true, then for the above 𝜀₀, there exists a 𝜈∈𝐑³⁺ such that𝑙𝑖𝑚𝑠𝑢𝑝_𝑡→∞𝑦(𝑡)<𝜇₁,(3.22)where (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) is the solution of (1.4) with the initial condition (𝑥₁₀,𝑥₂₀,𝑦₀)=𝜈. So there exists a constant 𝑇₃(>𝑇₂) such that𝑦(𝑡)<𝜇₁,𝑡≥𝑇₃,(3.23)then we derive that𝑥̇₁≥𝑥𝑎̇₂(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)+𝐿𝜇₁𝑝₁(𝑡)𝑥²¹,≥𝑐(𝑡)𝑥₁−𝑓(𝑡)+𝐿𝜇₁𝑝₂(𝑡)𝑥²²(3.24)for 𝑡≥𝑇₃. Let (𝑥_1𝜇1,𝑥_2𝜇1) be the solution of (3.16) with 𝜇=𝜇₁ and (𝑥_1𝜇1(𝑇₃),𝑥_2𝜇1(𝑇₃))=(𝑥₁(𝑇₃),𝑥₂(𝑇₃)), then by the vector comparison theorem, we obtain𝑥_𝑖(𝑡)≥𝑥_𝑖𝜇1(𝑡),𝑖=1,2,(3.25)𝑡≥𝑇₃. By the global asymptotic stability of (𝑥^∗1𝜇₁(𝑡),𝑥^∗2𝜇₁(𝑡)), for the given 𝜀₀>0, there exists 𝑇₄>𝑇₃ such that𝑥_𝑖𝜇1(𝑡)>𝑥^∗𝑖𝜇₁(𝑡)−𝜀⁰2,𝑡≥𝑇⁴,𝑖=1,2,(3.26) and hence, by (3.21), we get𝑥_𝑖(𝑡)>𝑥^∗𝑖(𝑡)−𝜀₀,𝑡≥𝑇₄,𝑖=1,2.(3.27)Since 0<𝑦(𝑡)<𝜇₁<𝜀₀ together with (1.4) and (1.7), we have𝑦̇≥𝑦𝑔(𝑡)+∑²_𝑖=1ℎ𝑦_𝑖(𝑡𝑡)₊𝜙_𝑖𝑡,𝑥^∗𝑖(𝑡)−𝜀₀𝑥^∗𝑖(𝑡)−𝜀₀²−𝑞(𝑡)𝜀₀,𝑡≠𝜏^𝑘,=1−𝑢^𝑘𝑦(𝑡),𝑡=𝜏^𝑘,𝑘∈𝐍.(3.28)Hence, it follows from Lemma 2.2 in [9, page 23] that𝑦(𝑡)≥𝑦₀₀_{<𝜏𝑘<𝑡}1−𝑢^𝑘exp^𝑡0𝜓_𝜀0(𝑠)𝑑𝑠,(3.29)that is,𝑦(𝑡)≥𝑦₀_{0<𝜏𝑘<𝜔}1−𝑢^𝑘exp(𝜔𝐀^𝜔𝜓^𝜀₀^[𝑡/𝜔].(3.30)By (3.14), we know that 𝑦(𝑡)→∞ as 𝑡→∞, which leads to a contradiction. This completes the proof.Lemma 3.5. Assume that (3.1) holds, then there exists a positive constant 𝛿_𝑦(𝛿_𝑦<𝑀_𝑦) such that any solution (𝑥₁,𝑥₂,𝑦) of system (1.4) with initial conditions satisfies 𝑙𝑖𝑚𝑖𝑛𝑓_𝑡→∞𝑦(𝑡)≥𝛿_𝑦.