Stability and Permanence of a Pest Management Model with Impulsive Releasing and Harvesting

Abstract

We formulate a pest management model with periodically releasing infective pests, immature and mature natural enemies, and harvesting pests and crops at two different fixed moments. Sufficient conditions ensuring the locally and globally asymptotical stability of the susceptible pest-eradication period solution are found by means of Floquet theory, small amplitude perturbation techniques, and multicomparison results. Furthermore, the permanence of system is also derived. By numerical analysis, we also show that impulsive releasing and harvesting at two different fixed moments can bring obvious effects on the dynamics of system, which also corroborates our theoretical results.

1. Introduction

As is known to all, pest outbreaks often cause serious ecological and economic problems. Therefore, how to effectively control insects and other arthropods has become an increasingly complex issue. Usually, chemical pesticides were taken as a relatively simple way to solve the pest-related problems, and some mathematical models on pest management with toxin (pesticide) input were studied in [1–4]. However, the overuse of chemical pesticides may create new ecological and sociological harm such as pesticide pollution and pesticide-resistant pest varieties and inflicts harmful effects on humans and so forth. Therefore, nonchemical use instead for pest control has become a hot topic in order to reduce pest density to tolerable levels and minimize the damage caused. For instance, biological control methods by periodically releasing infective pests or their natural enemies are often taken due to their advantage in the aspects of self-sustainable mechanism, lower environmental impact, and cost effectiveness.

Recently, some biocontrol models on pest management described by impulsive differential equation were proposed and the dynamics such as stability, permanence, periodicity, and bifurcation are deeply investigated (see also, e.g., [2–12]). In [5], an impulsive system to model the process of periodic releasing natural enemies and harvesting pest at different fixed time for pest control is considered, and the sufficient conditions on the existence and global stability of the periodic solution are derived for the given model. Georgescu et al. [6, 7] construct an integrated pest management model which relies on the simultaneous periodic release of infective pest individuals and of natural predators with age structure and obtain some sufficient conditions on the local and global stability, permanence, and bifurcation of the systems. However, most of the existing models on pest management scarcely take into account the factor on the relation between pest and its food (e.g., crop). In fact, farmers may harvest crops several times in process of its growth, which should cause a great impact on the density of the pest.

Motived by the above discussion, we construct a model of pest control by periodically releasing infective pests, immature and mature natural enemies, and harvesting pests and crops. To account for the discontinuity of release and harvest at different fixed moments, our model is based on impulsive differential equations. We analyze the dynamical behavior of the system by using the theory of impulsive differential equation introduced in [13–15].

The rest of this paper is organized as follows. A pest management model with impulsive releasing and harvesting is introduced in Section 2 and some useful preliminaries are given in Section 3. Section 4 deals with stability and permanence analysis of system. In this section, two sufficient conditions are deduced including the locally and globally asymptotical stability of the susceptible pest-eradication period solution, the permanence of system is also discussed. A simple example and conclusions are given in Section 5.

2. Model Description

In the following, to establish our pest management model, we rely on the following biological assumptions. (𝐴1) The pest population is divided into two classes, the susceptible and infective. The infective pests neither recover nor reproduce and infective pests cannot damage crops. The disease is transmitted from infective pests to susceptible pests and does not propagate to predators. (𝐴2) In the absence of susceptible pests, the crops have a logistic growth rate with intrinsic birth rate 𝑟 and carrying capacity 𝐾. (𝐴3) The predators (natural enemies) have an age structure, that is, immature and mature. Only the mature predators have the ability to feed on susceptible pests, but do not prey on infective pests and crops. (𝐴4) The functional response of the susceptible pest is described by the abstract function 𝑃₁, the functional response of the mature predator is described by the abstract function 𝑃₂, and the infection rate is described by the abstract function 𝑔, where 𝑃₁, 𝑃₂, and 𝑔 satisfy certain assumptions outlined below.

