Dynamic Analysis of an Impulsive Predator-Prey Model with Disease in Prey and Ivlev-Type Functional Response

Abstract

A predator-prey model with disease in prey, Ivlev-type functional response, and impulsive effects is proposed. By using Floquet theory and small amplitude perturbation skill, sufficient conditions of the existence and global stability of susceptible pest-eradication periodic solution are obtained. By impulsive comparison theorem, conditions ensuring the permanence of the system are established. Examples and simulation are given to show the complex dynamics for the key parameters.

1. Introduction

It is well known that many evolution processes are characterized by the fact that at certain moments their stage changes abruptly. For example, for integrated pest management (IPM) strategy on ecosystem, the predators are released periodically every time 𝑇, and periodic catching or spraying pesticides are also applied. Hence, the predator and prey experience a change of state abruptly. It is natural to assume that these processes act in the form of impulse. The effects of impulsion on the dynamics of predator-prey system have been investigated extensively, see [1–15].

On the other hand, in population dynamics, 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. Usually, functional response plays key role in the dynamics of predator-prey system [16–18]. Recently, many different functional responses are studied such as Holling-type [8, 11], Beddington-type [7, 12], and Watt-type [13, 14]. Ivlev-type functional response is the most common type of functional response among arthropod predators, and much progress has been seen in the study of predator-prey model with Ivlev-type functional response [19–21].

Considering the influence of periodic pesticide spraying on all species and periodically releasing predator at fixed different time, authors [1] proposed and studied the following predator-prey system with impulsive perturbation and Ivlev-type functional response: (𝑡)(𝑡)𝑥(𝑡)𝑥′=𝑎𝑥1−𝑏−ℎ𝑦(𝑡)1−𝑒^{−𝑐𝑥(𝑡)},𝑦′(𝑡)=𝑒𝑦(𝑡)1−𝑒^{−𝑐𝑥(𝑡)}(𝑡),()𝑥𝑡−𝑑𝑦𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₁𝑥(𝑡),𝑦𝑡⁺=1−𝑝₂𝑦(𝑡)()𝑥𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝑥(𝑡),𝑦𝑡⁺=𝑦(𝑡)+𝑞,𝑡=𝑛𝑇. (1.1) They obtained the local stability of prey-free periodic solution and permanence of the system.

However, for biological control, in addition to the approach to release natural enemies, another approach is to use microbial control with pathogens since diseases can be important natural controls of some pests. For example, insects can be infected by disease-causing organism such as bacteria and viruses. Under appropriate condition, these naturally occurring organisms may multiply to cause disease outbreak that can decimate an insect population. There is a large amount of literatures on applications of entomopathogens to suppress pests [22–26]. Then how does the disease in prey affect the dynamics of above system? Further, whether we can derive the global stability of the susceptible pest-eradication periodic solution?

Motivated by above discussion, in this paper, we are concerned with the following predator-prey model with complex influence of disease in prey, Ivlev-type functional response, and impulsive perturbation as follows: (𝑡)(𝑡)𝑆(𝑡)(𝑡)𝑆′=𝑟𝑆1−+𝐼𝐾−𝛼𝑆(𝑡)𝐼(𝑡)−𝑎𝑦(𝑡)1−𝑒^{−𝑐𝑆(𝑡)},𝐼^′(𝑡)=𝛼𝑆(𝑡)𝐼(𝑡)−𝑑₁𝐼(𝑡),𝑦′(𝑡)=𝜎𝑎𝑦(𝑡)1−𝑒^{−𝑐𝑆(𝑡)}−𝑑₂𝑦(𝑡),()𝑆𝑡𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₁𝑆(𝑡),𝐼𝑡⁺=1−𝑝₂𝐼(𝑡),𝑦𝑡⁺=1−𝑝₃𝑦(𝑡),()𝑆𝑡𝑡=𝑛+𝜄−1𝑇,₊(𝑡),𝐼𝑡=𝑆₊=𝐼(𝑡)+𝑞₁,𝑦𝑡⁺=𝑦(𝑡)+𝑞₂,𝑡=𝑛𝑇, (1.2) where 𝑆(𝑡) and 𝐼(𝑡) represent densities of susceptible prey (pest) population and infective prey (pest) population, respectively; 𝑦(𝑡) is the density of predator (natural enemy), and 𝑆(𝑡⁺)=lim_𝑣→𝑡+𝑆(𝑣), 𝐼(𝑡⁺)=lim_𝑣→𝑡+𝐼(𝑣), 𝑦(𝑡⁺)=lim_𝑣→𝑡+𝑦(𝑣) exist.

For (1.2), we give the following biological assumptions.(i)The growth rate for prey is 𝑟𝑆(𝑡)(1−(𝑆(𝑡)+𝐼(𝑡))/𝐾),𝑟>0 is intrinsic growth rate, and 𝐾>0 is the carrying capacity.(ii)There are diseases among prey population, and the prey population is divided into susceptible class and infective class. The incidence rate is classic bilinear as 𝛼𝑆(𝑡)𝐼(𝑡), and 𝛼 is the contact number per unit time for every infective prey with susceptible prey such that 𝑟/𝐾+𝛼≤𝑟.(iii)The predator only catches susceptible prey, and the predation functional response is Ivlev type. Parameters 𝑎, 𝑐 are positive constants, and 𝜎 is conversion rate from prey to predator.(iv)Parameters 𝑑₁, 𝑑₂ are death rates for infective prey and predator, respectively.(v)0≤𝑝_𝑖<1(𝑖=1,2,3) represents the fraction of prey and predator which due to the pesticide at 𝑡=(𝑛+𝜄−1)𝑇, 0<𝜄<1,𝑞₁,𝑞₂ are positive constants representing the release amount of infective prey and predator periodically at time 𝑡=𝑛𝑇 respectively, where 𝑛∈𝑍₊ (positive integer set), and 𝑇 is the period of impulsive effect.

By using Floquet theory, comparison method of impulsive differential equation, and numerical analysis skill, we aim to study the dynamics of (1.2) with disease in prey, Ivlev-type functional response, and impulsive effects.

The rest of the paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, by using the Floquet theory and small amplitude perturbation skill and comparison theorem of impulsive differential equation, the existence of susceptible pest-eradication periodic solution and permanence of system (1.2) are studied. In Section 4, some examples and numerical analysis are given to show the rich dynamics of (1.2). Finally, biological implications and a brief discussion are given in Section 5 to conclude this paper.

2. Preliminaries

Let 𝑅₊=[0,∞) and 𝑅³₊={𝑧=(𝑧₁,𝑧₂,𝑧₃)^𝑇∈𝑅³∶𝑧_𝑖∈𝑅₊,𝑖=1,2,3}. Denote by 𝑓=(𝑓₁,𝑓₂,𝑓₃)^𝑇 the map defined by right-hand sides of the first three equations of (1.2). Let 𝑉∶𝑅₊×𝑅³₊→𝑅₊, and then 𝑉 is said to belong to class 𝑉₀ if(i)𝑉 is continuous in ((𝑛−1)𝑇,(𝑛+𝜄−1)𝑇]×𝑅³₊, and ((𝑛+𝜄−1)𝑇,𝑛𝑇]×𝑅³₊, lim_{(𝑡,𝑠)→((𝑛+𝜄−1)𝑇+,𝑧)}𝑉(𝑡,𝑠)=𝑉((𝑛+𝜄−1)𝑇⁺,𝑧), and lim_{(𝑡,𝑠)→(𝑛𝑇+,𝑧)}𝑉(𝑡,𝑠)=𝑉(𝑛𝑇⁺,𝑧) exist, and(ii)𝑉 is locally Lipschitzian in 𝑧.