(3.31)Proof. Suppose that (3.31) is not true, there must exists a time sequence {𝑡⁽_)𝑡𝑘_𝑘̂}^∞𝑘=1⊂𝐑₊, 𝑘̂∈𝐙 such that𝑙𝑖𝑚𝑖𝑛𝑓_𝑡(𝑘̂_)𝑘→∞𝑦𝑡𝑘_𝑘̂<𝜚^𝑦(𝑘̂+1)²,(3.32)and by Lemma 3.4, we have 𝑙𝑖𝑚𝑠𝑢𝑝_𝑡(𝑘̂_)𝑘→∞𝑦(𝑡⁽₎𝑘_𝑘̂)≥𝜚_𝑦, 𝑘̂=1,2,…. Hence, for each 𝑘̂, we choose two time sequences {𝑠⁽₎𝑘_𝑞̂} and {𝑡⁽₎𝑘_𝑞̂}, satisfying 0<𝑠⁽₎𝑘₁̂<𝑡⁽₎𝑘₁̂<𝑠⁽₎𝑘₂̂<𝑡⁽₎𝑘₂̂<⋯<𝑠⁽₎𝑘_𝑞̂<𝑡⁽₎𝑘_𝑞̂<⋯ and 𝑠⁽₎𝑘_𝑞̂→∞ as 𝑞→∞, as well as𝑦𝑠⁽₎𝑘_𝑞̂=𝑘̂𝜚+^𝑦1,𝑦𝑡⁽₎𝑘_𝑞̂=_{0<𝜏𝑘<𝑡()}𝑘_𝑞̂1−𝑢^𝑘(𝑘̂+𝜚^𝑦1)²,(3.33)_{0<𝜏𝑘<𝑡()}𝑘_𝑞̂1−𝑢^𝑘(𝑘̂+𝜚^𝑦1)²<𝑦(𝑡)<𝑘̂𝜚+^𝑦1,𝑡∈𝑠⁽₎𝑘_𝑞̂,𝑡⁽₎𝑘_𝑞̂.(3.34)𝑘̂By Lemma 3.2, for a given positive integer 𝑇⁽𝑘̂₎>0, there exists 𝑥_𝑖(𝑡)≤𝑀_𝑥(𝑖=1,2) such that 𝑦(𝑡)≤𝑀_𝑦 and 𝑡≥𝑇⁽𝑘̂₎ for all 𝑠⁽₎𝑘_𝑞̂→∞. In view of 𝑞→∞ as 𝐾⁽₎𝑘₁̂, there exists a positive integer 𝑠⁽₎𝑘_𝑞̂>𝑇⁽𝑘̂₎ such that 𝑞≥𝐾⁽₎𝑘₁̂ as 𝑡≥𝑇⁽𝑘̂₎. Hence, for any 𝑦𝑦𝑡₊̇≥𝑦−|||𝑔(𝑡)|||−𝑀_𝑦𝑞(𝑡),𝑡≠𝜏_𝑘,=1−𝑢^𝑘𝑦(𝑡),𝑡=𝜏^𝑘,𝑘∈𝐍.(3.35), we have𝑠⁽₎𝑘_𝑞̂Integrating the above inequality from 𝑡⁽₎𝑘_𝑞̂ to 𝑞≥𝐾⁽₎𝑘₁̂, for any 𝑦(𝑡⁽₎𝑘_𝑞̂)≥𝑦(𝑠⁽₎𝑘_𝑞̂)_𝑠(𝑘̂_{)𝑞<𝜏𝑘<𝑡(}𝑘_)𝑞̂(1−𝑢_𝑘)exp^𝑡(_)𝑞𝑠𝑘₍̂₎𝑘_𝑞̂−|||𝑔(𝑡)|||−𝑀^𝑦𝑞(𝑡)𝑑𝑡.(3.36), then we have^𝑡(_)𝑞𝑠𝑘₍̂₎𝑘_𝑞̂|||𝑔(𝑡)|||+𝑀^𝑦𝑞(𝑡)𝑑𝑡>ln(𝑘̂+1)for𝑞≥𝐾⁽₎𝑘₁̂.