An Impulsive Two-Prey One-Predator System with Seasonal Effects

Abstract

In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls) for all species and the constant impulse (the biological control) for the predator at different fixed time but, the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical examples and conclusion are given.

1. Introduction

Recently, it is of great interest to study dynamical properties for impulsive perturbations in population dynamics. Impulsive prey-predator population systems have been discussed by a number of researchers [1–8] and, what is more, there are also many literatures on simple multispecies systems consisting of a three-species food chain with impulsive perturbations [7, 9–18]. Especially, two-prey and one-predator impulsive systems are drawing notice. For examples, Song and Li [13] studied dynamical behavior of a Holling type II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategies at fixed time. Zhang et al. [17, 18] studied a Lotka-Volterra type two-prey one-predator system with impulsive effect on the predator of a fixed moment.

It is necessary and important to consider systems with periodic ecological parameters which might be quite naturally exposed such as those due to seasonal effects of weather or food supply [19]. Indeed, it has been studied that dynamical systems with simple dynamical behavior may display complex dynamical behavior when they have periodic perturbations[20–22]. For this reason, in this paper, we consider the intrinsic growth rates 𝐴 of prey population as a periodically varying function of time due to seasonal variations. The seasonality is superimposed as follows[19–22]:𝐴₀=𝐴(1+𝜖sin(𝜔𝑡)), (1.1)where the parameter 𝜖(𝑖=1,2) represents the degree of seasonality, 𝜆=𝐴𝜖≥0 is the magnitude of the perturbation in 𝐴₀, and 𝜔 is the angular frequency of the fluctuation caused by seasonality. It is pertinent to point out that the forced ecosystem we are studying in this paper is similar to forced nonlinear oscillators in physics such as the Duffing oscillator.

Thus, we develop the Holling-type IV two-competitive-prey one-predator system with seasonality by introducing a proportional periodic impulsive poisoning(spraying pesticide) for all species and a constant periodic releasing, or immigrating, for the predator at different fixed time as follows:𝑥^′₁(𝑡)=𝑥₁(𝑡)(𝑎₁+𝜆₁sin(𝜔₁𝑡)−𝑏₁𝑥₁(𝑡)−𝛾₁𝑥₂𝑒(𝑡)−¹𝑦(𝑡)1+𝑐₁𝑥²₁𝑥(𝑡)),_′2(𝑡)=𝑥₂(𝑡)(𝑎₂+𝜆₂sin(𝜔₂𝑡)−𝑏₂𝑥₂(𝑡)−𝛾₂𝑥₁𝑒(𝑡)−²𝑦(𝑡)1+𝑐₂𝑥²₂(𝑡)),𝑦′(𝑡)=𝑦(𝑡)(−𝑎₃+𝛽₁𝑥₁(𝑡)1+𝑐₁𝑥²₁+𝛽(𝑡)²𝑥₂(𝑡)1+𝑐₂𝑥²₂(𝑡)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₁(𝑡)=−𝑝₁𝑥₁(𝑡),Δ𝑥₂(𝑡)=−𝑝₂𝑥₂(𝑡),Δ𝑦(𝑡)=−𝑝₃𝑦(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₁(𝑡)=0,Δ𝑥₂(𝑡)=0,Δ𝑦(𝑡)=𝑞,𝑡=𝑛𝑇,(𝑥₁(0⁺),𝑥₂(0⁺),𝑦(0⁺))=(𝑥₀₁,𝑥₀₂,𝑦₀), (1.2)where 𝑎_𝑖(𝑖=1,2) are intrinsic rates of increase, 𝑏_𝑖(𝑖=1,2) are the coefficients of intra-specific competition, 𝛾_𝑖(𝑖=1,2) are parameters representing competitive effects between two preys, 𝑒_𝑖(𝑖=1,2) are the per-capita rates of predation of the predator, 𝑐_𝑖(𝑖=1,2) are the half-saturation constants, 𝑎₃>0 denotes the death rate of the predator, 𝛽_𝑖(𝑖=1,2) are the rates of conversing prey into predator, 𝜆_𝑖(𝑖=1,2) are the magnitude, 𝜔_𝑖(𝑖=1,2) are the angular frequency, 𝜏,𝑇 are the period of spaying pesticides (harvesting) and the impulsive immigration or stock of the predator, respectively, 0≤𝑝₁,𝑝₂,𝑝₃<1 present the fraction of the preys and the predator which die due to the harvesting or pesticides, and 𝑞 is the size of immigration or stock of the predator.

In Section 2, we give some notations and lemmas. In Section 3, we show the boundedness of the system and take into account the local and global stabilities of two-prey-free periodic solutions by using Floquet theory for the impulsive equation, small amplitude perturbation skills and comparison techniques, and finally, prove that the system is permanent under some conditions. Moreover, we give the sufficient conditions under which one of the two prey extinct and the remaining two species are permanent. Numerical examples are given in Section 4.

2. Preliminaries

Let ℝ₊=[0,∞), ℝ^∗₊=(0,∞), and ℝ³₊={𝐱=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))∈ℝ³∶𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)≥0}. Denote ℕ the set of all of nonnegative integers and 𝑓=(𝑓₁,𝑓₂,𝑓₃)^𝑇 the right hand of the first three equations in (1.2). Let 𝑉∶ℝ₊×ℝ³₊→ℝ₊, then 𝑉 is said to belong to class 𝑉₀ if (1)𝑉 is continuous on ((𝑛−1)𝑇,(𝑛+𝜏−1)𝑇]×ℝ³₊∪((𝑛+𝜏−1)𝑇,𝑛𝑇]×ℝ³₊, and lim_{(𝑡,𝑦)→(𝑡0,𝐱)}𝑉(𝑡,𝑦)=𝑉(𝑡₀,𝐱) exists, where 𝑡₀=(𝑛+𝜏−1)𝑇⁺ and 𝑛𝑇⁺,(2)𝑉 is locally Lipschitzian in 𝐱.

Definition 2.1. Let 𝑉∈𝑉₀. For (𝑡,𝑥)∈((𝑛−1)𝑇,(𝑛+𝜏−1)𝑇]×ℝ³₊∪((𝑛+𝜏−1)𝑇,𝑛𝑇]×ℝ³₊, the upper right derivative of 𝑉 with respect to the impulsive differential system (1.2) is defined as𝐷⁺𝑉(𝑡,𝐱)=limsup_ℎ→0+1ℎ[𝑉(𝑡+ℎ,𝐱+ℎ𝑓(𝑡,𝐱))−𝑉(𝑡,𝐱)]. (2.1)The solution of system (1.2) is a piecewise continuous function 𝑋(𝑡)∶ℝ₊→𝑅³₊, 𝑋(𝑡) is continuous on ((𝑛−1)𝑇,(𝑛+𝜏−1)𝑇)∪((𝑛+𝜏−1)𝑇,𝑛𝑇)(𝑛∈ℕ,0≤𝜏≤1). Obviously, the smoothness properties of 𝑓 guarantee the global existence and uniqueness of solutions of system (1.2) [23, 24].Definition 2.2. The system (1.2) is permanent if there exist 𝑀≥𝑚>0 such that, for any solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) of system (1.2) with (𝑥₀₁,𝑥₀₂,𝑦₀)>0,𝑚≤lim_𝑡→∞inf𝑥₁(𝑡)≤lim_𝑡→∞sup𝑥₁(𝑡)≤𝑀,𝑚≤lim_𝑡→∞inf𝑥₂(𝑡)≤lim_𝑡→∞sup𝑥₂(𝑡)≤𝑀,𝑚≤lim_𝑡→∞inf𝑦(𝑡)≤lim_𝑡→∞sup𝑦(𝑡)≤𝑀. (2.2)We will use a comparison result of impulsive differential inequalities. Suppose that 𝑔∶ℝ₊×ℝ₊→ℝ satisfies the following hypotheses.