On the basis of the above assumptions, we establish the following impulsively controlled system: 𝑥^′(𝑡)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾−𝑃₁(𝑥(𝑡𝑆(𝑡),𝑆))_′(𝑡)=𝛽𝑃₁(𝑥(𝑡))𝑆(𝑡)−𝑔(𝐼(𝑡))𝑆(𝑡)−𝑃₂(𝑆(𝑡))𝑦_𝑀(𝑡)−𝑑_𝑆𝑆(𝑡),𝐼^′(𝑡)=𝑔(𝐼(𝑡))𝑆(𝑡)−𝑑_𝐼𝐼(𝑡),𝑦^′_𝐽(𝑡)=𝜆𝑃₂(𝑆(𝑡))𝑦_𝑀(𝑡)−𝑑_𝐽𝑦_𝐽(𝑡)−𝑚𝑦_𝐽(𝑡),𝑦^′_𝑀(𝑡)=𝑚𝑦_𝐽(𝑡)−𝑑_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡),(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥Δ𝑆=−𝑃_𝑆𝑆(𝑡),Δ𝐼(𝑡)=−𝑃_𝐼𝐼(𝑡),Δ𝑦_𝐽(𝑡)=−𝑃_𝐽𝑦_𝐽(𝑡),Δ𝑦_𝑀(𝑡)=−𝑃_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡)(𝑡)𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,Δ𝑆=0,Δ𝐼=𝛿_𝐼,Δ𝑦_𝐽(𝑡)=𝛿_𝐽,Δ𝑦_𝑀(𝑡)=𝛿_𝑀,𝑡=𝑛𝑇,𝑛∈ℕ,(1) where 𝑥(𝑡) represents the density of the crop at time 𝑡, 𝑆(𝑡) represents the density of the susceptible pest at time 𝑡, 𝐼(𝑡) represents the density of the infective pest at time 𝑡, 𝑦_𝐽(𝑡) and 𝑦_𝑀(𝑡) represent the density of the immature and mature predator at time 𝑡, respectively; 𝑟 is the logistic intrinsic growth rate of the crop in the absence of the susceptible pest, 𝐾 is its carrying capacity; 0<𝛽, 𝜆≤1 represent the conversion rate at which ingested preys in excess of what is needed for maintenance is translated into predator population increase; 𝑚 is the rate at which the immature predators become the mature predators. 𝑑_𝑆, 𝑑_𝐼, 𝑑_𝐽, 𝑑_𝑀>0 are the death rates of the susceptible pest population, infective pest population, and of the immature and mature predator population, respectively; Δ𝑥(𝑡)=𝑥(𝑡⁺)−𝑥(𝑡), Δ𝑆(𝑡)=𝑆(𝑡⁺)−𝑆(𝑡), Δ𝐼(𝑡)=𝐼(𝑡⁺)−𝐼(𝑡), Δ𝑦_𝐽(𝑡)=𝑦_𝐽(𝑡⁺)−𝑦_𝐽(𝑡), Δ𝑦_𝑀(𝑡)=𝑦_𝑀(𝑡⁺)−𝑦_𝑀(𝑡); 𝑇 is the period of the impulsive effect; 𝛿(0≤𝛿<(1−𝑒^−𝑟𝑇)/2) is the harvesting rate of crop population; 0≤𝑃_𝑆, 𝑃_𝐼, 𝑃_𝐽, 𝑃_𝑀<1 denote the transfer rate of susceptible pest population, infective pest population, immature and mature predator population at every impulsive period (𝑛+𝜏−1)𝑇(𝑛∈ℕ,0<𝜏<1), respectively; 𝛿_𝐼, 𝛿_𝐽, 𝛿_𝑀>0 represents the amount of infective pests, immature and mature predators, respectively, which are released at every impulsive period 𝑛𝑇(𝑛∈ℕ), respectively; Also, 𝑃₁(⋅), 𝑃₂(⋅), 𝑔(⋅)∈𝐻, here 𝐻={𝑓∶𝑅→𝑅∣𝑓(0)=0, 𝑓^′(𝑥)>0 and 𝑓^′′(𝑥)≤0 for all 𝑥>0}.