(3.37)Obviously, it follows from (3.33) that𝑔(𝑡)Hence, in view of the periodicity of 𝑞(𝑡) and 𝑡⁽₎𝑘_𝑞̂−𝑠⁽₎𝑘_𝑞̂→∞as𝑘̂→∞,𝑞≥𝐾⁽₎𝑘₁̂.(3.38), we get𝑇By (3.14), (3.33), and (3.38), there are positive constants 𝑁₀ and 𝑦𝑠⁽₎𝑘_𝑞̂=_{0<𝜏𝑘≤𝑠()}𝑘_𝑞̂𝑡1−𝑢^𝑘𝑘̂𝜚+^𝑦1<𝜀⁰,(3.39)₍₎𝑘_𝑞̂−𝑠⁽₎𝑘_𝑞̂>2𝑇,(3.40)_{0<𝜏𝑘<𝜅}1−𝑢^𝑘exp^𝜅0𝜓_𝜀0(𝑡)𝑑𝑡>1(3.41) such that𝑘̂≥𝑁₀,𝑞≥𝐾⁽₎𝑘₁̂𝜅>𝑇for 𝑦(𝑡)<𝜀₀,𝑡∈𝑠⁽₎𝑘_𝑞̂,𝑡⁽₎𝑘_𝑞̂(3.42) for 𝑘̂≥𝑁₀,𝑞≥𝐾⁽𝑘₎₁̂., and for all 𝑥̇₁≥𝑥𝑎̇₂(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)+𝐿𝜀₀𝑝₁(𝑡)𝑥²¹,≥𝑐(𝑡)𝑥₁−𝑓(𝑡)+𝐿𝜀₀𝑝₂(𝑡)𝑥²².(3.43). However, (3.39) implies that(𝑥_1𝜀0,𝑥_2𝜀0)for 𝜇=𝜀₀ Then, we get(𝑥_1𝜀0(𝑇₃),𝑥_2𝜀0(𝑇₃))=(𝑥₁(𝑇₃),𝑥₂(𝑇₃))Let 𝑥_𝑖(𝑡)≥𝑥_𝑖𝜀0(𝑡),𝑖=1,2,𝑡∈𝑠⁽₎𝑘_𝑞̂,𝑡⁽₎𝑘_𝑞̂.(3.44) be the solution of (3.16) with lim_𝑞→∞𝑠⁽₎𝑘_𝑞̂=∞ and 𝑘̂, then by the vector comparison theorem, we obtain𝐾⁽₎𝑘₂̂>𝐾⁽₎𝑘₁̂From 𝑞≥𝐾⁽₎𝑘₂̂ and Lemmas 3.2 and 3.3, we obtain that for any 𝜌_𝑥≤𝑥_𝑖𝑠⁽₎𝑘_𝑞̂≤𝑀^𝑥,𝑖=1,2.(3.45), there is a 𝜇=𝜀₀ such that for any 𝜔,(𝑥^∗1𝜀₀(𝑡),𝑥^∗2𝜀₀(𝑡))
For (𝑥^∗1𝜀₀(𝑡),𝑥^∗2𝜀₀(𝑡)), (3.16) has a globally asymptotically stable positive 𝜀₀>0-periodic solution 𝑇₅>𝑇₄. By the global asymptotic stability of 𝑇₅, for the given 𝑘̂, there exists 𝑞, and 𝑥_𝑖𝜀0(𝑡)>𝑥^∗𝑖𝜀₀(𝑡)−𝜀⁰2,𝑡≥𝑇⁵+𝑠⁽₎𝑘_𝑞̂,𝑖=1,2,𝑞≥𝐾⁽₎𝑘₂̂,(3.46) is dependent of any 𝑥_𝑖𝜀0(𝑡)>𝑥^∗𝑖(𝑡)−𝜀₀,𝑡≥𝑇₅+𝑠⁽₎𝑘_𝑞̂,𝑖=1,2,𝑞≥𝐾⁽₎𝑘₂̂.