Definition 2.1. Let 𝑉∈𝑉₀, then for (𝑡,𝑧)∈((𝑛−1)𝑇,(𝑛+𝜄−1)𝑇]×𝑅³₊ and (𝑡,𝑧)∈((𝑛+𝜄−1)𝑇,𝑛𝑇]×𝑅³₊, the upper right derivative of 𝑉(𝑡,𝑧) with respect to system (1.2) is defined as 𝐷⁺𝑉(𝑡,𝑧)=limsup_ℎ→0+1ℎ𝑉(𝑡+ℎ),𝑧+ℎ𝑓(𝑡,𝑧)−𝑉(𝑡,𝑧). (2.1)

Definition 2.2. System (1.2) is said to be permanent if there exist positive constants 𝑚 and 𝑀 with 𝑀≥𝑚>0 such that each positive solution (𝑆(𝑡),𝐼(𝑡),𝑦(𝑡)) satisfying 𝑚≤𝑆(𝑡), 𝐼(𝑡),?and 𝑦(𝑡)≤𝑀 for all 𝑡 is sufficiently large.
The solution of system (1.2) is continuously differentiable on ((𝑛−1)𝑇,𝑛+𝜄−1)𝑇] and ((𝑛+𝜄−1)𝑇,𝑛𝑇], 𝑛∈𝑍₊. Obviously, the global existence and uniqueness of solutions to system (1.2) are guaranteed by the smoothness properties of function 𝑓. For more details see [27].

Lemma 2.3. Suppose that (𝑆(𝑡),𝐼(𝑡),𝑦(𝑡)) is a solution of system (1.2) with 𝑆(0⁺)≥0,𝐼(0⁺)≥0, 𝑦(0⁺)≥0. Then 𝑆(𝑡)≥0, 𝐼(𝑡)≥0, 𝑦(𝑡)≥0 for all 𝑡≥0. Furthermore, 𝑆(𝑡)>0,?𝐼(𝑡)>0,𝑦(𝑡)>0 if 𝑆(0⁺)>0, 𝐼(0⁺)>0, 𝑦(0⁺)>0.

Lemma 2.4 (see [28]). Let 𝑉∈𝑉₀. Assume that 𝐷⁺𝑉()((()𝑉𝑡𝑡,𝑧≤𝑔𝑡,𝑉𝑡,𝑧)),𝑡≠𝑛𝑇,𝑡≠𝑛+𝜄−1𝑇,𝑡,𝑧⁺≤𝜓_𝑛(𝑉(𝑉𝑡𝑡,𝑧)),𝑡=𝑛𝑇,𝑡,𝑧⁺≤𝜙_𝑛+𝜄(𝑉(𝑡,𝑧)),𝑡=(𝑛+𝜄−1)𝑇, (2.2) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous in ((𝑛−1)𝑇,(𝑛+𝜄−1)𝑇]×𝑅₊, and ((𝑛+𝜄−1)𝑇,𝑛𝑇]×𝑅₊, lim_{(𝑡,𝑠)→((𝑛+𝜄−1)𝑇+,𝑧)}𝑔(𝑡,𝑠)=𝑔((𝑛+𝜄−1)𝑇⁺,𝑧), and lim_{(𝑡,𝑠)→(𝑛𝑇+,𝑧)}𝑔(𝑡,𝑠)=𝑔(𝑛𝑇⁺,𝑧) exist; 𝜓_𝑛∶𝑅₊→𝑅₊ and 𝜙_𝑛+𝜄∶𝑅₊→𝑅₊ are nondecreasing. Let 𝑟(𝑡) be the maximal solution of the scalar impulsive differential equation 𝑑𝑢(𝑡)((𝑡()𝑢𝑡𝑑𝑡=𝑔𝑡,𝑢)),𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=𝜓_𝑛(𝑢(𝑡𝑢𝑡)),𝑡=𝑛𝑇,₊=𝜙_𝑛+𝜄(𝑢(𝑡()𝑢0)),𝑡=𝑛+𝜄−1𝑇,₊=𝑢₀, (2.3) existing on [0,∞). Then 𝑉(0⁺,𝑧(0⁺))≤𝑢₀ implies that 𝑉(𝑡,𝑧(𝑡))≤𝑟(𝑡) for 𝑡≥0, where 𝑧(𝑡)=(𝑆(𝑡),𝐼(𝑡),𝑦(𝑡)) is any solution of system (1.2). Assume that all the inequalities “≤” in system (2.2) are replaced by “≥” in the preceding equations, and let 𝜌(𝑡) be the minimal solution of (2.3) existing on [0,∞). Then 𝑉(0⁺,𝑧(0⁺))≥𝑢₀ implies that 𝑉(𝑡,𝑧(𝑡))≥𝜌(𝑡) for 𝑡≥0.

Lemma 2.5. There exists a positive constant 𝑀 such that 𝑆(𝑡)≤𝑀, 𝐼(𝑡)≤𝑀, 𝑦(𝑡)≤𝑀 for each solution of system (1.2) with positive initial values, where 𝑡 is sufficiently large.

Proof. Define 𝑉(𝑡)=𝜎𝑆(𝑡)+𝜎𝐼(𝑡)+𝑦(𝑡), then it is clear that 𝑉∈𝑉₀. For the continuity points of (1.2), that is, 𝑡≠(𝑛+𝜄−1)𝑇 and 𝑡≠𝑛𝑇, then 𝐷⁺𝑉(𝑡)+𝑑𝑉(𝑡)=𝑉′(𝑡)+𝑑𝑉(𝑡), by simple computation, we have 𝐷⁺𝑉(𝑡)+𝑑𝑉(𝑡)=𝜎(𝑑+𝑟)𝑆(𝑡)−𝜎𝐼(𝑡)𝑑₁−𝑑−𝑑₂𝑦(𝑡)−(𝑡𝑆(𝑡)(𝑡−𝑑𝜎𝑟𝑆)(+𝐼))𝐾≤𝜎(𝑑+𝑟)𝑆(𝑡)−𝜎𝑟𝑆²(𝑡)𝐾≤𝑀₁, (2.4) where 𝑑=min{𝑑₁,𝑑₂} and 𝑀₁=𝜎𝐾(𝑑+𝑟)²/4𝑟.
When 𝑡=(𝑛+𝜄−1)𝑇, by system (1.2) directly, we have 𝑉(𝑛+𝜄−1)𝑇⁺≤𝑉((𝑛+𝜄−1)𝑇). (2.5) Similarly, when 𝑡=𝑛𝑇, we have 𝑉𝑛𝑇⁺≤𝑉(𝑛𝑇)+𝜎𝑞₁+𝑞₂. (2.6)
According to Lemma 2.4, we can obtain that 𝑉(𝑡)≤𝑉0⁺−𝑀₁𝑑𝑒^−𝑑𝑡+𝜎𝑞₁+𝑞₂𝑒^𝑑𝑇1−𝑒^−𝑑𝑡𝑒^𝑑𝑇+𝑀−1𝑑¹⟶𝑀₁𝑑+𝜎𝑞₁+𝑞₂𝑒^𝑑𝑇𝑒^𝑑𝑇.−1 (2.7) Thus, 𝑉(𝑡) is uniformly ultimately bounded from above. By the definition of 𝑉(𝑡), we follow the conclusion immediately. This completes the proof.