Some familiar examples of functions 𝑓∈𝐻 in the biological literature include (𝐹1)𝑓₁(𝑥)=𝑎𝑥, with 𝑎>0; (𝐹2)𝑓₂(𝑥)=𝑎𝑥/(1+𝑏𝑥), with 𝑎,𝑏>0; (𝐹3)𝑓₃(𝑥)=𝑎(1−𝑒^−𝑏𝑥), with 𝑎,𝑏>0, where functions (𝐹1) and (𝐹2) are known as Holling type functional responses (see, [16–26]), and (𝐹3) belongs to Ivlev type functional responses (see, [27–30]).

3. Preliminaries

In this section, we will give some definitions and lemmas, which will be useful for our main results. Let ℝ₊=[0,∞) and ℝ⁵₊={𝑋=(𝑥(𝑡),𝑆(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡))∈ℝ⁵∣𝑥(𝑡),𝑆(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡)≥0}. Denote 𝑓=(𝑓_𝑥,𝑓_𝑆,𝑓_𝐼,𝑓_𝐽,𝑓_𝑀)^𝑇  the map defined by the right hand of the first five equations in system (1). Let 𝑉∶ℝ₊×ℝ⁵₊→ℝ₊, then 𝑉∈𝑉₀ if (1)𝑉 is continuous in ((𝑛−1)𝑇,(𝑛+𝜏−1)𝑇]×ℝ⁵₊, ((𝑛+𝜏−1)𝑇,𝑛𝑇]×ℝ⁵₊ and for each 𝑥∈𝑅⁵₊,𝑛∈ℕ, lim_{(𝑡,𝑦)→((𝑛+𝜏−1)𝑇}⁺_,𝑥)𝑉(𝑡,𝑦)=𝑉((𝑛+𝜏−1)𝑇⁺,𝑥) and lim_{(𝑡,𝑦)→(𝑛𝑇}⁺_,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑛𝑇⁺,𝑥) exist. (2)𝑉 is locally Lipschitzian 𝑥.

Definition 1. Letting 𝑉∈𝑉₀, one defines the upper right derivative of 𝑉 with respect to the impulsive differential system (1) at (𝑡,𝑥)∈((𝑛−1)𝑇,(𝑛+𝜏−1)𝑇]×ℝ⁵₊ and ((𝑛+𝜏−1)𝑇,𝑛𝑇]×ℝ⁵₊ by 𝐷⁺𝑉(𝑡,𝑥)=limsup_ℎ→0+1ℎ𝑉𝑡+ℎ,𝑥+ℎ𝑓(𝑡,𝑥)−𝑉(𝑡,𝑥).(2)

Definition 2. The system (1) is said to be permanent if there are positive constants 𝑚,𝑀>0 and a finite time 𝑇₀ such that all solutions of (1) with initial values 𝑥(0⁺), 𝑆(0⁺), 𝐼(0⁺), 𝑦_𝐽(0⁺), 𝑦_𝑀(0⁺), 𝑚≤𝑥(𝑡), 𝑆(𝑡), 𝐼(𝑡), 𝑦_𝐽(𝑡), 𝑦_𝑀(𝑡)≤𝑀 hold for all 𝑡≥𝑇₀, where 𝑚 and 𝑀  are independent of initial value, 𝑇₀ may depend on initial value.

Remark 3. The global existence and uniqueness of system (1) is guaranteed by the smoothness properties of 𝑓 (for details, see [13, 14]).