(3.47) and 𝑁₁≥𝑁₀ such that𝑡⁽₎𝑘_𝑞̂−𝑠⁽₎𝑘_𝑞̂≥2𝑇and hence, by (3.21), we get𝑘̂≥𝑁₁By (3.44), there is an 𝑞≥𝐾⁽₎𝑘₂̂ such that 𝑇≥𝑇₅ for all 𝑥_𝑖(𝑡)≥𝑥^∗𝑖(𝑡)−𝜀₀,𝑖=1,2(3.48) and 𝑡∈[𝑇+𝑠⁽₎𝑘_𝑞̂,𝑡⁽₎𝑘_𝑞̂], where 𝑘̂≥𝑁₁. Hence, from (3.44) and (3.47), we obtain𝑞≥𝐾⁽₎𝑘₂̂for all 𝑡∈[𝑇+𝑠⁽₎𝑘_𝑞̂,𝑡⁽₎𝑘_𝑞̂], 𝑘≥𝑁₁, and 𝑞≥𝐾⁽₎𝑘₂̂. Since, for any 𝑦̇≥𝑦𝑔(𝑡)+∑²_𝑖=1ℎ_𝑖(𝑡)𝜙_𝑖𝑡,𝑥^∗𝑖(𝑡)−𝜀₀𝑥^∗𝑖(𝑡)−𝜀₀²−𝑞(𝑡)𝜀₀,𝑡≠𝜏^𝑘,𝑡∈𝑇+𝑠_𝑞𝑘̂,𝑡⁽₎𝑘_𝑞̂𝑦𝑡₊,=1−𝑢^𝑘𝑦(𝑡),𝑡=𝜏^𝑘,𝑘∈𝐍.(3.49), 𝑇+𝑠⁽₎𝑘_𝑞̂ and 𝑡⁽₎𝑘_𝑞̂, by (1.4) and (3.42), we have 𝑘≥𝑁₁Integrating from 𝑞≥𝐾⁽₎𝑘₂̂ to 𝑦𝑡⁽𝑘̂_𝑞≥𝑦𝑠⁽₎𝑘_𝑞̂+𝑇_𝑠(𝑘̂_{)𝑞+𝑇<𝜏𝑘<𝑡(}𝑘_)𝑞̂1−𝑢^𝑘exp^𝑡(_)𝑞𝑠𝑘₍̂𝑘̂_)𝑞+𝑇𝜓_𝜀0(𝑡)𝑑𝑡.(3.50) for any _0<_{𝜏𝑘<𝑡()}𝑘_𝑞̂1−𝑢^𝑘(𝑘̂+𝜚^𝑦1)²≥_{0<𝜏𝑘≤𝑡()}𝑘_𝑞̂1−𝑢^𝑘(𝑘̂+𝜚^𝑦1)²_𝑠(𝑘̂_{)𝑞+𝑇<𝜏𝑘<𝑡(}𝑘_)𝑞̂1−𝑢^𝑘exp^𝑡(_)𝑞𝑠𝑘₍̂>𝑘̂_)𝑞+𝑇𝜓_𝜀0(𝑡)𝑑𝑡_{0<𝜏𝑘<𝑡(}₎𝑘_𝑞̂1−𝑢^𝑘(𝑘̂+𝜚^𝑦1)².(3.51) and 𝐀_𝜔𝑔+²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖≤0,(3.52), we obtain𝑦(𝑡)Hence, by (3.34), (3.40), and (3.41), we finally derive that𝐀_𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖>0,_𝑞𝑘∏₌₁1−𝑢^𝑘≤exp−𝜔𝐀^𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖.(3.53)This leads to a contradiction. This completes the proof.According to Lemmas 3.2–3.5, we can directly prove the sufficient part of the theorem. Next, we are ready to show the necessity of this theorem.