(H)𝑔 is continuous on ((𝑛−1)𝑇,(𝑛+𝜏−1)𝑇]×ℝ₊∪((𝑛+𝜏−1)𝑇,𝑛𝑇]×ℝ₊ and the limit lim_{(𝑡,𝑦)→(𝑡0,𝑥)}𝑔(𝑡,𝑦)=𝑔(𝑡₀,𝑥) exists, where 𝑡₀=(𝑛+𝜏−1)𝑇⁺ and 𝑛𝑇⁺, and is finite for 𝑥∈ℝ₊ and 𝑛∈ℕ.Lemma 2.3 (see [24]). Suppose𝑉∈𝑉₀ and𝐷⁺𝑉(𝑡,𝐱)≤𝑔(𝑡,𝑉(𝑡,𝐱)),𝑡≠(𝑛+𝜏−1)𝑇,𝑛𝑇,𝑉(𝑡,𝐱(𝑡⁺))≤𝜓¹_𝑛(𝑉(𝑡,𝐱)),𝑡=(𝑛+𝜏−1)𝑇,𝑉(𝑡,𝐱(𝑡⁺))≤𝜓²_𝑛(𝑉(𝑡,𝐱)),𝑡=𝑛𝑇, (2.3)where 𝑔∶ℝ₊×ℝ₊→ℝ satisfies (𝐻), and 𝜓¹_𝑛,𝜓²_𝑛∶ℝ₊→ℝ₊ are nondecreasing for all 𝑛∈ℕ. Let 𝑟(𝑡) be the maximal solution for the impulsive Cauchy problem𝑢′(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑡≠(𝑛+𝜏−1)𝑇,𝑛𝑇,𝑢(𝑡⁺)=𝜓¹_𝑛(𝑢(𝑡)),𝑡=(𝑛+𝜏−1)𝑇,𝑢(𝑡⁺)=𝜓²_𝑛(𝑢(𝑡)),𝑡=𝑛𝑇,𝑢(0⁺)=𝑢₀≥0, (2.4)defined on [0,∞). Then, 𝑉(0⁺,𝐱₀)≤𝑢₀ implies that 𝑉(𝑡,𝐱(𝑡))≤𝑟(𝑡),𝑡≥0, where 𝐱(𝑡) is any solution of (2.3).We now indicate a special case of Lemma 2.3 which provides estimations for the solution of a system of differential inequalities. For this, we let PC(ℝ₊,ℝ)(PC¹(ℝ₊,ℝ)) denote the class of real piecewise continuous (real piecewise continuously differentiable) functions defined on ℝ₊.Lemma 2.4 (see [24]). Let the function𝑢(𝑡)∈𝑃𝐶¹(ℝ⁺,ℝ) satisfy the inequalities𝑑𝑢𝑑𝑡≤𝑓(𝑡)𝑢(𝑡)+ℎ(𝑡),𝑡≠𝜏_𝑘,𝑡>0,𝑢(𝜏⁺_𝑘)≤𝛼_𝑘𝑢(𝜏_𝑘)+𝜃_𝑘,𝑘≥0,𝑢(0⁺)≤𝑢₀, (2.5)where 𝑓,ℎ∈𝑃𝐶(ℝ₊,ℝ) and 𝛼_𝑘≥0,𝜃_𝑘 and 𝑢₀ are constants and (𝜏_𝑘)_𝑘≥0 is a strictly increasing sequence of positive real numbers. Then, for 𝑡>0,𝑢(𝑡)≤𝑢₀(_{0<𝜏𝑘<𝑡}𝛼_𝑘)exp(^𝑡₀𝑓(𝑠)𝑑𝑠)+^𝑡₀(_{𝑠≤𝜏𝑘<𝑡}𝛼_𝑘)exp(^𝑡_𝑠𝑓(𝛾)𝑑𝛾)ℎ(𝑠)𝑑𝑠+_{0<𝜏𝑘<𝑡}(_{𝜏𝑘<𝜏𝑗<𝑡}𝛼_𝑗)exp(^𝑡_𝜏𝑘𝑓(𝛾)𝑑𝛾)𝜃_𝑘. (2.6) Similar result can be obtained when all conditions of the inequalities in the Lemmas 2.3 and 2.4 are reversed. Using Lemma 2.4, it is possible to prove that the solutions of the Cauchy problem (2.4) with strictly positive initial value remain strictly positive.Lemma 2.5. The positive octant(ℝ^∗₊)³ is an invariant region for system (1.2).Proof. Let (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡))∶[0,𝑡₀)→ℝ³ be a solution of system (1.2) with a strictly positive initial value (𝑥₀₁,𝑥₀₂,𝑦₀). By Lemma 2.4, we can obtain that, for 0≤𝑡<𝑡₀,𝑥₁(𝑡)≥𝑥₀₁(1−𝑝₁)^[𝑡/𝑇]exp(^𝑡₀𝑔₁𝑥(𝑠)𝑑𝑠),₂(𝑡)≥𝑥₀₂(1−𝑝₂)^[𝑡/𝑇]exp(^𝑡₀𝑔₂(𝑠)𝑑𝑠),𝑦(𝑡)≥𝑦₀(1−𝑝₃)^[𝑡/𝑇]exp(^𝑡₀𝑔₃(𝑠)𝑑𝑠), (2.7)where 𝑔₁(𝑠)=𝑎₁−𝜆₁−𝑏₁𝑥₁(𝑠)−𝛾₁𝑥₂(𝑠)−𝑒₁𝑦(𝑠), 𝑔₂(𝑠)=𝑎₂−𝜆₂−𝑏₂𝑥₂(𝑠)−𝛾₂𝑥₁(𝑠)−𝑒₂𝑦(𝑠), and 𝑔₃(𝑠)=−𝑎₃. Thus, 𝑥₁(𝑡),𝑥₂(𝑡) and 𝑦(𝑡) remain strictly positive on [0,𝑡₀).

Now, we give the basic properties of an impulsive differential equation as follows:𝑦′(𝑡)=−𝑎₃𝑦(𝑡),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑦(𝑡)=−𝑝₃𝑦(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑦(𝑡)=𝑞,𝑡=𝑛𝑇. (2.8)System (2.8) is a periodically forced linear system. It is easy to obtain that𝑦^∗(𝑡)=𝑞exp(−𝑎₃(𝑡−(𝑛−1)𝑇))1−(1−𝑝₃)exp(−𝑎₃𝑇),(𝑛−1)𝑇<𝑡≤(𝑛+𝜏−1)𝑇,𝑞(1−𝑝₃)exp(−𝑎₃(𝑡−(𝑛−1)𝑇))1−(1−𝑝₃)exp(−𝑎₃𝑇),(𝑛+𝜏−1)𝑇<𝑡≤𝑛𝑇, (2.9)𝑦^∗(0⁺)=𝑦^∗(𝑛𝑇⁺)=𝑞/(1−(1−𝑝₃)exp(−𝑎₃𝑇)), 𝑦^∗((𝑛+𝜏−1)𝑇⁺)=(𝑞(1−𝑝₃)exp(−𝑎₃𝜏𝑇))/(1−(1−𝑝₃)exp(−𝑎₃𝑇)) is a positive periodic solution of (2.8). Moreover, we can obtain that𝑦(𝑡)=(1−𝑝₃)^𝑛−1(𝑦(0⁺)−𝑞(1−𝑝₃)𝑒^−𝑇1−(1−𝑝₃)exp(−𝑎₃𝑇))exp(−𝑎₃𝑡)+𝑦^∗(𝑡),(𝑛−1)𝑇<𝑡≤(𝑛+𝜏−1)𝑇,(1−𝑝₃)^𝑛(𝑦(0⁺)−𝑞(1−𝑝₃)𝑒^−𝑇1−(1−𝑝₃)exp(−𝑎₃𝑇))exp(−𝑎₃𝑡)+𝑦^∗(𝑡),(𝑛+𝜏−1)𝑇<𝑡≤𝑛𝑇 (2.10)is a solution of (2.8). From (2.9) and (2.10), we get easily the following result.Lemma 2.6. lim_𝑡→∞|𝑦(𝑡)−𝑦^∗(𝑡)|=0 for all solutions 𝑦(𝑡) of (2.8) with 𝑦(0⁺)≥0.Therefore, system (1.2) has a two-prey-free periodic solution(0,0,𝑦^∗(𝑡)). (2.11)