Lemma 4 (see [15]). Let 𝑉∶ℝ₊×ℝ^𝑛→ℝ^𝑚₊ satisfy 𝑉_𝑖∈𝑉₀,𝑖=1,2,…,𝑚, and assume that 𝐷⁺𝑉((𝑡≤(≥)𝑔(((𝑡()𝑉𝑡𝑡,𝑥))𝑡,𝑉𝑡,𝑥))),𝑡≠𝑘+𝜏−1𝑇,𝑘𝑇,𝑡,𝑥⁺≤(≥)𝜓^𝜏_𝑘(𝑉((𝑡()𝑉𝑡𝑡,𝑥))),𝑡=𝑘+𝜏−1𝑇,𝑡,𝑥⁺≤(≥)𝜓_𝑘(𝑉((𝑡𝑥0𝑡,𝑥))),𝑡=𝑘𝑇,𝑘∈ℕ,₊=𝑥₀,(3) where 𝑔∶ℝ₊×ℝ^𝑚₊→ℝ^𝑚₊ is continuous in ((𝑘−1)𝑇,(𝑘+𝜏−1)𝑇]×ℝ^𝑚 and ((𝑘+𝜏−1)𝑇,𝑘𝑇]×ℝ^𝑚, for each 𝑝∈ℝ^𝑚,𝑘=1,2,…, the limit lim_{(𝑡,𝑞)→((𝑘+𝜏−1)𝑇}⁺_,𝑝)𝑔(𝑡,𝑞)=𝑔((𝑘+𝜏−1)𝑇⁺,𝑝) and lim_{(𝑡,𝑞)→((𝑘−1)𝑇}⁺_,𝑝)𝑔(𝑡,𝑞)=𝑔((𝑘−1)𝑇⁺,𝑝) exists. 𝑔(𝑡,𝑞) is quasimonotone nondecreasing in 𝑞. 𝜓^𝜏_𝑘,𝜓_𝑘∶ℝ^𝑚₊→ℝ^𝑚₊ is nondecreasing for all 𝑘∈ℕ. Let 𝜃(𝑡) be the maximal  (𝑚𝑖𝑛𝑖𝑚𝑎𝑙) solution of the following impulsive differential equation on [0,∞): 𝑤^′(𝑡)((𝑡()𝑤𝑡=𝑔𝑡,𝑤)),𝑡≠𝑘+𝜏−1𝑇,𝑘𝑇,₊=𝜓^𝜏_𝑘(𝑤(𝑡()𝑤𝑡)),𝑡=𝑘+𝜏−1𝑇,₊=𝜓_𝑘(𝑤(𝑡𝑤0)),𝑡=𝑘𝑇,𝑘∈ℕ,₊=𝑤₀.(4)
Then for any solution 𝑥(𝑡) of the system (3), 𝑉(0⁺,𝑥₀)≤  (≥)𝑤₀ implies that 𝑉(𝑡,𝑥(𝑡))≤(≥)𝜃(𝑡) for all 𝑡≥0.

Lemma 5 (see [13, 15]). Consider the following system: 𝑣^′(𝑡)≤(≥)𝑝(𝑡)𝑣(𝑡)+𝑞(𝑡),𝑡≠𝑡_𝑘,𝑣𝑡⁺_𝑘≤(≥)𝑑_𝑘𝑣𝑡_𝑘+𝑏_𝑘,𝑡=𝑡_𝑘𝑣0,𝑘∈ℕ,₊≤(≥)𝑣₀,(5) where 𝑝,𝑞∈𝑃𝐶(ℝ₊,ℝ) and 𝑑_𝑘≥0, 𝑣₀ and 𝑏_𝑘 are constants. Suppose that (𝐴1) the sequence 𝑡_𝑘 satisfies 0≤𝑡₁≤𝑡₂<⋯, with lim_𝑡→∞𝑡_𝑘=∞; (𝐴2)𝑣∈𝑃𝐶^′(ℝ₊,ℝ) and 𝑣(𝑡) is left-continuous at 𝑡_𝑘,𝑘∈ℕ.
Then, for 𝑡>0, 𝑣(𝑡)≤(≥)𝑣₀𝑒^{∫𝑡0𝑝(𝑠)𝑑𝑠}_{0<𝑡𝑘<𝑡}𝑑_𝑘+_{0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝑑_𝑗𝑒^{∫𝑡𝑡𝑘𝑝(𝑠)𝑑𝑠}𝑏_𝑘+^𝑡_{0𝑠<𝑡𝑘<𝑡}𝑑_𝑘𝑒^{∫𝑡𝑠𝑝(𝜏)𝑑𝜏}𝑞(𝑠)𝑑𝑠.(6)

Lemma 6. There exists a constant 𝑀=max{(1/𝜆)((𝐿/𝑑)+(𝜌𝑒^𝑑𝑇/(𝑒^𝑑𝑇−1))),𝐾}>0, such that 𝑥(𝑡), 𝑆(𝑡), 𝐼(𝑡), 𝑦_𝐽(𝑡), 𝑦_𝑀(𝑡)≤𝑀 for each solution of (1) with 𝑡 large enough.