Consider the case oflim_𝑡→∞𝑦(𝑡)=0.(3.54)it is then easy to derive that the predator population 0<𝜖<1 must be extinct because of sterilization and impulsive harvesting. Suppose that𝜖₁>0We will show that𝐀_𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖+𝜖₁𝑥^∗𝑖+𝜖₁²−𝑞𝜖>0,_𝑞𝑘∏₌₁1−𝑢^𝑘<exp−𝜔𝐀^𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖+𝜖₁𝑥^∗𝑖+𝜖₁²−𝑞𝜖.(3.55)In fact, from (1.7) and (3.53), we know that for any given 𝑞(𝑡)>0, there exists 𝑡≥0 such that𝑥̇₁≤𝑥𝑎̇₂(𝑡)𝑥₂−𝑏(𝑡)𝑥₁−𝑑(𝑡)𝑥²¹,≤𝑐(𝑡)𝑥₁−𝑓(𝑡)𝑥²²,(3.56)Note that 𝜖₁>0 for 𝑇⁽¹⁾>0. Since𝑥_𝑖(𝑡)≤𝑥^∗𝑖(𝑡)+𝜖₁for𝑡>𝑇⁽¹⁾,𝑖=1,2,(3.57)for the given ^𝑞_𝑘=11−𝑢^𝑘<exp−𝜔𝐀^𝜔𝑔+²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^𝑖𝑥^𝑖²−𝑞𝜖.(3.58), it is easy to show that there exists 𝑇⁽²⁾(>𝑇⁽¹⁾) such that𝑦(𝑇⁽²⁾)<𝜖we then have𝜖≤𝑦(𝑡)≤𝑦𝑇⁽¹⁾_{𝑇(1)<𝜏𝑘<𝑡}(1−𝑢_𝑘)exp^𝑡𝑇₍₁₎𝑔(𝑠)+²_𝑖=1ℎ_𝑖(𝑠)𝜙_𝑖𝑠,𝑥^𝑖(𝑠)𝑥^2𝑖(𝑠)−𝑞(𝑠)𝜖a𝑑s𝑠𝑡→→∞0.(3.59)We now show that there must exist 𝜖≤0 such that 𝑀(𝜖)=max_{𝑡∈[0,𝜔]}{𝑔(𝑡)+∑^2𝑖=1ℎ_𝑖(𝑡)𝜙_𝑖(𝑠,𝑥_𝑖(𝑡))𝑥^2𝑖(𝑡)+𝑞(𝑡)𝜖}. Otherwise, by the last two equations in system (1.4), we have𝑀(𝜖)This implies that 𝑦(𝑡)≤𝜖exp𝑀𝜖𝜔for𝑡≥𝑇⁽²⁾.(3.60), which is a contradiction.

Let 𝑇⁽³⁾>𝑇⁽²⁾. Note that 𝑇⁽³⁾∈(𝑇⁽²⁾+𝑃₁𝜔,𝑇⁽²⁾+(𝑃₁+1)𝜔], is bounded. We then show that𝑃₁∈𝐙₊={0,1,2,…}Otherwise, there exists 𝑦(𝑇⁽³⁾)>𝜖exp(𝑀(𝜖)𝜔). such that 𝜖exp𝑀(𝜀)𝜔<=𝑦𝑦𝑇𝑇⁽³⁾₍₂₎≤𝑦𝑇₍₂₎_{𝑇(2)<𝜏𝑘<𝑇(3)}1−𝑢^𝑘exp^𝑇(3)_𝑇(2)𝑔(𝑠)+²_𝑖=1ℎ_𝑖(𝑠)𝜙_𝑖𝑠,𝑥^𝑖(𝑠)𝑥^2𝑖(𝑠)−𝑞(𝑠)𝜖𝑑𝑠^𝑞_𝑘=11−𝑢^𝑘exp^𝜔0𝑔(𝑠)+²_𝑖=1ℎ_𝑖(𝑠)𝜙_𝑖𝑠,𝑥^𝑖(𝑠)𝑥^2𝑖(𝑠)−𝑞(𝑠)𝜖𝑑𝑠^𝑃1×_{𝑇(2)+𝑃1𝜔<𝜏𝑘<𝑇(3)}1−𝑢^𝑘exp^𝜔0𝑔(𝑠)+²_𝑖=1<𝜖exp𝑀(𝜀)𝜔,ℎ_𝑖(𝑠)𝜙_𝑖𝑠,𝑥^𝑖(𝑠)𝑥^2𝑖(𝑠)−𝑞(𝑠)𝜖𝑑𝑠(3.61) in which 𝜖, and 𝑦(𝑡)→0 By (3.58), we have𝑡→∞which leads to a contradiction. This implies that (3.60) holds. In view of the arbitrariness of 𝑦, we get that 𝐀_𝜔𝑔+²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖≤0,(3.62) as 𝐀_𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖>0,_𝑞𝑘∏₌₁1−𝑢^𝑘≤exp−𝜔𝐀^𝜔𝑔+∑²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖.(3.63). This completes the proof.