Lemma 2.6. For the following system: (𝑡)(𝑡)()𝑢𝑡𝑢′=−𝑑𝑢,𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=𝑢(𝑡)()𝑢𝑡1−𝑝,𝑡=𝑛+𝜄−1𝑇,₊=𝑢(𝑡)+𝑞,𝑡=𝑛𝑇. (2.8) System (2.8) has a positive periodic solution 𝑢^∗(𝑡) and for every solution 𝑢(𝑡) of system (2.8), |𝑢(𝑡)−𝑢^∗(𝑡)|→0 as 𝑡→∞, where 𝑢^∗(𝑡)=𝑞𝑒^{−𝑑(𝑡−(𝑛−1)𝑇)}1−1−𝑝𝑒^−𝑑𝑇,()()𝑞𝑒𝑛−1𝑇<𝑡≤𝑛+𝜄−1𝑇,1−𝑝^{−𝑑(𝑡−(𝑛−1)𝑇)}1−1−𝑝𝑒^−𝑑𝑇,()𝑢𝑛+𝜄−1𝑇<𝑡≤𝑛𝑇,_∗0⁺=𝑢^∗𝑛𝑇⁺=𝑞1−1−𝑝𝑒^−𝑑𝑇,𝑢^∗(𝑛+𝜄−1)𝑇⁺=𝑢^∗𝑛𝑇⁺=𝑞1−𝑝𝑒^{−𝑑𝜄𝑇}1−1−𝑝𝑒^−𝑑𝑇. (2.9)

Proof. It is easily verified that 𝑢^∗(𝑡) is a periodic solution of system (2.8) with the given initial values. For the solution 𝑢(𝑡) of system (2.8), we can derive that 𝑢(𝑡)=1−𝑝^𝑛−1𝑢0⁺−𝑢^∗0⁺𝑒^−𝑑𝑡+𝑢^∗(𝑡),𝑡∈((𝑛−1)𝑇,(𝑛+𝜄−1)𝑇],1−𝑝^𝑛𝑢0⁺−𝑢^∗0⁺𝑒^−𝑑𝑡+𝑢^∗(𝑡),𝑡∈((𝑛+𝜄−1)𝑇,𝑛𝑇]. (2.10) Therefore, lim_𝑡→∞|𝑢(𝑡)−𝑢^∗(𝑡)|=0. This completes the proof.

3. Extinction and Permanence

For (1.2), if 𝑆(𝑡)≡0 for all 𝑡≥0, we have the following subsystem of (1.2): 𝐼^′(𝑡)=−𝑑₁𝐼(𝑡),𝑦^′(𝑡)=−𝑑₂𝑦(𝑡),()𝐼𝑡𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₂𝐼(𝑡),𝑦𝑡⁺=1−𝑝₃𝑦(𝑡)()𝐼𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝐼(𝑡)+𝑞₁,𝑦𝑡⁺=𝑦(𝑡)+𝑞₂,𝑡=𝑛𝑇. (3.1) For subsystem (3.1), by assumption (iii), there is no relation between 𝐼(𝑡) and 𝑦(𝑡). By Lemma 2.6, we have the following conclusion.

Lemma 3.1. System (3.1) has a unique positive periodic solution 𝐼^∗(𝑡)=𝑞₁𝑒^{−𝑑1(𝑡−(𝑛−1)𝑇)}1−1−𝑝₂𝑒^−𝑑1𝑇,()()𝑞𝑛−1𝑇<𝑡≤𝑛+𝜄−1𝑇,₁1−𝑝₂𝑒^{−𝑑1(𝑡−(𝑛−1)𝑇)}1−1−𝑝₂𝑒^−𝑑1𝑇,()𝑦𝑛+𝜄−1𝑇<𝑡≤𝑛𝑇,_∗(𝑡)=𝑞₂𝑒^{−𝑑2(𝑡−(𝑛−1)𝑇)}1−1−𝑝₃𝑒^−𝑑2𝑇,()()𝑞𝑛−1𝑇<𝑡≤𝑛+𝜄−1𝑇,₂1−𝑝₃𝑒^{−𝑑2(𝑡−(𝑛−1)𝑇)}1−1−𝑝₃𝑒^−𝑑2𝑇,(𝑛+𝜄−1)𝑇<𝑡≤𝑛𝑇, (3.2) with initial values 𝐼^∗0⁺=𝐼^∗𝑛𝑇⁺=𝑞₁1−1−𝑝₂𝑒^−𝑑1𝑇,𝐼^∗(𝑛+𝜄−1)𝑇⁺=𝐼^∗𝑛𝑇⁺=𝑞₁1−𝑝₂𝑒^{−𝑑1𝜄𝑇}1−1−𝑝₂𝑒^−𝑑1𝑇,𝑦^∗0⁺=𝑦^∗𝑛𝑇⁺=𝑞₂1−1−𝑝₃𝑒^−𝑑2𝑇,𝑦^∗(𝑛+𝜄−1)𝑇⁺=𝑦^∗𝑛𝑇⁺=𝑞₂1−𝑝₃𝑒^{−𝑑2𝜄𝑇}1−1−𝑝₃𝑒^−𝑑2𝑇. (3.3)

Next, we investigate the stability of the susceptible pest-eradication periodic solution (0,𝐼^∗(𝑡),𝑦^∗(𝑡)) of system (1.2).