Proof. Since 𝑥^′(𝑡)≤𝑟𝑥(1−(𝑥(𝑡)/𝐾)),  then 𝑥^′(𝑡)|_{𝑥(𝑡)=𝐾}≤0, and 𝑥((𝑛+𝜏−1)𝑇⁺)≤𝑥((𝑛+𝜏−1)𝑇)(0<𝛿<1), so 𝑥(𝑡)≤𝐾 for 𝑡 large enough. Let us define 𝑉(𝑡)∈𝑉₀ by 𝑉(𝑡)=𝜆𝛽𝑥(𝑡)+𝜆𝑆(𝑡)+𝜆𝐼(𝑡)+𝑦_𝐽(𝑡)+𝑦_𝑀(𝑡) and denote 𝑑=min{𝑑_𝑆,𝑑_𝐼,𝑑_𝐽,𝑑_𝑀}. Then, it is obvious that 𝑑𝑉(𝑡)(𝑡)()𝑥(𝑡)−𝑑𝑡+𝑑𝑉≤𝜆𝛽𝑟+𝑑𝜆𝛽𝑟𝑥²(𝑡)𝐾,𝑡≠(𝑛+𝜏−1)𝑇,𝑡≠𝑛𝑇.(7) Since the right-hand side (7) is bounded from above by 𝐿=𝐾𝜆𝛽(𝑟+𝑑)²/4𝑟, it follows that 𝑑𝑉(𝑡)(𝑡)()𝑑𝑡+𝑑𝑉≤𝐿,𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇.(8) When 𝑡=(𝑛+𝜏−1)𝑇 and 𝑡=𝑛𝑇, it is easy to obtain that 𝑉(𝑛+𝜏−1)𝑇⁺)𝑇),𝑉≤𝑉((𝑛+𝜏−1𝑛𝑇⁺=𝑉(𝑛𝑇)+𝜆𝛿_𝐼+𝛿_𝐽+𝛿_𝑀.(9) Then, by Lemma 5, we can obtain that 𝑉(𝑡)≤𝑉(0)𝑒^−𝑑𝑡+^𝑡₀𝐿𝑒^{−𝑑(𝑡−𝑠)}𝑑𝑠+_{0<𝑘𝑇<𝑡}𝜌𝑒^{−𝑑(𝑡−𝑘𝑇)}⟶𝐿𝑑+𝜌𝑒^𝑑𝑇𝑒^𝑑𝑇−1,𝑡⟶∞,(10) where 𝜌=𝜆𝛿_𝐼+𝛿_𝐽+𝛿_𝑀. So it follows that 𝑉(𝑡) is uniformly bounded on [0,∞). The proof is completed.

Lemma 7 (see [31]). Let one consider the following impulsive control subsystem: 𝑥^′(𝑡)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾()(𝑡)(𝑡)(),𝑡≠𝑛+𝜏−1𝑇,Δ𝑥=−𝛿𝑥,𝑡=𝑛+𝜏−1𝑇.(11)
Suppose 𝛿^∗₀=1−𝑒^−𝑟𝑇. Then one has the following results. (1)If 𝛿>𝛿^∗₀, then the trivial periodic solution of system (11) is locally asymptotically stable. (2)If 𝛿<𝛿^∗₀, then the system (11) has a unique positive periodic solution 𝑥^∗(𝑡), which is globally asymptotically stable, where 𝑥^∗(𝑡)=𝐾1−𝛿−𝑒^−𝑟𝑇1−𝛿−𝑒^−𝑟𝑇+𝛿𝑒^{−𝑟(𝑡−(𝑛+𝜏−1)𝑇)},)()𝑇]𝑥𝑡∈((𝑛+𝜏−1𝑇,𝑛+𝜏,𝑛∈ℕ,_∗0⁺=𝑥^∗𝑛𝑇⁺=𝐾(1−𝛿)𝑒^𝑟𝑇−1𝑒^𝑟𝜏𝑇−1+(1−𝛿)𝑒^𝑟𝑇−𝑒^𝑟𝜏𝑇.(12)