3. Main Results

Theorem 3.1. The periodic solution(0,0,𝑦^∗(𝑡)) of system (1.2) is globally asymptotically stable if for 𝑖=1,2,(𝑎_𝑖+𝜆_𝑖𝑏)𝑇−^2𝑖𝑒_𝑖𝑞Ψ𝑏²_𝑖+𝑐_𝑖(𝑎_𝑖+𝜆_𝑖)²1<ln1−𝑝_𝑖, (3.1)where Ψ=(1−(1−𝑝₃)exp(−𝑎₃𝑇)−𝑝₃exp(−𝑎₃𝜏𝑇))/𝑎₃(1−(1−𝑝₃)exp(−𝑎₃𝑇)).Proof. First, we will prove the local stability of the periodic solution (0,0,𝑦^∗(𝑡)). For this, consider the following impulsive differential equation:𝑥^′₁₁(𝑡)=𝑥₁₁(𝑡)(𝑎₁+𝜆₁−𝑏₁𝑥₁₁(𝑡)−𝛾₁𝑥₁₂𝑒(𝑡)−¹𝑦₁(𝑡)1+𝑐₁𝑥²₁₁𝑥(𝑡)),_′12(𝑡)=𝑥₁₂(𝑡)(𝑎₂+𝜆₂−𝑏₂𝑥₁₂(𝑡)−𝛾₂𝑥₁₁𝑒(𝑡)−²𝑦₁(𝑡)1+𝑐₂𝑥²₁₂𝑦(𝑡)),_′1(𝑡)=𝑦₁(𝑡)(−𝑎₃+𝛽₁𝑥₁₁(𝑡)1+𝑐₁𝑥²₁₁+𝛽(𝑡)²𝑥₁₂(𝑡)1+𝑐₂𝑥²₁₂(𝑡)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₁₁(𝑡)=−𝑝₁𝑥₁₁(𝑡),Δ𝑥₁₂(𝑡)=−𝑝₂𝑥₁₂(𝑡),Δ𝑦₁(𝑡)=−𝑝₃𝑦₁(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₁₁(𝑡)=0,Δ𝑥₁₂(𝑡)=0,Δ𝑦₁(𝑡)=𝑞,𝑡=𝑛𝑇,(𝑥₁₁(0⁺),𝑥₁₂(0⁺),𝑦₁(0⁺))=(𝑥₀₁,𝑥₀₂,𝑦₀). (3.2)Then, 0≤𝑥₁(𝑡)≤𝑥₁₁(𝑡), 0≤𝑥₂(𝑡)≤𝑥₁₂(𝑡), and 0≤𝑦(𝑡)≤𝑦₁(𝑡) by Lemma 2.3, where (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) and (𝑥₁₁(𝑡),𝑥₁₂(𝑡),𝑦₁(𝑡)) are solutions of systems (1.2) and (3.2), respectively. Thus we will show the local stability of the solution (0,0,𝑦^∗₁(𝑡)) of system (3.2), where 𝑦^∗₁(𝑡)=𝑦^∗(𝑡). The local stability of the two-pest-free periodic solution (0,0,𝑦^∗₁(𝑡)) may be determined by considering the behavior of small amplitude perturbations of the solution. Let (𝑥₁₁(𝑡),𝑥₁₂(𝑡),𝑦₁(𝑡)) be any solution of system (3.2). Define 𝑢(𝑡)=𝑥₁₁(𝑡),𝑣(𝑡)=𝑥₁₂(𝑡),𝑤(𝑡)=𝑦(𝑡)−𝑦^∗₁(𝑡). Then, they may be written as𝑢(𝑡)𝑣(𝑡)𝑤(𝑡)=Φ(𝑡)𝑢(0)𝑣(0)𝑤(0), (3.3)where Φ(𝑡) satisfies𝑑Φ=𝑎𝑑𝑡¹+𝜆₁−𝑒₁𝑦^∗₁(𝑡)000𝑎₂+𝜆₂−𝑐₂𝑦^∗₁𝛽(𝑡)0₁𝑦^∗₁(𝑡)𝛽₂𝑦^∗₁(𝑡)−𝑎₃Φ(𝑡) (3.4)and Φ(0)=𝐼, the identity matrix. So the fundamental solution matrix isΦ(𝑡)=exp(^𝑡₀𝑎₁+𝜆₁−𝑒₁𝑦^∗₁(𝑠)𝑑𝑠)000exp(^𝑡₀𝑎₂+𝜆₂−𝑒₂𝑦^∗₁(𝑠)𝑑𝑠)0exp(^𝑡₀𝛽₁𝑦^∗₁(𝑠)𝑑𝑠)exp(^𝑡₀𝛽₂𝑦^∗₁(𝑠)𝑑𝑠)exp(^𝑡₀−𝑎₃𝑑𝑠). (3.5)The resetting impulsive conditions of system (3.2) become𝑢((𝑛+𝜏−1)𝑇⁺)𝑣((𝑛+𝜏−1)𝑇⁺)𝑢((𝑛+𝜏−1)𝑇⁺)=1−𝑝₁0001−𝑝₂0001−𝑝₃,𝑢((𝑛+𝜏−1)𝑇)𝑣((𝑛+𝜏−1)𝑇)𝑤((𝑛+𝜏−1)𝑇)𝑢(𝑛𝑇⁺)𝑣(𝑛𝑇⁺)𝑤(𝑛𝑇⁺)=.100010001𝑢(𝑛𝑇)𝑣(𝑛𝑇)𝑤(𝑛𝑇) (3.6)Note that all eigenvalues of𝑆=1−𝑝₁0001−𝑝₂0001−𝑝₃100010001Φ(𝑇) (3.7)are 𝜇₁=(1−𝑝₁)exp(∫^𝑇₀𝑎₁+𝜆₁−𝑒₁𝑦^∗₁(𝑡)𝑑𝑡), 𝜇₂=(1−𝑝₂)exp(∫^𝑇₀𝑎₂+𝜆₂−𝑒₂𝑦^∗₁(𝑡)𝑑𝑡), and 𝜇₃=(1−𝑝₃)exp(−𝑎₃𝑇)<1. Since(𝑎₁+𝜆₁)𝑇−𝑒₁𝑞Ψ<(𝑎₁+𝜆₁𝑏)𝑇−²¹𝑒₁𝑞Ψ𝑏²₁+𝑐₁(𝑎₁+𝜆₁)²,(𝑎₂+𝜆₂)𝑇−𝑒₂𝑞Ψ<(𝑎₂+𝜆₂𝑏)𝑇−²²𝑒₂𝑞Ψ𝑏²₂+𝑐₂(𝑎₂+𝜆₂)²,∫^𝑇₀𝑦^∗₁(𝑡)𝑑𝑡=𝑞(1−(1−𝑝₃)exp(−𝑎₃𝑇)−𝑝₃exp(−𝑎₃𝜏𝑇))𝑎₃(1−(1−𝑝₃)exp(−𝑎₃,𝑇)) (3.8)we obtain from (3.1) that the conditions |𝜇₁|<1 and |𝜇₂|<1 hold. Therefore, from the Floquet theory [23], we obtain (0,0,𝑦^∗(𝑡)) is locally stable.
Now, to prove the global stability of the two-prey-free periodic solution, let (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) be a solution of system (1.2). From (3.1), we can take a sufficiently small number 𝜖₁>0 satisfying𝜁=(1−𝑝₁)exp((𝑎₁+𝜆₁𝑏)𝑇+²¹𝑒₁(𝜖₁𝑇−𝑞Ψ)𝑏²₁+𝑐₁((𝑎₁+𝜆₁)²+2(𝑎₁+𝜆₁)𝑏₁𝜖₁+𝑏²₁𝜖²₁))<1. (3.9)It follows from the first equation in (1.2) that 𝑥^′₁(𝑡)≤𝑥₁(𝑡)(𝑎₁+𝜆₁−𝑏₁𝑥₁(𝑡)) for 𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇 and 𝑥₁(𝑡⁺)=(1−𝑝₁)𝑥₁(𝑡)≤𝑥₁(𝑡) for 𝑡=(𝑛+𝜏−1)𝑇. Then, from Lemma 2.3, we have 𝑥₁(𝑡)≤𝑢(𝑡), where 𝑢(𝑡) is a solution of the following impulsive differential equation:𝑢′(𝑡)=𝑢(𝑡)(𝑎₁+𝜆₁−𝑏₁𝑢(𝑡)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑢(𝑡)=0,𝑡=𝑛𝑇,𝑡=(𝑛+𝜏−1)𝑇,𝑢(0⁺)=𝑥₁(0⁺). (3.10)Since 𝑢(𝑡)→(𝑎₁+𝜆₁)/𝑏₁ as 𝑡→∞, 𝑥₁(𝑡)≤(𝑎₁+𝜆₁)𝑏₁+𝜖 for any 𝜖>0 with 𝑡 large enough. For simplicity, we may assume that 𝑥₁(𝑡)≤(𝑎₁+𝜆₁)/𝑏₁+𝜖₁ for all 𝑡>0. Similarly, we get 𝑥₂(𝑡)≤(𝑎₂+𝜆₂)/𝑏₂+𝜖₂ for any 𝜖₂>0 and 𝑡>0. Since 𝑦^′(𝑡)≥−𝑎₃𝑦(𝑡) for 𝑡≠𝑛𝑇,(𝑛+𝜏−1)𝑇, we can obtain from Lemmas 2.3 and 2.6 that𝑦(𝑡)>𝑦^∗(𝑡)−𝜖₁, (3.11)for 𝑡 sufficiently large. Without loss of generality, we may suppose that (3.11) holds for all 𝑡≥0. From (1.2), and (3.11) we obtain𝑥^′₁(𝑡)≤𝑥₁(𝑡)(𝑎₁+𝜆₁−𝑒₁(𝑦^∗(𝑡)−𝜖₁)1+𝑐₁(((𝑎₁+𝜆₁)/𝑏₁)+𝜖₁)²),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₁(𝑡)=−𝑝₁𝑥₁(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₁(𝑡)=0,𝑡=𝑛𝑇. (3.12)Integrating (3.12) on ((𝑛+𝜏−1)𝑇,(𝑛+𝜏)𝑇], we get𝑥₁((𝑛+𝜏)𝑇)≤𝑥₁((𝑛+𝜏−1)𝑇⁺)exp(^{(𝑛+𝜏)𝑇}_{(𝑛+𝜏−1)𝑇}𝑎₁+𝜆₁−𝑒₁(𝑦^∗(𝑡)−𝜖₁)1+𝑐₁(((𝑎₁+𝜆₁)/𝑏₁)+𝜖₁)²𝑑𝑡)=𝑥₁((𝑛+𝜏−1)𝑇)𝜁, (3.13)and thus 𝑥₁((𝑛+𝜏)𝑇)≤𝑥₁(𝜏𝑇)𝜁^𝑛 which implies that 𝑥₁((𝑛+𝜏)𝑇)→0 as 𝑛→∞. Further, we obtain, for 𝑡∈((𝑛+𝜏−1)𝑇,(𝑛+𝜏)𝑇],𝑥₁(𝑡)≤𝑥₁((𝑛+𝜏−1)𝑇+)exp(^𝑡_{(𝑛+𝜏−1)𝑇}𝑎₁+𝜆₁−𝑒₁(𝑦^∗(𝑡)−𝜖₁)1+𝑐₁(((𝑎₁+𝜆₁)/𝑏₁)+𝜖₁)²𝑑𝑡)≤𝑥₁((𝑛+𝜏−1)𝑇)exp((𝑎₁+𝜆₁+𝑒₁𝜖₁)𝑇), (3.14)which implies that 𝑥₁(𝑡)→0 as 𝑡→∞. Similarly, we obtain 𝑥₂(𝑡)→0 as 𝑡→∞. Now, take sufficiently small positive numbers 𝜖₃ and 𝜖₄ satisfying 𝛽₁𝜖₃+𝛽₂𝜖₄<𝑎₃ to prove that 𝑦(𝑡)→𝑦^∗(𝑡) as 𝑡→∞. Without loss of generality, we may assume that 𝑥₁(𝑡)≤𝜖₃ and 𝑥₂(𝑡)≤𝜖₄ for all 𝑡≥0. It follows from the third equation in (1.2) that, for 𝑡≠(𝑛+𝜏−1)𝑇 and 𝑡≠𝑛𝑇,𝑦′(𝑡)≤𝑦(𝑡)(−𝑎₃+𝛽₁𝜖₃+𝛽₂𝜖₄). (3.15)Thus, by Lemma 2.3, we induce that 𝑦(𝑡)≤𝑦^{0𝑥002𝑑𝑐∗}(𝑡), where ^0𝑥0𝑦^{02𝑑𝑐∗}(𝑡) is the solution of (2.8) with 𝑎₃ changed into 𝑎₃−𝛽₁𝜖₃−𝛽₂𝜖₄. Therefore, by taking sufficiently small 𝜖₃ and 𝜖₄, we obtain from Lemma 2.6 and (3.11) that 𝑦(𝑡) tends to 𝑦^∗(𝑡) as 𝑡→∞.Let 𝐹_𝑖(𝑇)=(𝑎_𝑖+𝜆_𝑖)𝑇−(𝑏²_𝑖𝑒_𝑖𝑞Ψ/(𝑏²_𝑖+𝑐_𝑖(𝑎_𝑖+𝜆_𝑖)²))+ln(1−𝑝_𝑖) for 𝑖=1,2. Then 𝐹_𝑖(0)=ln(1−𝑝_𝑖)<0, 𝐹_𝑖(𝑇)→∞, and 𝐹_𝑖^′′(𝑇)>0. Thus 𝐹_𝑖(𝑇) has a unique positive solution 𝑇_𝑖.Corollary 3.2. The periodic solution(0,0,𝑦^∗(𝑡)) of system (1.2) is globally asymptotically stable if 𝑇<min{𝑇₁,𝑇₂}.From the proof of Theorem 3.1, we can easily get the following corollary.Corollary 3.3. The periodic solution(0,0,𝑦^∗(𝑡)) of system (1.2) is locally stable if(𝑎₁+𝜆₁)𝑇−𝑒₁1𝑞Ψ<ln1−𝑝₁,(𝑎₂+𝜆₂)𝑇−𝑒₂1𝑞Ψ<ln1−𝑝₂. (3.16) Furthermore, the periodic solution of system (1.2) may remain globally stable even if there are no the seasonal effects on system (1.2).Corollary 3.4. Suppose that𝜆₁=𝜆₂=0. Then, the periodic solution (0,0,𝑦^∗(𝑡)) of system (1.2) is globally asymptotically stable if𝑎₁𝑏𝑇−²¹𝑒₁𝑞Ψ𝑏²₁+𝑐₁𝑎²₁1<ln1−𝑝₁,𝑎₂𝑏𝑇−²²𝑒₂𝑞Ψ𝑏²₂+𝑐₂𝑎²₂1<ln1−𝑝₂. (3.17)Now, we show that all solutions of system (1.2) are uniformly bounded.Theorem 3.5. There is an𝑀>0 such that 𝑥₁(𝑡)≤𝑀,𝑥₂(𝑡)≤𝑀, and 𝑦(𝑡)≤𝑀 for all 𝑡 large enough, where (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) is a solution of system (1.2).Proof. Let (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) be a solution of (1.2) with 𝑥₀₁,𝑥₀₂,𝑦₀≥0 and let 𝐹(𝑡)=(𝛽₁/𝑒₁)𝑥₁(𝑡)+(𝛽₂/𝑒₂)𝑥₂(𝑡)+𝑦(𝑡) for 𝑡>0. Then, if 𝑡≠𝑛𝑇 and 𝑡≠(𝑛+𝜏−1)𝑇, we obtain that (𝑑𝐹(𝑡)/𝑑𝑡)+𝛿𝐹(𝑡)=−(𝑏₁𝛽₁/𝑒₁)𝑥²₁(𝑡)+(𝛽₁/𝑒₁)(𝑎₁+𝜆₁sin(𝜔₁𝑡)+𝛿)𝑥₁(𝑡)−(𝛽₁𝛾₁/𝑒₁)𝑥₁(𝑡)𝑥₂(𝑡)−(𝑏₂𝛽₂/𝑒₂)𝑥²₂(𝑡)+(𝛽₂/𝑒₂)(𝑎₂+𝜆₂sin(𝜔₂𝑡)+𝛿)𝑥₂(𝑡)−(𝛽₂𝛾₂/𝑒₂)𝑥₁(𝑡)𝑥₂(𝑡)+(𝛿−𝑎₃)𝑦(𝑡). From choosing 0<𝛿₀<𝑎₃, we have, for 𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇 and 𝑡>0,𝑑𝐹(𝑡)𝑑𝑡+𝛿₀𝑏𝐹(𝑡)≤−¹𝛽₁𝑒₁𝑥²₁𝛽(𝑡)+¹𝑒₁(𝑎₁+𝜆₁+𝛿₀)𝑥₁𝑏(𝑡)−²𝛽₂𝑒₂𝑥²₂𝛽(𝑡)+²𝑒₂(𝑎₂+𝜆₂+𝛿₀)𝑥₂(𝑡). (3.18)As the right-hand side of (3.18) is bounded from above by 𝑀₀=(𝛽₁(𝑎₁+𝜆₁+𝛿₀)²/4𝑏₁𝑒₁)+(𝛽₂(𝑎₂+𝜆₂+𝛿₀)²/4𝑏₂𝑒₂), it follows that𝑑𝐹(𝑡)𝑑𝑡+𝛿₀𝐹(𝑡)≤𝑀₀,𝑡≠𝑛𝑇,𝑛≠(𝑛+𝜏−1)𝑇,𝑡>0. (3.19)If 𝑡=𝑛𝑇, then Δ𝐹(𝑡)=𝑞 and if 𝑡=(𝑛+𝜏−1)𝑇, then Δ𝐹(𝑡)≤−𝑝𝐹(𝑡), where 𝑝=min{𝑝₁,𝑝₂,𝑝₃}. From Lemma 2.4, we get that𝐹(𝑡)≤𝐹₀(1−𝑝)^{[𝑡/𝑘𝑇]}exp(^𝑡₀−𝛿₀+𝑑𝑠)_𝑡0(1−𝑝)^{[(𝑡−𝑠)/𝑘𝑇]}exp(^𝑡_𝑠−𝛿₀𝑑𝛾)𝑀₀+𝑑𝑠_{[𝑡/𝑘𝑇]𝑗=1}(1−𝑝)^{[(𝑡−𝑘𝑇)/𝑗𝑇]}exp(^𝑡_𝑘𝑇−𝛿₀𝑑𝛾)𝑞≤𝐹₀exp(−𝛿₀𝑀𝑡)+⁰𝛿₀(1−exp(−𝛿₀𝑡))+𝑞exp(𝛿₀𝑇)exp(𝛿₀,𝑇)−1 (3.20)where 𝐹₀=(𝛽₁/𝑒₁)𝑥₀₁+(𝛽₂/𝑒₂)𝑥₀₂+𝑦₀. Since the limit of the right-hand side of (3.20) as 𝑡→∞ is𝑀₀𝛿₀+𝑞exp(𝛿₀𝑇)exp(𝛿₀𝑇)−1<∞, (3.21)it easily follows that 𝐹(𝑡) is bounded for sufficiently large 𝑡. Therefore, 𝑥₁(𝑡),𝑥₂(𝑡) and 𝑦(𝑡) are bounded by a constant 𝑀 for sufficiently large 𝑡.Theorem 3.6. System (1.2) is permanent if 𝑎₃>max{((𝑎₁−𝜆₁)𝛽₁/𝑏₁),((𝑎₂−𝜆₂)𝛽₂/𝑏₂)},(𝑎₁−𝜆₁−𝛾₁(𝑎₂−𝜆₂)𝑏₂)𝑇−𝑒₁𝑞Θ₂1>ln1−𝑝₁,(𝑎₂−𝜆₂−𝛾₂(𝑎₁−𝜆₁)𝑏₁)𝑇−𝑒₂𝑞Θ₁1>ln1−𝑝₂, (3.22)where Θ_𝑖=1−(1−𝑝₃)exp((−𝑎₃+(𝛽_𝑖(𝑎_𝑖−𝜆_𝑖)/𝑏_𝑖))𝑇)−𝑝₃exp((−𝑎₃+(𝛽_𝑖(𝑎_𝑖−𝜆_𝑖)/𝑏_𝑖))𝜏𝑇)(𝑎₃−(𝛽_𝑖(𝑎_𝑖−𝜆_𝑖)/𝑏_𝑖))(1−(1−𝑝₃)exp((−𝑎₃+(𝛽_𝑖(𝑎_𝑖−𝜆_𝑖)/𝑏_𝑖,))𝑇))𝑖=1,2. (3.23)