Remark 3.6. From the proof of Theorem 3.1 above, we note that the predator population 𝑥₁(𝑡) in system (1.4) will be extinct provided that𝑥₂(𝑡)or excessive harvesting, that is,𝑡

4. Discussion

Our result could provide a useful insight into the conservation of beneficial animals, especially rare animals. As an example, we depict the case of Oreolalax omeimontis tadpole, Oreolalax omeimontis, and red-eared slider (T. scripta elegans). Oreolalax omeimontis is a rare species of frog found near Mt. Omei in Sichuan (China). The red-eared slider is a native of the Mississippi Valley area of the United States [8, 20]. Since the 1970s, large numbers of red-eared sliders have been produced on turtle farms in the USA for the international pet trade. Red-eared Turtles are traded as pet animals and have been introduced to many countries. They are omnivorous and will eat insects, tadpoles, crayfish, shrimp, worms, snails, amphibians and small fish, as well as aquatic plants, and they hardly may be controlled by a natural enemy. In our model, the variables 𝑦(𝑡) and 𝑡 represent the density of Oreolalax omeimontis tadpole and Oreolalax omeimontis at time 𝑢_𝑘(𝑘=1,2⋯𝑞), respectively. The variable 𝑦 describes the density of the red-eared slider at time 𝑥₁. It is well known that Oreolalax omeimontis is beneficial to humans because they eat so many insect pests. Ironically, although red-eared sliders have been widely introduced throughout the world, its detrimental effects have been reported by many researchers [21, 22]. The red-eared slider is one of main enemies of Oreolalax omeimontis. To protect these beneficial and rare toads, we must control the amount of red-eared sliders in the habitat of Oreolalax omeimontis. From a control point of view, our aim is to keep red-eared sliders at an acceptably low level with a minimum use of artificial control measures, not to eradicate all red-eared sliders. Hence, in the above example, the problem of nonextinction of populations becomes a description of reasonable harvesting rates 𝑥₂.

According to our main result, system (1.4) is permanent if and only if the growth of the predator 𝜔 by foraging prey populations 𝑢_𝑘(𝑘=1,2,⋯𝑞) and 𝜔 plus its intrinsic rate of increase is positive on average during the period ^𝑞_𝑘=11−𝑢^𝑘>exp−𝜔𝐀^𝜔𝑔+²_𝑖=1ℎ_𝑖𝜙_𝑖𝑡,𝑥^∗𝑖𝑥^∗2𝑖.(4.1), and harvesting rates 𝑔+∑^2𝑖=1ℎ_𝑖𝜙_𝑖(𝑡,𝑥^∗𝑖)𝑥^∗2_𝑖<0 of red-eared sliders during the period 0≤𝑡≤𝜔. are small enough to satisfy thatThese seem to be reasonable from a biological point of view; but it should be noted that condition (3.1) allows the predator to be for some time intervals among That is, under reasonable harvesting rates, the predator can survive together with prey populations even if the growth rate of the former is negative for some seasons during a period. We hope that our result can be used to help protect beneficial animals found in their habitats.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (10671001).