Remark 8. From Lemma 7, we have (1𝑎) if 𝛿^∗₀>2𝛿, then 𝑥^∗(𝑡)>𝐾/2 for all 𝑡≥0; (2𝑎) if 𝑡∈((𝑛−1)𝑇,𝑛𝑇], 𝑛∈ℕ, then the periodic solution 𝑥^∗(𝑡) can be rewritten in the form 𝑥^∗(𝑡)=𝐾1−𝛿−𝑒^−𝑟𝑇1−𝛿−𝑒^−𝑟𝑇+𝛿𝑒^{−𝑟(𝑇−𝜏𝑇)}𝑒^{−𝑟(𝑡−(𝑛−1)𝑇)},)()𝑇],𝐾𝑡∈((𝑛−1𝑇,𝑛+𝜏−11−𝛿−𝑒^−𝑟𝑇1−𝛿−𝑒^−𝑟𝑇+𝛿𝑒^{−𝑟(𝑡−(𝑛+𝜏−1)𝑇)},𝑡∈((𝑛+𝜏−1)𝑇,𝑛𝑇],𝑛∈ℕ.(13)

Lemma 9. Let one consider the following impulsive control subsystem: 𝑧^′(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)()(𝑡)𝑧0=𝑎−𝑑𝑧,𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑧=−𝑝𝑧,𝑡=𝑛+𝜏−1𝑇,Δ𝑧=𝛿,𝑡=𝑛𝑇,𝑛∈ℕ,₊=𝑧₀,(14) where 𝑎(𝑡) is a 𝑇-periodic 𝑃𝐶(ℝ₊,ℝ) function. 𝑝,𝑑 are the positive real constants and 𝑝<1. Then system (14) has a unique T-periodic solution 𝑧^∗(𝑡), and for each solution 𝑧(𝑡) of (14), 𝑧(𝑡)→𝑧^∗(𝑡) as 𝑡→∞, where 𝑧^∗(𝑡)=𝑒^{−𝑑(𝑡−(𝑛−1)𝑇)}𝑧^∗0⁺+₀^{𝑡−(𝑛−1)𝑇}𝑎(𝑠)𝑒^𝑑𝑠,)()𝑇],𝑧𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_∗(𝑡)=𝑒^{−𝑑(𝑡−(𝑛−1)𝑇)}𝑧^∗𝜏𝑇⁺𝑒^𝑑𝜏𝑇+^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠,)],𝑧𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗0⁺=1−𝑝∫₀^𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+∫^𝑇_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠𝑒^−𝑑𝑇+𝛿1−1−𝑝𝑒^−𝑑𝑇,𝑧^∗𝜏𝑇⁺=1−𝑝∫₀^𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+𝑒^−𝑑𝑇∫^𝑇_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+𝛿𝑒^{−𝑑𝜏𝑇}1−1−𝑝𝑒^−𝑑𝑇.(15)