Theorem 3.2. Let (𝑆(𝑡),𝐼(𝑡),𝑦(𝑡)) be any solution of (1.2). Then (0,𝐼^∗(𝑡),𝑦^∗(𝑡)) is globally asymptotically stable provided that 𝐻₁𝑟∶𝑟𝑇−𝐾𝑞+𝛼¹1−𝑝₂𝑒^−𝑑1𝑇+𝑝₂𝑒^{−𝑑1𝜄𝑇}−1𝑑₁1−𝑝₂𝑒^−𝑑1𝑇𝑞−1−𝑎𝑐²1−𝑝₃𝑒^−𝑑2𝑇+𝑝₃𝑒^{−𝑑2𝜄𝑇}−1𝑑₂1−𝑝₃𝑒^−𝑑2𝑇1−1<ln1−𝑝₁. (3.4)

Proof. First, by using Floquet theory, we show the local stability of periodic solution (0,𝐼^∗(𝑡),𝑦^∗(𝑡)) of (1.2). Considering the behavior of small amplitude perturbation, let 𝑆(𝑡)=𝑢(𝑡),𝐼(𝑡)=𝑣(𝑡)+𝐼^∗(𝑡),𝑦(𝑡)=𝑤(𝑡)+𝑦^∗(𝑡), (3.5) where 𝑢(𝑡),𝑣(𝑡), and 𝑤(𝑡) are all small perturbations. By using Taylor expansion and after neglecting higher-order terms, (1.2) can be linearized, and the linearized equations read as 𝑢^′(𝑡)=𝑟−𝑟𝐼^∗(𝑡)𝐾−𝛼𝐼^∗(𝑡)−𝑎𝑐𝑦^∗(𝑡)𝑢(𝑡),𝑣^′(𝑡)=𝛼𝐼^∗(𝑡)𝑢(𝑡)−𝑑₁𝑣(𝑡),𝑤^′(𝑡)=−𝑎𝑐𝜎𝑦^∗(𝑡)𝑢(𝑡)−𝑑₂𝑤(𝑡),()𝑢𝑡𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₁𝑢(𝑡),𝑣𝑡⁺=1−𝑝₂𝑣(𝑡),𝑤(𝑡)=1−𝑝₃𝑤(𝑡)()𝑢𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝑢(𝑡),𝑣𝑡⁺=𝑣(𝑡)+𝑞₁,𝑤𝑡⁺=𝑤(𝑡)+𝑞₂,𝑡=𝑛𝑇. (3.6) Suppose that Φ(𝑡) is the fundamental solution matrix of system (3.6), and then Φ(𝑡) satisfies 𝑑Φ(𝑡)=𝑑𝑡𝑟−𝑟𝐼^∗(𝑡)𝐾−𝛼𝐼^∗(𝑡)−𝑎𝑐𝑦^∗(𝑡)00𝛼𝐼^∗(𝑡)−𝑑₁0𝑎𝑐𝜎𝑦^∗(𝑡)0−𝑑₂Φ(𝑡), (3.7)Φ(0)=𝐼₃ is the identical matrix. Then the resetting impulsive conditions of (1.2) become 𝑢(𝑛+𝜄−1)𝑇⁺𝑣(𝑛+𝜄−1)𝑇⁺𝑤(𝑛+𝜄−1)𝑇⁺=1−𝑝₁0001−𝑝₂0001−𝑝₃𝑢)𝑇)𝑣)𝑇)𝑤𝑇,𝑢((𝑛+𝜄−1((𝑛+𝜄−1((𝑛+𝜄−1))𝑛𝑇⁺𝑣𝑛𝑇⁺𝑤𝑛𝑇⁺=𝑢()𝑣()𝑤().100010001𝑛𝑇𝑛𝑇𝑛𝑇 (3.8) By Floquet theory, the local stability of (0,𝐼^∗(𝑡),𝑦^∗(𝑡)) is determined by the eigenvalues of 𝑁=1−𝑝₁0001−𝑝₂0001−𝑝₃Φ(𝑡)100010001, (3.9) where Φ(𝑡)=1−𝑝₁𝑒^{∫𝑇0(𝑟−𝑟𝐼∗(𝑡)/𝐾−𝛼𝐼∗(𝑡)−𝑎𝑐𝑦∗(𝑡))𝑑𝑡}∗001−𝑝₂𝑒^−𝑑1𝑇0∗∗01−𝑝₃𝑒^−𝑑2𝑇. (3.10) The exact expression of ∗ and ∗∗ are omitted since they are not required in the analysis that follows. The eigenvalues of 𝑁 are 𝜆₁=1−𝑝₁𝑒^{∫𝑇0(𝑟−𝑟𝐼∗(𝑡)/𝐾−𝛼𝐼∗(𝑡)−𝑎𝑐𝑦∗(𝑡))𝑑𝑡},𝜆₂=1−𝑝₂𝑒^−𝑑1𝑇,𝜆₃=1−𝑝₃𝑒^−𝑑2𝑇. (3.11) Obviously, 𝜆₂<1,𝜆₃<1. It follows that |𝜆₁|<1 if and only if 𝑟𝑇−^𝑇₀𝑟𝐼^∗(𝑡)𝐾+𝛼𝐼^∗(𝑡)+𝑎𝑐𝑦^∗(𝑡)1𝑑𝑡<ln1−𝑝₁. (3.12) By computation, we have ^𝑇₀𝐼^∗(𝑡)𝑞𝑑𝑡=¹1−𝑝₂𝑒^−𝑑1𝑇+𝑝₂𝑒^{−𝑑1𝜄𝑇}−1𝑑₁1−𝑝₂𝑒^−𝑑1𝑇,−1_𝑇0𝑦^∗(𝑡)𝑞𝑑𝑡=²1−𝑝₃𝑒^−𝑑2𝑇+𝑝₃𝑒^{−𝑑2𝜄𝑇}−1𝑑₂1−𝑝₃𝑒^−𝑑2𝑇.−1 (3.13) Therefore, we can derive that |𝜆₁|<1 if and only if (𝐻₁) holds true. Thus, by Floquet theory, under condition (𝐻₁), the positive periodic solution (0,𝐼^∗(𝑡),𝑦^∗(𝑡)) is locally asymptotically stable.
Next, we prove the globally attractive property.
Choose 𝜀₁>0 such that 1−𝑝₁exp^𝑇₀𝑟𝑟−𝐾+𝛼𝐼^∗(𝑡)−𝜀₁−𝑎𝑐𝑦^∗(𝑡)−𝜀₁𝑑𝑡≜𝛿<1. (3.14) Besides, we have 𝐼^′(𝑡)=𝛼𝑆(𝑡)𝐼(𝑡)−𝑑₁𝐼(𝑡)≥−𝑑₁𝐼(𝑡). (3.15) From Lemma 3.1 and comparison theorem of impulsive equation, for 𝑡 sufficiently large, we have 𝐼(𝑡)≥𝐼^∗(𝑡)−𝜀₁. (3.16) Similarly, for 𝑡 sufficiently large, we have 𝑦(𝑡)≥𝑦^∗(𝑡)−𝜀₁. (3.17) For simplification, we suppose that (3.16) and (3.17) hold for all 𝑡≥0. Hence, it follows that 𝑆^′(𝑡)(𝑡)𝑆(𝑡)(𝑡)=𝑟𝑆1−+𝐼𝐾−𝛼𝑆(𝑡)𝐼(𝑡)−𝑎𝑦(𝑡)1−𝑒^{−𝑐𝑆(𝑡)}(𝑡𝑟≤𝑆)𝑟−𝐾+𝛼𝐼^∗(𝑡)−𝜀₁−𝑎𝑐𝑦^∗(𝑡)−𝜀₁()𝑆𝑡,𝑡≠𝑛+𝜄−1𝑇,₊=1−𝑝₁𝑆(𝑡),𝑡=(𝑛+𝜄−1)𝑇. (3.18) Integrating (3.18) on ((𝑛+𝜄−1)𝑇),(𝑛+𝜄)𝑇] leads to 𝑆((𝑛+𝜄)𝑇)≤𝑆(𝑛+𝜄−1)𝑇⁺exp_()𝑇^{(𝑛+𝜄)𝑇}_{𝑛+𝜄−1}𝑟𝑟−𝐾+𝛼𝐼^∗(𝑡)−𝜀₁−𝑎𝑐𝑦^∗(𝑡)−𝜀₁)𝑇𝑑𝑡=𝑆((𝑛+𝜄−1)1−𝑝₁×exp_()𝑇^{(𝑛+𝜄)𝑇}_{𝑛+𝜄−1}𝑟𝑟−𝐾+𝛼𝐼^∗(𝑡)−𝜀₁−𝑎𝑐𝑦^∗(𝑡)−𝜀₁)𝑇)𝑑𝑡=𝑆((𝑛+𝜄−1𝛿. (3.19) Thus 𝑆((𝑛+𝜄)𝑇)≤𝑆(𝜄𝑇)𝛿^𝑛. In virtue of the assumption 𝛿<1, we can follow that 𝑆((𝑛+𝜄)𝑇)→0 as 𝑛→∞. Noting that 0<𝑆(𝑡)<𝑆((𝑛+𝜄−1)𝑇)(1−𝑝₁)𝑒^𝑟𝑇 for (𝑛+𝜄−1)𝑇≤𝑡≤(𝑛+𝜄)𝑇, hence 𝑆(𝑡)→0 as 𝑛→∞.
Next, we prove 𝐼(𝑡)→𝐼^∗(𝑡) as 𝑡→∞. For an arbitrary positive constant 𝜀 small enough such that 0<𝜀<𝑑₁/𝛼, since 0<𝑆(𝑡)<𝜀 for 𝑡 large enough, without loss of generality, we assume that 0<𝑆(𝑡)<𝜀 hold for all 𝑡≥0. Then −𝑑₁𝐼(𝑡)≤𝐼^′(𝑡)=𝛼𝑆(𝑡)𝐼(𝑡)−𝑑₁𝐼(𝑡)≤𝛼𝜀−𝑑₁𝐼(𝑡). (3.20) By Lemma 3.1 and comparison theorem again, for all 𝜀′>0, there exists 𝑇>0, for all 𝑡>𝑇, and we obtain 𝐼^∗(𝑡)−𝜀′≤𝐼(𝑡)≤𝐼^Δ(𝑡)+𝜀′, (3.21) where 𝐼^∗(𝑡) is defined in Lemma 3.1, and 𝐼^Δ(𝑡) is the solution of the following system: 𝐼(𝑡)=𝛼𝜀−𝑑₁𝐼(𝑡)()𝐼𝑡,𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₂𝐼(𝑡)()𝐼𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝐼(𝑡)+𝑞₁,𝑡=𝑛𝑇, (3.22) that is, 𝐼^Δ(𝑡)=𝑞₁𝑒^{(−𝑑1+𝛼𝜀)(𝑡−(𝑛−1)𝑇)}1−1−𝑝₂𝑒^{(−𝑑1+𝛼𝜀)𝑇},()()𝑞𝑛−1𝑇<𝑡≤𝑛+𝜄−1𝑇,₁1−𝑝₂𝑒^{(−𝑑1+𝛼𝜀)(𝑡−(𝑛−1)𝑇)}1−1−𝑝₂𝑒^{(−𝑑1+𝛼𝜀)𝑇},(𝑛+𝜄−1)𝑇<𝑡≤𝑛𝑇. (3.23) Let 𝜀→0, it follows that 𝐼^∗(𝑡)−𝜀′<𝐼(𝑡)<𝐼^∗(𝑡)+𝜀′ (3.24) holds for 𝑡 sufficiently large. Note that 𝜀′>0 is a constant small enough, then letting 𝜀′→0, and we have 𝐼(𝑡)→𝐼^∗(𝑡) as 𝑡→∞. By the same method, we can similarly derive that 𝑦(𝑡)→𝑦^∗(𝑡) as 𝑡→∞. This completes the proof.