Proof. Let (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) be a solution of system (1.2) with (𝑥₀₁,𝑥₀₂,𝑦₀)>0. From Theorem 3.5, we may assume that 𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)≤𝑀 and 𝑀>max{(𝑎₁/𝑒₁),(𝑎₂/𝑒₂)}. Thus, we only need to prove the existence of the lower bound 𝑚. For this, we consider the following impulsive differential equation:𝑥^′₂₁(𝑡)=𝑥₂₁(𝑡)(𝑎₁−𝜆₁−𝑏₁𝑥₂₁(𝑡)−𝛾₁𝑥₂₂𝑒(𝑡)−¹𝑦₂(𝑡)1+𝑐₁𝑥²₂₁𝑥(𝑡)),_′22(𝑡)=𝑥₂₂(𝑡)(𝑎₂−𝜆₂−𝑏₂𝑥₂₂(𝑡)−𝛾₂𝑥₂₁𝑒(𝑡)−²𝑦₂(𝑡)1+𝑐₂𝑥²₂₂𝑦(𝑡)),_′2(𝑡)=𝑦₂(𝑡)(−𝑎₃+𝛽₁𝑥₂₁(𝑡)1+𝑐₁𝑥²₁₁+𝛽(𝑡)²𝑥₂₂(𝑡)1+𝑐₂𝑥²₂₂(𝑡)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=−𝑝₁𝑥₂₁(𝑡),Δ𝑥₂₂(𝑡)=−𝑝₂𝑥₂₂(𝑡),Δ𝑦₂(𝑡)=−𝑝₃𝑦₂(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=0,Δ𝑥₂₂(𝑡)=0,Δ𝑦₂(𝑡)=𝑞,𝑡=𝑛𝑇,(𝑥₂₁(0⁺),𝑥₂₂(0⁺),𝑦₂(0⁺))=(𝑥₀₁,𝑥₀₂,𝑦₀). (3.24)Then, 𝑥₁(𝑡)≥𝑥₂₁(𝑡), 𝑥₂(𝑡)≥𝑥₂₂(𝑡) and 𝑦(𝑡)≥𝑦₂(𝑡) by Lemma 2.3, where (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)), and (𝑥₂₁(𝑡),𝑥₂₂,𝑦₂(𝑡)) are solutions of systems (1.2) and (3.24), respectively. So, we will show that 𝑥₂₁(𝑡)≥𝑚₁,𝑥₂₂(𝑡)≥𝑚₁, and 𝑦₂(𝑡)≥𝑚₁. As in the proof of Theorem 3.1, we can show that 𝑥₂₁(𝑡)≤((𝑎₁−𝜆₁)/𝑏₁)+𝜖₁ and 𝑥₂₂(𝑡)≤((𝑎₂−𝜆₂)/𝑏₂)+𝜖₂ for 𝑡>0. Let 𝑚=(𝑞(1−𝑝₃)exp(−𝑎₃𝑇)/1−(1−𝑝₃)exp(−𝑎₃𝑇))−𝜖 for 𝜖>0. Since 𝑦₂(𝑡)≥−𝑎₃𝑦(𝑡) for 𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇, it follows from Lemmas 2.3 and 2.6 that 𝑦₂(𝑡)>𝑦^∗(𝑡)−𝜖 and hence 𝑦₂(𝑡)>𝑚 for sufficiently large 𝑡. Thus we only need to find 𝑚₁ and 𝑚₂ such that 𝑥₂₁(𝑡)≥𝑚₁ and 𝑥₂₂(𝑡)≥𝑚₂ for 𝑡 large enough. We will do this in the following two steps.
Step 1. First, take sufficiently small positive numbers 𝑚₁ and 𝑚₂ such that 𝑚₁<(1/𝛽₁)(𝑎₃−𝛽₂(((𝑎₂−𝜆₂)/𝑏₂)+𝜖₂)), 𝑚₂<(1/𝛽₂)(𝑎₃−𝛽₁(((𝑎₁−𝜆₁)/𝑏₁)+𝜖₁)) and 𝛽₁𝑚₁+𝛽₂𝑚₂<𝑎₃. We will prove, there exist 𝑡₁,𝑡₂∈(0,∞) such that 𝑥₂₁(𝑡₁)≥𝑚₁ and 𝑥₂₂(𝑡₂)≥𝑚₂. Suppose not. Then that, we have only the following three cases:
(i)there exists a 𝑡₂>0 such that 𝑥₂₂(𝑡₂)≥𝑚₂, but 𝑥₂₁(𝑡)<𝑚₁, for all 𝑡>0;(ii)there exists a 𝑡₁>0 such that 𝑥₂₁(𝑡₁)≥𝑚₁, but 𝑥₂₂(𝑡)<𝑚₂, for all 𝑡>0;(iii)𝑥₂₁(𝑡)<𝑚₁ and 𝑥₂₂(𝑡)<𝑚₂ for all 𝑡>0.
Case (i): from (3.22) we can take 𝜂₁>0 small enough such that𝜙₁=(1−𝑝₁)exp((𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁(𝑎₂−𝜆₂𝑏₂+𝜖₂)−𝑒₁𝜂₁)𝑇−𝑒₁𝑞Θ₂)>1. (3.25) We obtain from the condition of case (i) that 𝑦′(𝑡)≤𝑦₂(𝑡)(−𝑎₃+𝛽₁𝑥₂₁(𝑡)+𝛽₂𝑥₂₂(𝑡))≤𝑦₂(𝑡)(−𝑎₃+𝛽₁𝑚₁+𝛽₂(((𝑎₂−𝜆₂)/𝑏₂)+𝜖₂))≡𝐸₁𝑦₂(𝑡) for 𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇, where 𝐸₁=−𝑎₃+𝛽₁𝑚₁+𝛽₂(((𝑎₂−𝜆₂)/𝑏₂)+𝜖₂)<0. Thus we have 𝑦₂(𝑡)≤𝑢(𝑡) and 𝑢(𝑡)→𝑢^∗(𝑡) as 𝑡→∞, where 𝑢(𝑡) is a solution of system𝑢′(𝑡)=𝐸₁𝑢(𝑡),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑢(𝑡)=−𝑝₃𝑢(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑢(𝑡)=𝑞,𝑡=𝑛𝑇,𝑢(0⁺)=𝑦(0⁺𝑢),_∗(𝑡)=𝑞exp(𝐸₁(𝑡−(𝑛−1)𝑇))1−(1−𝑝₃)exp(𝐸₁𝑇),(𝑛−1)𝑇<𝑡≤(𝑛+𝜏−1)𝑇,𝑞(1−𝑝₃)exp(𝐸₁(𝑡−(𝑛−1)𝑇))1−(1−𝑝₃)exp(𝐸₁𝑇),(𝑛+𝜏−1)𝑇<𝑡≤𝑛𝑇. (3.26)Therefore, we can take a 𝑇₁>0 such that 𝑦(𝑡)≤𝑢(𝑡)<𝑢^∗(𝑡)+𝜂₁ for 𝑡>𝑇₁. Thus we get𝑥^′₂₁(𝑡)≥𝑥₂₁(𝑡)(𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁(𝑎₂−𝜆₂𝑏₂+𝜖₂)−𝑒₁(𝑢^∗(𝑡)+𝜂₁)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=−𝑝₁𝑥₂₁(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=0,𝑡=𝑛𝑇, (3.27)for 𝑡>𝑇₁. Let 𝑁₁∈ℕ be such that (𝑁₁+𝜏−1)𝑇≥𝑇₁. Integrating (3.17) on ((𝑛+𝜏−1)𝑇,(𝑛+𝜏)𝑇],𝑛≥𝑁₁, we can obtain that 𝑥₂₁((𝑛+𝜏)𝑇)≥𝑥₂₁((𝑛+𝜏−1)𝑇)(1−𝑝₁)exp(∫^{(𝑛+𝜏)𝑇}_{(𝑛+𝜏−1)𝑇}𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁(((𝑎₂−𝜆₂)/𝑏₂)+𝜖₂)−𝑒₁(𝑣^∗(𝑡)+𝜂₁)𝑑𝑡=𝑥₂₁((𝑛+𝜏−1)𝑇)𝜙₁. Thus 𝑥₂₁((𝑁₁+𝑘+𝜏)𝑇)≥𝑥₂₁((𝑁₁+𝜏)𝑇)𝜙^𝑘₁→∞ as 𝑘→∞, which is a contradiction to the boundedness of 𝑥₂₁(𝑡).
Case (ii): the same argument as the case (i) can be applied. So we omit it.