Proof. First, it is easy to obtain that 𝑧(𝑡)=𝑒^−𝑑𝑡𝑧0⁺+^𝑡₀𝑎(𝑠)𝑒^𝑑𝑠(],𝑧(𝑡)𝑑𝑠,𝑡∈0,𝜏𝑇=𝑒^{−𝑑(𝑡−𝜏𝑇)}𝑧𝜏𝑇⁺+𝑒^{−𝑑𝑡𝑡}_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠,𝑡∈(𝜏𝑇,𝑇].(16) Since the T-periodicity requirement, we have 𝑧^∗𝜏𝑇⁺=𝑒^{−𝑑𝜏𝑇}𝑧^∗0⁺+₀^𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠,𝑧𝑑𝑠1−𝑝_∗0⁺=𝑒^{−𝑑(𝑇−𝜏𝑇)}𝑧^∗𝜏𝑇⁺+𝑒^{−𝑑𝑇𝑇}_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+𝛿.(17) By (17), we can obtain that 𝑧^∗0⁺=1−𝑝∫₀^𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+∫^𝑇_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠𝑒^−𝑑𝑇+𝛿1−1−𝑝𝑒^−𝑑𝑇,𝑧^∗𝜏𝑇⁺=1−𝑝∫₀^𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+𝑒^−𝑑𝑇∫^𝑇_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠𝑑𝑠+𝛿𝑒^{−𝑑𝜏𝑇}1−1−𝑝𝑒^−𝑑𝑇.(18) So, we will obtain the T-periodic solution of (14): 𝑧^∗(𝑡)=𝑒^{−𝑑(𝑡−(𝑛−1)𝑇)}𝑧^∗0⁺+₀^{𝑡−(𝑛−1)𝑇}𝑎(𝑠)𝑒^𝑑𝑠,)()𝑇],𝑧𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_∗(𝑡)=𝑒^{−𝑑(𝑡−(𝑛−1)𝑇)}𝑧^∗𝜏𝑇⁺𝑒^𝑑𝜏𝑇+^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑎(𝑠)𝑒^𝑑𝑠,)].𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇(19) Let 𝑍(𝑡)=𝑧(𝑡)−𝑧^∗(𝑡). Substituting 𝑍(𝑡) into (14), we have 𝑍^′(𝑡)(𝑡)()(𝑡)(𝑡)()(𝑡)𝑍0=−𝑑𝑍,𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑍=−𝑝𝑍,𝑡=𝑛+𝜏−1𝑇,Δ𝑍=0,𝑡=𝑛𝑇,𝑛∈ℕ,₊=𝑧₀−𝑧^∗0⁺.(20) Then, 𝑍(𝑡)=𝑍(0⁺)𝑒^−𝑑𝑡∏_{0<(𝑛+𝜏−1)𝑇<𝑡}(1−𝑝)→0, as 𝑡→∞. The proof is completed.

4. Main Results

4.1. Local and Global Stability

In this section, we will study the existence and stability of the system (1) susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)). To this purpose, it is seen first that when 𝑆(𝑡)=0, system (1) can be rewritten in the form 𝑥^′(𝑡)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾,𝐼^′(𝑡)=−𝑑_𝐼𝐼(𝑡),𝑦^′_𝐽(𝑡)=−𝑑_𝐽+𝑚𝑦_𝐽(𝑡),𝑦^′_𝑀(𝑡)=𝑚𝑦_𝐽(𝑡)−𝑑_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡),(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥Δ𝐼=−𝑃_𝐼𝐼(𝑡),Δ𝑦_𝐽(𝑡)=−𝑃_𝐽𝑦_𝐽(𝑡),Δ𝑦_𝑀(𝑡)=−𝑃_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡)𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,Δ𝐼=𝛿_𝐼,Δ𝑦_𝐽(𝑡)=𝛿_𝐽,Δ𝑦_𝑀(𝑡)=𝛿_𝑀,𝑡=𝑛𝑇,(21) which describes the dynamics of system (1) in the absence of the susceptible pest population. So, when 𝑡∈((𝑛−1)𝑇,𝑛𝑇](𝑛∈ℕ), we can calculate the T-periodic solution of (21) by Lemmas 7 and 9. It is seen that 𝑥^∗(𝑡)=𝐾1−𝛿−𝑒^−𝑟𝑇1−𝛿−𝑒^−𝑟𝑇+𝛿𝑒^{−𝑟(𝑇−𝜏𝑇)}𝑒^{−𝑟(𝑡−(𝑛−1)𝑇)},)()𝑇],𝐾𝑡∈((𝑛−1𝑇,𝑛+𝜏−11−𝛿−𝑒^−𝑟𝑇1−𝛿−𝑒^−𝑟𝑇+𝛿𝑒^{−𝑟(𝑡−(𝑛+𝜏−1)𝑇)},)],𝐼𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇(22)_∗(𝑡)=𝛿_𝐼𝑒^{−𝑑𝐼(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐼𝑒^{−𝑑𝐼𝑇},)()𝑇],𝛿𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_𝐼1−𝑃_𝐼𝑒^{−𝑑𝐼(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐼𝑒^{−𝑑𝐼𝑇},)],𝑦𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇(23)_∗𝐽(𝑡)=𝛿_𝐽𝑒^{−(𝑚+𝑑𝐽)(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇},)()𝑇],𝛿𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_𝐽1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇},)],𝑦𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇(24)_∗𝑀(𝑡)=𝑒^{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}𝑦^∗_𝑀0⁺(()𝑇,)()𝑇],𝑒+𝐴𝑡−𝑛−1)𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}𝑦^∗_𝑀𝜏𝑇⁺𝑒^{𝑑𝑀𝜏𝑇}(()𝑇),)],𝑦+𝐵𝑡−𝑛−1𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇(25)_∗𝑀0⁺=1−𝑃_𝑀𝐴(𝜏𝑇)+𝐵(𝑇)𝑒^{−𝑑𝑀𝑇}+𝛿_𝑀1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇},𝑦(26)_∗𝑀𝜏𝑇⁺=1−𝑃_𝑀𝐴(𝜏𝑇)+𝑒^{−𝑑𝑀𝑇}𝐵(𝑇)+𝛿_𝑀𝑒^{−𝑑𝑀𝜏𝑇}1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇},(27) where 𝐴(𝑡)=𝑚𝛿_𝐽𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝑡}−11−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑑_𝑀−𝑚+𝑑_𝐽,(],𝐵(𝑡)=𝑡∈0,𝜏𝑇𝑚𝛿_𝐽1−𝑃_𝐽𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝑡}−𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝜏𝑇}1−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑑_𝑀−𝑚+𝑑_𝐽,𝑡∈(𝜏𝑇,𝑇].(28) To discuss the locally asymptotical stability of the susceptible pest-eradication periodic solution, we now introduce the Floquet theory for a linear impulsive control system: 𝜔^′(𝑡)=𝐴(𝑡)𝜔(𝑡),𝑡≠𝜏_𝑘,Δ𝜔(𝑡)=𝐵_𝑘𝜔(𝑡),𝑡=𝜏_𝑘,𝑘∈ℕ,(29) under the following conditions: H1:𝐴(⋅)∈PC(ℝ,𝑀_𝑛(ℝ)) and 𝐴(𝑡+𝑇)=𝐴(𝑡) for 𝑡≥0. H2:𝐵_𝑘∈𝑀_𝑛,  det(𝐼_𝑛+𝐵_𝑘)≠0, 𝜏_𝑘<𝜏_𝑘+1 for 𝑘∈ℕ, and 𝐼_𝑛 denotes the 𝑛×𝑛 real identity matrix. H3: There is a 𝑞∈ℕ such that 𝐵_𝑘+𝑞=𝐵_𝑘,  𝜏_𝑘+𝑞=𝜏_𝑘+𝑇 for 𝑘∈ℕ.