Finally, we study the permanence of system (1.2).

Theorem 3.3. System (1.2) is permanent if 𝐻₂𝑟∶𝑟𝑇−𝐾𝑞+𝛼¹1−𝑝₂𝑒^−𝑑1𝑇+𝑝₂𝑒^{−𝑑1𝜄𝑇}−1𝑑₁1−𝑝₂𝑒^−𝑑1𝑇𝑞−1−𝑎𝑐²1−𝑝₃𝑒^−𝑑2𝑇+𝑝₃𝑒^{−𝑑2𝜄𝑇}−1𝑑₂1−𝑝₃𝑒^−𝑑2𝑇1−1>ln1−𝑝₁. (3.25)

Proof. By Lemma 2.5, without loss of generality, we suppose that 𝑆(𝑡)≤𝑀, 𝐼(𝑡)≤𝑀 and 𝑦(𝑡)≤𝑀 for all 𝑡≥0, where 𝑀 is a constant satisfying 𝑀>𝑟/𝑎𝑐.
On the other hand, from (3.16) and (3.17), we can obtain 𝐼(𝑡)≥𝐼^∗(𝑡)−𝜀₁≥𝐼^∗0⁺1−𝑝₂𝑒^−𝑑1𝑇−𝜀₁≜𝑚₁𝑦(𝑡)>0,≥𝑦^∗(𝑡)−𝜀≥𝑦^∗0⁺1−𝑝₃𝑒^−𝑑2𝑇−𝜀₁≜𝑚₂>0 (3.26) for 𝑡 large enough. Therefore, 𝐼(𝑡) and 𝑦(𝑡) are ultimately positively bounded from below. Hence, we only need to prove that there exists a constant 𝑚₃>0 such that 𝑆(𝑡)>𝑚₃ for 𝑡 sufficiently large. By the proof of Theorem 3.2, assumption (𝐻₂) is equivalent to ^𝑇₀𝑟−𝑟𝐼^∗(𝑡)𝐾−𝛼𝐼^∗(𝑡)−𝑎𝑐𝑦^∗(𝑡)1𝑑𝑡>ln1−𝑝₁. (3.27) By the density of real number, we can select positive constant 𝜀 and 𝜁 small enough with 0<𝜁<min{𝑑₁/𝛼,𝑑₂/𝑎𝑐𝜎} such that 𝜂≜1−𝑝₁𝑒^{∫𝑇0(𝑟−𝑟𝜁/𝐾−(𝑟/𝐾+𝛼)(𝐼(𝑡)+𝜀)−𝑎𝑐(𝑦(𝑡)+𝜀))𝑑𝑡}>1, (3.28) where 𝐼(𝑡)=𝑞₁𝑒^{(−𝑑1+𝛼𝜁)(𝑡−(𝑛−1)𝑇)}1−1−𝑝₂𝑒^{(−𝑑1+𝛼𝜁)𝑇},()()𝑞𝑛−1𝑇<𝑡≤𝑛+𝜄−1𝑇,₁1−𝑝₂𝑒^{(−𝑑1+𝛼𝜁)(𝑡−(𝑛−1)𝑇)}1−1−𝑝₂𝑒^{(−𝑑1+𝛼𝜁)𝑇},(𝑛+𝜄−1)𝑇<𝑡≤𝑛𝑇,𝑦(𝑡)=𝑞₂𝑒^{(−𝑑2+𝑎𝑐𝜎𝜁)(𝑡−(𝑛−1)𝑇)}1−1−𝑝₃𝑒^{(−𝑑2+𝑎𝑐𝜎𝜁)𝑇},()()𝑞𝑛−1𝑇<𝑡≤𝑛+𝜄−1𝑇,₂1−𝑝₃𝑒^{(−𝑑2+𝑎𝑐𝜎𝜁)(𝑡−(𝑛−1)𝑇)}1−1−𝑝₃𝑒^{(−𝑑2+𝑎𝑐𝜎𝜁)𝑇},(𝑛+𝜄−1)𝑇<𝑡≤𝑛𝑇. (3.29)(3.30) We claim that 𝑆(𝑡)<𝜁 cannot hold for all 𝑡≥0. Otherwise, we can follow from (1.2) that 𝐼^′(𝑡)=𝛼𝑆(𝑡)𝐼(𝑡)−𝑑₁𝐼(𝑡)≤𝛼𝜁−𝑑₁𝐼(𝑡)()𝐼𝑡,𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₂𝐼(𝑡)()𝐼𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝐼(𝑡)+𝑞₁𝑦,𝑡=𝑛𝑇,_′(𝑡)=𝜎𝑎𝑦(𝑡)1−𝑒^{−𝑐𝑆(𝑡)}−𝑑₂𝑦(𝑡)≤𝑎𝑐𝜎𝜁−𝑑₂𝑦(𝑡)()𝑦𝑡,𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₃𝑦(𝑡)()𝑦𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝑦(𝑡)+𝑞₂,𝑡=𝑛𝑇. (3.31)(3.32) By comparison theorem of impulsive differential equation, then there exists a 𝑇₁>0 such that 𝐼(𝑡)≤𝐼(𝑡)+𝜀,𝑦(𝑡)≤𝑦(𝑡)+𝜀,∀𝑡≥0, (3.33) where 𝐼(𝑡),𝑦(𝑡) are defined in (3.29) and (3.30), respectively; that is, they are the solutions of systems (3.31) and (3.32), respectively. Then, for 𝑡>𝑇₁, we have 𝑆^′(𝑡)≥𝑆(𝑡)𝑟−𝑟𝜁𝐾−𝑟𝐾+𝛼𝐼(𝑡)+𝜀−𝑎𝑐𝑦(𝑡)()𝑆𝑡+𝜀,𝑡≠𝑛+𝜄−1𝑇,𝑡≠𝑛𝑇,₊=1−𝑝₁𝑆(𝑡)()𝑆𝑡,𝑡=𝑛+𝜄−1𝑇,₊=𝑆(𝑡),𝑡=𝑛𝑇. (3.34) Let 𝑛₁∈𝑍₊ and (𝑛₁+𝜄−1)𝑇>𝑇₁. Integrating (3.34) on ((𝑛+𝜄−1)𝑇,(𝑛+𝜄)𝑇),𝑛≥𝑛₁, we have 𝑆(𝑛+𝜄)𝑇≥𝑆(𝑛+𝜄−1)𝑇⁺exp_()𝑇^{(𝑛+𝜄)𝑇}_{𝑛+𝜄−1}𝑟−𝑟𝜁𝐾−𝑟𝐾+𝛼𝐼(𝑡)+𝜀−𝑎𝑐𝑦(𝑡)()𝑇+𝜀=𝑆𝑛+𝜄−11−𝑝₁×exp_()𝑇^{(𝑛+𝜄)𝑇}_{𝑛+𝜄−1}𝑟−𝑟𝜁𝐾−𝑟𝐾+𝛼𝐼(𝑡)+𝜀−𝑎𝑐𝑦(𝑡)()+𝜀=𝑆𝑛+𝜄−1𝑇𝜂. (3.35) Thus, 𝑆((𝑛₁+𝑛+𝜄)𝑇)≥𝑆(𝑛₁+𝜄)𝑇⋅𝜂^𝑛→∞ as 𝑡→∞, which is a contradiction to the boundedness of 𝑆(𝑡). Hence, there exists a 𝑡₁>0 such that 𝑆(𝑡₁)≥𝜁. Then there are two cases.