Case (iii): we choose 𝜂₂>0 sufficiently small so that𝜙₂=(1−𝑝₁)exp((𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁𝑚₂)𝑇−𝑒₁(𝑞Ψ+𝜂₂𝑇))>1. (3.28) Then we obtain 𝑦^′₂(𝑡)≤𝑦₂(𝑡)(−𝑎₃+𝛽₁𝑚₁+𝛽₂𝑚₂)≡𝐸₂𝑦₂(𝑡) for 𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇, where 𝐸₂=−𝑎₃+𝛽₁𝑚₁+𝛽₂𝑚₂<0. It follows from Lemmas 2.3 and 2.6 that 𝑦₂(𝑡)≤𝑤(𝑡) and 𝑤(𝑡)→𝑤^∗(𝑡) as 𝑡→∞, where 𝑤(𝑡) is a solution of the following system:𝑤′(𝑡)=𝐸₂𝑤(𝑡),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑤(𝑡)=−𝑝₃𝑤(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑤(𝑡)=𝑞,𝑡=𝑛𝑇,𝑤(0⁺)=𝑦(0⁺𝑤),_∗(𝑡)=𝑞exp(𝐸₂(𝑡−(𝑛−1)𝑇))1−(1−𝑝₃)exp(𝐸₂𝑇),(𝑛−1)𝑇<𝑡≤(𝑛+𝜏−1)𝑇,𝑞(1−𝑝₃)exp(𝐸₂(𝑡−(𝑛−1)𝑇))1−(1−𝑝₃)exp(𝐸₂𝑇),(𝑛+𝜏−1)𝑇<𝑡≤𝑛𝑇. (3.29)Thus there exists a 𝑇₂>0 such that 𝑦₂(𝑡)≤𝑤(𝑡)<𝑤^∗(𝑡)+𝜂₂ for 𝑡>𝑇₂ and𝑥^′₂₁(𝑡)≥𝑥₂₁(𝑡)(𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁𝑚₂−𝑒₁(𝑤^∗(𝑡)+𝜂₂)),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=−𝑝₁𝑥₂₁(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=0,𝑡=𝑛𝑇, (3.30)for 𝑡>𝑇₂. Let 𝑁₂∈ℕ be such that (𝑁₂+𝜏−1)𝑇≥𝑇₂. Integrating (3.30) on ((𝑛+𝜏−1)𝑇,(𝑛+𝜏)𝑇],𝑛≥𝑁₂, we can obtain that 𝑥₂₁((𝑛+𝜏)𝑇)≥𝑥₂₁((𝑛+𝜏−1)𝑇)(1−𝑝₁)exp(∫^(𝑛+𝜏)_{(𝑛+𝜏−1)𝑇}𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁𝑚₂−𝑒₁(𝑤^∗(𝑡)+𝜂₂)𝑑𝑡)=𝑥₂₁((𝑛+𝜏−1)𝑇)𝜙₂. Similarly, we have 𝑥₂₁((𝑁₂+𝑘+𝜏)𝑇)≥𝑥₂₁((𝑁₂+𝜏)𝑇)𝜙^𝑘₂→∞ as 𝑘→∞, which is a contradiction to the boundedness of 𝑥₂₁(𝑡). Therefore, there exist 𝑡₁>0 and 𝑡₂>0 such that 𝑥₂₁(𝑡₁)≥𝑚₁ and 𝑥₂₂(𝑡₂)≥𝑚₂.
Step 2. If 𝑥₂₁(𝑡)≥𝑚₁ for all 𝑡≥𝑡₁, then we are done. If not, we may let 𝑡^∗=inf_𝑡>𝑡1{𝑥₂₁(𝑡)<𝑚₁}. Then, 𝑥₂₁(𝑡)≥𝑚₁ for 𝑡∈[𝑡₁,𝑡^∗] and, by the continuity of 𝑥₂₁(𝑡), we have 𝑥₂₁(𝑡^∗)=𝑚₁. In this step, we have only to consider two possible cases.
(i) Suppose that 𝑡^∗=(𝑛₁+𝜏−1)𝑇 for some 𝑛₁∈ℕ. Then, (1−𝑝₁)𝑚₁≤𝑥₂₁(𝑡^∗+)=(1−𝑝₁)𝑥₂₁(𝑡^∗)<𝑚₁. Select 𝑛₂,𝑛₃∈ℕ such that (𝑛₂−1)𝑇>ln(𝜂₁/(𝑀+𝑞))/𝐸₁ and (1−𝑝₁)^𝑛2𝜙^𝑛3₁exp(𝑛₂𝜎𝑇)>(1−𝑝₁)^𝑛2𝜙^𝑛3₁exp((𝑛₂+1)𝜎𝑇)>1, where 𝜎=𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁(((𝑎₂−𝜆₂)/𝑏₂)+𝜖₂)−𝑒₁𝑀<0. Let 𝑇^′=𝑛₂𝑇+𝑛₃𝑇. In this case, we will show that there exists 𝑡₃∈(𝑡^∗,𝑡^∗+𝑇^′] such that 𝑥₂₁(𝑡₃)≥𝑚₁. Otherwise, by (2.10) and (3.26) with 𝑣(𝑛₁𝑇⁺)=𝑦(𝑛₁𝑇⁺), we have𝑣(𝑡)=(1−𝑝₃)^{𝑛−(𝑛1+1)}(𝑣(𝑛₁𝑇⁺)−𝑞(1−𝑝₃)exp(−𝑇)1−(1−𝑝₃)exp(𝐸₁)𝑇)exp(𝐸₁(𝑡−𝑛₁𝑇))+𝑣^∗(𝑡),(𝑛−1)𝑇<𝑡≤(𝑛+𝜏−1)𝑇,(1−𝑝₃)^{(𝑛−𝑛1)}(𝑣(𝑛₁𝑇⁺)−𝑞(1−𝑝₃)exp(−𝑇)1−(1−𝑝₃)exp(𝐸₁)𝑇)exp(𝐸₁(𝑡−𝑛₁𝑇))+𝑣^∗(𝑡),(𝑛+𝜏−1)𝑇<𝑡≤𝑛𝑇, (3.31)and 𝑛₁+1≤𝑛≤𝑛₁+1+𝑛₂+𝑛₃. So we get |𝑣(𝑡)−𝑣^∗(𝑡)|≤(𝑀+𝑞)exp(𝐸₁(𝑡−𝑛₁𝑇))<𝜂₁ and 𝑦₂(𝑡)≤𝑣(𝑡)≤𝑣^∗(𝑡)+𝜂₁ for 𝑛₁𝑇+(𝑛₂−1)𝑇≤𝑡≤𝑡^∗+𝑇^′, which implies that (3.27) holds for 𝑡∈[𝑡^∗+𝑛₂𝑇,𝑡^∗+𝑇^′]. As in step 1, we have𝑥₂₁(𝑡^∗+𝑇′)≥𝑥₂₁(𝑡^∗+𝑛₂𝑇)𝜙^𝑛3₁. (3.32)Since 𝑦₂(𝑡)≤𝑀, we have𝑥^′₂₁(𝑡)≥𝑥₂₁(𝑡)(𝑎₁−𝜆₁−𝑏₁𝑚₁−𝛾₁(𝑎₂−𝜆₂𝑏₂+𝜖₂)−𝑒₁𝑀)=𝜎𝑥₂₁(𝑡),𝑡≠𝑛𝑇,𝑡≠(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=−𝑝₁𝑥₂₁(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑥₂₁(𝑡)=0,𝑡=𝑛𝑇, (3.33)for 𝑡∈[𝑡^∗,𝑡^∗+𝑛₂𝑇]. Integrating (3.33) on [𝑡^∗,𝑡^∗+𝑛₂𝑇], we have𝑥₂₁((𝑡^∗+𝑛₂𝑇))≥𝑚₁exp(𝜎𝑛₂𝑇)>𝑚₁. (3.34)Thus 𝑥₂₁(𝑡^∗+𝑇^′)≥𝑚₁(1−𝑝₁)^𝑛2exp(𝜎𝑛₂𝑇)𝜙^𝑛3₁>𝑚₁ which is a contradiction. Now, let 𝑡 = inf_𝑡>𝑡∗{𝑥₂₁(𝑡)≥𝑚₁}. Then, 𝑥₂₁(𝑡)≤𝑚₁ for 𝑡^∗≤𝑡<𝑡 and 𝑥₂₁(𝑡)=𝑚₁. So, we have, for 𝑡∈[𝑡^∗,𝑡), 𝑥₂₁(𝑡)≥𝑚₁(1−𝑝₁)^𝑛2+𝑛3exp(𝜎(𝑛₂+𝑛₃)𝑇)≡𝑚₁. For 𝑡>𝑡^∗ the same argument can be continued since 𝑥₂₁(𝑡)≥𝑚₁. Hence 𝑥₂₁(𝑡)≥𝑚₁ for all 𝑡>𝑡₁.
(ii) 𝑡^∗≠(𝑛+𝜏−1)𝑇,𝑛∈ℕ. Suppose that 𝑡^∗∈((𝑛^′₁+𝜏−1)𝑇,(𝑛^′₁+𝜏)𝑇) for some 𝑛^′₁∈ℕ. There are two possible cases for 𝑡∈(𝑡^∗,(𝑛^′₁+𝜏)𝑇). Firstly, if 𝑥₂₁(𝑡)≤𝑚₁ for all 𝑡∈(𝑡^∗,(𝑛^′₁+𝜏)𝑇), similar to case (i), we can prove there must be a 𝑡^′₃∈[(𝑛^′₁+𝜏)𝑇,(𝑛^′₁+𝜏)𝑇+𝑇′] such that 𝑥₂₁(𝑡^′₃)≥𝑚₁. Here we omit it. Let 𝑡̂=inf_𝑡>𝑡∗{𝑥₂₁(𝑡)≥𝑚₁}. Then, 𝑥₂₁(𝑡)≤𝑚₁ for 𝑡∈(𝑡^∗,𝑡)̂ and 𝑥₂₁(𝑡)̂=𝑚₁. For 𝑡∈(𝑡^∗,𝑡)̂, we have 𝑥₂₁(𝑡)≥𝑚₁(1−𝑝₁)^𝑛2+𝑛3exp(𝜎(𝑛₂+𝑛₃+1)𝑇)=𝑚₁. So, 𝑚₁<𝑚₁ and 𝑥₂₁(𝑡)≥𝑚₁ for 𝑡∈(𝑡^∗,𝑡)̂. For 𝑡>𝑡^∗ the same argument can be continued since 𝑥₂₁(𝑡)̂≥𝑚₁. Hence 𝑥₂₁(𝑡)≥𝑚₁ for all 𝑡>𝑡₁. Secondly, if there exists a 𝑡∈(𝑡^∗,(𝑛^′₁+𝜏)𝑇) such that 𝑥₂₁(𝑡)≥𝑚₁. Let 𝑡̆=inf_𝑡>𝑡∗{𝑥₂₁(𝑡)≥𝑚₁}. Then 𝑥₂₁(𝑡)≤𝑚₁ for 𝑡∈(𝑡^∗,𝑡)̆ and 𝑥₂₁(𝑡)̆=𝑚₁. For 𝑡∈(𝑡^∗,𝑡)̆, we have 𝑥₂₁(𝑡)≥𝑥₂₁(𝑡^∗)exp(𝜎(𝑡−𝑡^∗))≥𝑚₁exp(𝜎𝑇)>𝑚₁. This process can be continued since 𝑥₂₁(𝑡)̆≥𝑚₁, and have 𝑥₂₁(𝑡)≥𝑚₁ for all 𝑡>𝑡₁. Similarly, we can show that 𝑥₂₂(𝑡)≥𝑚₂ for all 𝑡>𝑡₂. This completes the proof.