Case??1 (𝑆(𝑡)≥𝜁 for all 𝑡≥𝑡₁). Then let 𝑚₃=𝜁, our aim is obtained. Otherwise, we consider Case 2.
Case??2. We consider those solutions which leave the region 𝑅={(𝑆(𝑡),𝐼(𝑡),𝑦(𝑡))∈𝑅³₊∶𝑆(𝑡)<𝜁} and reenter it again. Let 𝑡^∗=inf_𝑡≥𝑡1{𝑆(𝑡)>𝜁}, then there are two possible cases for 𝑡^∗.
Subcase??2.1 (𝑡^∗=(𝑛₁+𝜄−1)𝑇, 𝑛₁∈𝑍₊). Then for 𝑡∈[𝑡₁,𝑡^∗], 𝑆(𝑡)≥𝜁 and (1−𝑝₁)𝜁≤𝑆(𝑡^∗+)=(1−𝑝₁)𝑆(𝑡^∗)<𝜁. Select 𝑛₂,𝑛₃∈𝑍₊ such that 𝑛₂−1𝑇>ln𝜀/𝑀+𝑞₁𝛼𝜁−𝑑₁,1−𝑝₁^𝑛2𝑒^{𝑛2𝜂1𝑇}𝜂^𝑛3>1−𝑝₁^𝑛2𝑒^{(𝑛2+1)𝜂1𝑇}𝜂^𝑛3>1, (3.36)(3.37) where 𝜂₁=𝑟−𝑟𝜁/𝐾−(𝑟/𝐾+𝛼)𝑀−𝑎𝑐𝑀<0.
Let 𝑇=𝑛₂𝑇+𝑛₃𝑇, then we claim that there must exist a 𝑡₂∈(𝑡^∗,𝑡^∗+𝑇) such that 𝑆(𝑡₂)>𝜁. Otherwise, considering system (3.31), we can derive that 𝐼(𝑡)=1−𝑝₂^{𝑛−(𝑛1+1)}𝐼𝑛₁𝑇⁺−𝑞₁1−1−𝑝₂𝑒^{(−𝑑1+𝛼𝜀)𝑇}𝑒^{(−𝑑1+𝛼𝜁)(𝑡−𝑛1𝑇)}+𝐼(𝑡),(𝑛−1)𝑇<𝑡≤(𝑛+𝜄−1)𝑇,1−𝑝₂^𝑛−𝑛1𝐼𝑛₁𝑇⁺−𝑞₁1−1−𝑝₂𝑒^{(−𝑑1+𝛼𝜀)𝑇}𝑒^{(−𝑑1+𝛼𝜁)(𝑡−𝑛1𝑇)}+𝐼(𝑡),(𝑛+𝜄−1)𝑇<𝑡≤𝑛𝑇, (3.38) and 𝑛₁+1≤𝑛≤𝑛₁+𝑛₂+𝑛₃.
From (3.36) and (3.38), we have 𝐼(𝑡)−𝐼(𝑡)<𝑀+𝑞₁𝑒^{(𝛼𝜁−𝑑1)(𝑡−𝑛1𝑇)}<𝜀. (3.39) That is, 𝐼(𝑡)≤𝐼(𝑡)+𝜀,𝑛₁+𝑛₂−1𝑇≤𝑡≤𝑡^∗+𝑇. (3.40) Similarly, we can derive that 𝑦(𝑡)≤𝑦(𝑡)+𝜀,for𝑛₁+𝑛₂−1𝑇≤𝑡≤𝑡^∗+𝑇, (3.41) which implies that (3.34) holds for 𝑡^∗+𝑛₂𝑇≤𝑡≤𝑡^∗+𝑇. In view of the discussion after (3.34), we have 𝑆(𝑡^∗+𝑇)≥𝑆(𝑡^∗+𝑛₂𝑇)𝜂^𝑛3. Integrating the following system (3.42) on [𝑡^∗,𝑡^∗+𝑛₂𝑇], 𝑆^′(𝑡)≥𝑆(𝑡)𝑟−𝑟𝜁𝐾−𝑟𝐾()𝑆𝑡+𝛼𝑀−𝑎𝑐𝑀,𝑡≠𝑛+𝜄−1𝑇,₊=1−𝑝₁𝑆(𝑡),𝑡=(𝑛+𝜄−1)𝑇, (3.42) we have 𝑆𝑡^∗+𝑛₂𝑇≥𝜁1−𝑝₁^𝑛2𝑒^{𝑛2𝜂1𝑇}. (3.43) It follows from (3.37) and (3.43) that 𝑆𝑡^∗+𝑇≥𝜁1−𝑝₁^𝑛2𝑒^{𝑛2𝜂1𝑇}𝜂^𝑛3>𝜁, (3.44) which is a contradiction to the above assumption, then our claim is true.
Let 𝑡=inf_{𝑡≥𝑡∗}{𝑆(𝑡)>𝜁}, then for 𝑡∈(𝑡^∗,𝑡), 𝑥(𝑡)≤𝜁, 𝑥(𝑡)=𝜁. Therefore, 𝑆(𝑡)≥𝜁1−𝑝₁^𝑛2+𝑛3𝑒^{(𝑛2+𝑛3)𝜂1𝑇}≜𝑚₃. (3.45) For 𝑡>𝑡, the same argument can be continued since 𝑆(𝑡)≥𝜁.
Subcase??2.2 (𝑡^∗≠(𝑛₁+𝜄−1)𝑇,𝑛₁∈𝑍₊). Then 𝑆(𝑡)≥𝜁 for 𝑡∈[𝑡₁,𝑡^∗] and 𝑆(𝑡^∗)=𝜁.
Suppose 𝑡^∗∈((𝑛^′₁+𝜄−1)𝑇, (𝑛^′₁+𝜄)𝑇), 𝑛^′₁∈𝑍₊. Then either 𝑆(𝑡)≤𝜁 for all 𝑡∈(𝑡^∗,(𝑛^′₁+𝜄)𝑇), or there exists a 𝑡∈(𝑡^∗,(𝑛^′₁+𝜄)𝑇) such that 𝑆(𝑡)>𝜁.
If 𝑆(𝑡)≤𝜁 for all 𝑡∈(𝑡^∗,(𝑛^′₁+𝜄)𝑇), similar to the former discussion, there must be a 𝑡^′₂∈[(𝑛^′₁+𝜄)𝑇,(𝑛^′₁+𝜄)𝑇+𝑇] such that 𝑆(𝑡^′₂)>𝜁. Let 𝑡̃=inf_𝑡>𝑡∗{𝑆(𝑡)>𝜁}, and then 𝑆(𝑡)̃≤𝜁 for 𝑡∈(𝑡^∗,𝑡]̃. Therefore, 𝑆(𝑡)≥𝜁(1−𝑝₁)^𝑛2+𝑛3𝑒^{(𝑛2+𝑛3+1)𝜂1𝑇}≜𝑚₃. That is, 𝑆(𝑡)≥𝑚₃ holds for 𝑡∈(𝑡^∗,𝑡]̃. For 𝑡>𝑡̃, due to 𝑆(𝑡)̃≥𝜁, the same argument can be continued.
If there exists a 𝑡∈(𝑡^∗,(𝑛^′₁+𝜄)𝑇) such that 𝑆(𝑡)>𝜁, the same argument in Subcase 2.1 can also be continued, and one can follow the conclusion easily. We omit it here.
Incorporating all the cases above, we deduce that system (1.2) is permanent. The proof is complete.

Remark 3.4. For system (1.2), if there is no disease in prey, then Theorems 3.2 and 3.3 reduce to the corresponding results of [1], while Theorems 3.2 and 3.3 imply that disease in prey affects dynamics of (1.2), which will be shown by simulation in Section 4. It is interesting and valuable for biological control. Specially, the globally asymptotical stability of susceptible pest-eradication periodic solution is studied here, but authors [1] only give locally asymptotical stability of the prey-free periodic solution. Therefore, we improve and generalize the main results of [1].

4. Examples and Simulation

Theorems 3.2 and 3.3 show that dynamics of system(1.2) is affected by complicated factors such as contact number per unit time for every infective prey 𝛼, impulsive period 𝑇, releasing amount of infective prey 𝑞₁, and releasing amount of predator 𝑞₂. In this section, by numerical analysis, we show the effects of parameters 𝛼,𝑇,𝑞₁, and 𝑞₂ on dynamics of (1.2), respectively. For example, take 𝑟=4, 𝑘=8, 𝑎=2, 𝑐=0.5, 𝑝₁=0.5, 𝑝₂=𝑝₃=0, 𝑞₁=𝑞₂=2, 𝑑₁=0.2, 𝑑₂=0.4, 𝑇=4, and 𝑆(0)=𝐼(0)=𝑦(0)=2. If 𝛼=0.55, then Theorem 3.2 implies that system (1.2) has a susceptible pest-eradication periodic solution, which is globally asymptotically stable. If 𝛼=0.4, then Theorem 3.3 implies that (1.2) is permanent. By simulation, the results can be seen directly, see Figures 1 and 2, respectively. Actually, by simulation, there exists a critical value 𝛼≈0.531, and if 𝛼<0.531, then the system is permanent; otherwise, it has a susceptible pest-eradication periodic solution, see Figures 3, 4, and 5, respectively.

Figure 1 

Existence of the pest-eradication periodic solution of (1.2) with 𝑟=4, 𝑘=8, 𝑎=2, 𝑐=0.5, 𝑝1=0.5, 𝑝2=𝑝3=0, 𝑞1=𝑞2=2, 𝑑1=0.2, 𝑑2=0.4, 𝑇=4, 𝑆(0)=𝐼(0)=𝑦(0)=2, 𝛼=0.55.