Corollary 3.7. Suppose that𝜆₁=𝜆₂=0. Then, system (1.2) is permanent if 𝑎₃>max{(𝑎₁𝛽₁/𝑏₁),(𝑎₂𝛽₂/𝑏₂)}, (𝑎₁−(𝛾₁𝑎₂/𝑏₂))𝑇−𝑒₁𝑞Θ₂>ln1/(1−𝑝₁),and(𝑎₂−(𝛾₂𝑎₁/𝑏₁))𝑇−𝑒₂𝑞Θ₁>ln1/(1−𝑝₂).

It follows from Theorems 3.1 and 3.6 that one of the two prey extinct and the remaining two species are permanent under some conditions. Corollary 3.8. Let(𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) be any solution of system (1.2). Then, 𝑥₁ and 𝑦(𝑡) are permanent, and 𝑥₂(𝑡)→0 as 𝑡→∞ provided that 𝑎₃>(𝑎₂−𝜆₂)𝛽₂/𝑏₂, (𝑎₁−𝜆₁−(𝛾₁(𝑎₂−𝜆₂)/𝑏₂))𝑇−𝑒₁𝑞Θ₂>ln(1/(1−𝑝₁)), and (𝑎₂+𝜆₂)𝑇−(𝑏²₂𝑒₂𝑞Ψ/(𝑏²₂+𝑐₂(𝑎₂+𝜆₂)²))<ln(1/(1−𝑝₂)).Corollary 3.9. Let(𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) be any solution of system (1.2). Then, 𝑥₂ and 𝑦(𝑡) are permanent, and 𝑥₁(𝑡)→0 as 𝑡→∞ provided that 𝑎₃>(𝑎₁−𝜆₁)𝛽₁/𝑏₁, (𝑎₁+𝜆₁)𝑇−(𝑏²₁𝑒₁𝑞Ψ/(𝑏²₁+𝑐₁(𝑎₁+𝜆₁)²))<ln1/(1−𝑝₁) and (𝑎₂−𝜆₂−(𝛾₂(𝑎₁−𝜆₁)/𝑏₁))𝑇−𝑒₂𝑞Θ₁>ln1/(1−𝑝₂).