Figure 2 

Permanence of (1.2) with 𝑟=4, 𝑘=8, 𝑎=2, 𝑐=0.5, 𝑝1=0.5, 𝑝2=𝑝3=0, 𝑞1=𝑞2=2, 𝑑1=0.2, 𝑑2=0.4, 𝑇=4, 𝑆(0)=𝐼(0)=𝑦(0)=2, 𝛼=0.4.

Figure 3 

Bifurcation diagrams of susceptible pest for (1.2) with 𝑟=4, 𝑘=8, 𝑎=2, 𝑐=0.5, 𝑝1=0.5, 𝑝2=𝑝3=0, 𝑞1=𝑞2=2, 𝑑1=0.2, 𝑑2=0.4, 𝑇=4, 𝑆(0)=𝐼(0)=𝑦(0)=2. For each value of 𝛼, system (1.2) is integrated over 500 pulsing cycles, and the stroboscopic measurements of 𝑆(𝑡) are plotted, 𝛼 is over [0,1].

Figure 4 

Bifurcation diagrams of infective pest for (1.2) with 𝑟=4, 𝑘=8, 𝑎=2, 𝑐=0.5, 𝑝1=0.5, 𝑝2=𝑝3=0, 𝑞1=𝑞2=2, 𝑑1=0.2, 𝑑2=0.4, 𝑇=4, 𝑆(0)=𝐼(0)=𝑦(0)=2. For each value of 𝛼, system (1.2) is integrated over 500 pulsing cycles, and the stroboscopic measurements of 𝐼(𝑡) are plotted, 𝛼 is over [0,1].

Figure 5 

Bifurcation diagrams of predator for (1.2) with 𝑟=4, 𝑘=8, 𝑎=2, 𝑐=0.5, 𝑝1=0.5, 𝑝2=𝑝3=0, 𝑞1=𝑞2=2, 𝑑1=0.2, 𝑑2=0.4, 𝑇=4, 𝑆(0)=𝐼(0)=𝑦(0)=2. For each value of 𝛼, system (1.2) is integrated over 500 pulsing cycles, and the stroboscopic measurements of 𝑦(𝑡) are plotted, 𝛼 is over [0,1].

Take 𝑟=2, 𝐾=4, 𝑎=2, 𝑐=0.5, 𝛼=0.6, 𝑝₁=0.5, 𝑝₂=0.3, 𝑝₃=0.6,?𝜎=0.3,?𝜄=0.5,?𝑑₁=0.2,?𝑑₂=0.4,? and 𝑆(0)=𝐼(0)=𝑦(0)=1. For impulsive period 𝑇, the releasing amount of infective prey 𝑞₁, and releasing amount of predator 𝑞₂, respectively, Theorems 3.2 and 3.3 imply that system (1.2) exhibits a variety of dynamic behaviors such as cycles, periodic doubling cascade, chaos, and so on. Take susceptible pest as an example, and the numerical analysis of susceptible pest about parameters 𝑇,?𝑞₁, and 𝑞₂ shows that dynamical behaviors of the system are very complex, see Figures 6, 7, and 8. Similarly, dynamics of infective pest and predator may be complex too. It is omitted.

Figure 6 

Bifurcation diagrams of susceptible pest for (1.2) with 𝑟=2, 𝐾=4, 𝑎=2, 𝑐=0.5, 𝛼=0.6, 𝑝1=0.5,  𝑝2=0.3,  𝑝3=0.6,  𝜎=0.3,  𝜄=0.5,  𝑑1=0.2, 𝑑2=0.4, 𝑞1=2, 𝑞2=4, 𝑆(0)=𝐼(0)=𝑦(0)=1. For each value of 𝑇, system (1.2) is integrated over 400 pulsing cycles, and the last 400 stroboscopic measurements of 𝑆(𝑡) are plotted, and 𝑇 is over [10,50].

Figure 7 

Bifurcation diagrams of susceptible pest for (1.2) with 𝑟=2, 𝐾=4, 𝑎=2, 𝑐=0.5, 𝛼=0.6, 𝑝1=0.5,  𝑝2=0.3,  𝑝3=0.6,  𝜎=0.3,  𝜄=0.5,  𝑑1=0.2, 𝑑2=0.4, 𝑞2=4, 𝑇=10, 𝑆(0)=𝐼(0)=𝑦(0)=1. For each value of 𝑇, system (1.2) is integrated over 500 pulsing cycles, and the last 400 stroboscopic measurements of 𝑆(𝑡) are plotted, and 𝑞1 is over [0,5].

Figure 8 

Bifurcation diagrams of susceptible pest for (1.2) with 𝑟=2, 𝐾=4, 𝑎=2, 𝑐=0.5, 𝛼=0.6, 𝑝1=0.5,  𝑝2=0.3,  𝑝3=0.6,  𝜎=0.3,  𝜄=0.5,  𝑑1=0.2, 𝑑2=0.4, 𝑞1=2, 𝑇=10, 𝑆(0)=𝐼(0)=𝑦(0)=1. For each value of 𝑇, system (1.2) is integrated over 800 pulsing cycles, and the last 400 stroboscopic measurements of 𝑆(𝑡) are plotted, and 𝑞2 is over [0,5].

5. Discussion

In this paper, by using Floquet theory and small amplitude perturbation technique of impulsive differential equation, we consider a predator-prey model with complex influence of disease in prey, Ivlev-type functional response, and impulsive perturbation such as periodic pesticide spraying on all species and periodically releasing predator at fixed different time. Conditions ensuring the existence and global stability of the extinction of susceptible prey as well as the permanence of this system are established.

Our results show that the susceptible prey is either extinct or permanent, that is, when the stability of extinct equilibrium is lose, the system becomes permanent. We conclude that, to reduce pest population, a strategy should aim to make the inequality in Theorem 3.2 hold. This can be done by increasing the value of parameters 𝛼, 𝑇, and 𝑝₁ or decreasing the value of parameters 𝑞₁ and 𝑞₂, that is, releasing more infective pests, using more effective pesticides for the prey and increasing numbers of releasing of infective prey and predator (the natural enemy of the pest). By numerical analysis, small changes of parameters 𝛼, 𝑇, 𝑝_𝑖, or 𝑞_𝑖?(𝑖=1,2) may make much difference on the dynamics of (1.2). This indicates that pest control strategy is a very complex issue, and integrated pest management may be more effective than those when only spraying pesticides is used or only infective pest is released or only natural enemy is released. Hence, integrated pest management may be a prior strategy for pest management.

Finally, many authors [15, 29–31] pay much attention to study the existence of epidemic equilibrium points, dynamical effects from delays, and stochastic impulse in recent years. Then for (1.2), how about the potential presence or not of epidemic equilibrium points of the prey and the possible extensions to the incorporation of delays? On the other hand, if the predator is released stochastically, how does it affect the dynamics of (1.2)? we leave these interesting problems as our future work.

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and constructive suggestions, which are very useful for improving the quality of this paper. The paper is supported by National Natural Science Foundation of China (11161015) and Doctoral Foundation of Guilin University of Technology (2010).