4. Numerical Examples

In this section, we will give some examples.

Case 1. If we take 𝑎₁=1.1,𝑎₂=1.0,𝑎₃=0.7,𝑏₁=1.0,𝑏₂=1.2,𝑐₁=0.9,𝑐₂=1.0,𝑒₁=0.3,𝑒₂=0.2,𝛾₁=0.1,𝛾₂=0.1,𝛽₁=0.8,𝛽₂=1.0,𝑝₁=0.7,𝑝₂=0.8,𝑝₃=0.001,𝜏=0.6,𝑇=1,𝑞=5,𝜆₁=0.5, and 𝜆₂=0.6, then these parameters satisfy the condition of Theorem 3.1. Thus the periodic solution (0,0,𝑦^∗(𝑡)) is globally asymptotically stable. (See Figure 1.)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

(a)–(c) Time series of system (1.2) with an initial value (2,3,1), (d)–(f) time series of system (1.2) with an initial value (100,100,100).

Case 2. Let 𝑎₁=5,𝑎₂=4,𝑎₃=0.7,𝑏₁=1,𝑏₂=1.2,𝑐₁=0.9,𝑐₂=1,𝑒₁=0.3,𝑒₂=0.2,𝛾₁=0.1,𝛾₂=0.1,𝛽₁=0.1,𝛽₂=0.2,𝑝₁=0.3,𝑝₂=0.4,𝑝₃=0.001,𝜏=0.6,𝑇=4,𝑞=2,𝜆₁=4.0 and 𝜆₂=3. Then, from Theorem 3.6, we know that system (1.2) is permanent, (see Figure 2).

(a)

(b)

(c)

(d)

Figure 2 

(a)–(c) Time series. (d) The trajectory of system (1.2) with an initial value (2,3,1).

Case 3. From Corollary 3.8, we obtain that 𝑥₁ and 𝑦(𝑡) are permanent, and 𝑥₂(𝑡)→0 as 𝑡→∞, where (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) is a solution of system (1.2) with 𝑎₁=5,𝑎₂=1.3,𝑎₃=0.7,𝑏₁=1,𝑏₂=0.8,𝑐₁=0.9,𝑐₂=1,𝑒₁=0.3,𝑒₂=1,𝛾₁=0.1,𝛾₂=0.1,𝛽1=0.8,𝛽₂=0.9,𝑝₁=0.5,𝑝₂=0.8,𝑝₃=0.001,𝜏=0.6,𝜆₁=1,𝜆2=0.5,𝑇=1, and 𝑞=2. Figure 3 exhibits this phenomenon.

(a)

(b)

(c)

(d)

Figure 3 

(a)–(c) Time series. (d) The trajectory of system (1.2) with an initial value (2,3,1).

Case 4. It follows from Corollary 3.9 that 𝑥₂ and 𝑦(𝑡) are permanent, and 𝑥₁(𝑡)→0 as 𝑡→∞, where (𝑥₁(𝑡),𝑥₂(𝑡),𝑦(𝑡)) is a solution of system (1.2) with 𝑎₁=1,𝑎₂=5,𝑎₃=0.8,𝑏₁=0.8,𝑏₂=0.8,𝑐₁=0.8,𝑐₂=1,𝑒₁=1,𝑒₂=0.2,𝛾₁=0.1,𝛾₂=0.1,𝛽₁=0.9,𝛽₂=0.8,𝑝₁=0.8,𝑝₂=0.5,𝑝₃=0.001,𝜏=0.6,𝑇=1,𝑞=3,𝜆₁=0.5 and 𝜆₂=1. (see Figure 4.)

(a)

(b)

(c)

(d)

Figure 4 

(a)–(c) Time series. (d) The trajectory of system (1.2) with an initial value (2,3,1).

5. Conclusion

In this paper, based on a Holling-type IV two-prey one-predator system, we established an impulsive differential equation to model the seasonal effects, the process of a proportional periodic impulsive harvesting, and a constant periodic releasing of the predator at different fixed time. Using the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we proved that there exists a globally (locally) stable two-prey-free periodic solution if the conditions of Theorem 3.1 (Corollary 3.3) are satisfied. Also, we established permanence conditions of system (1.2) via the method of comparison including multiple Liapunov functions when the conditions of Theorem 3.6 are satisfied. From main Theorems, we easily obtained the sufficient conditions under which one of the two preys is extinct and whic of the remaining two species are permanent. In addition, we gave some examples by using numerical simulations.