Let Ψ(𝑡) be a fundamental matrix of (29), then there is a unique reversible matrix 𝑀∈𝑀_𝑛(ℝ) such that Ψ(𝑡+𝑇)=Ψ(𝑡)𝑀 for all 𝑡∈ℝ, which is called the monodromy matrix of (29) corresponding to Ψ. All monodromy matrices of (29) are similar and they have the same eigenvalues 𝜆₁,𝜆₂,…,𝜆_𝑛, which are called the Floquet multipliers of (29).

Lemma 10 (see [13] (Floquet theory)). Let the conditions H1–H3 hold. Then system (29) have the following properties (1)stable if and only if all Floquet multipliers 𝜆_𝑖(1≤𝑖≤𝑛) of (29) satisfy |𝜆_𝑖|≤1 and moreover, to those 𝜆_𝑖 for which |𝜆_𝑖|=1, there correspond simple elementary divisors; (2)asymptotically stable if and only if all Floquet multipliers 𝜆_𝑖(1≤𝑖≤𝑛) of (29) satisfy |𝜆_𝑖|<1; (3)unstable if there is a Floquet multipliers 𝜆_𝑖(1≤𝑖≤𝑛) of (29) such that |𝜆_𝑖|>1.

In the following, we present two main results with the locally and globally asymptotical stability of the susceptible pest-eradication periodic solution (𝑥^∗(𝑡), 0, 𝐼^∗(𝑡), 𝑃^∗_𝐽(𝑡), 𝑃^∗_𝑀(𝑡)).

Theorem 11. If 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)𝑑𝑡−^𝑇₀𝑔𝐼^∗(𝑡)𝑑𝑡−𝑃^′₂(0)^𝑇₀𝑦^∗_𝑀(𝑡)𝑑𝑡−𝑑_𝑆1𝑇<ln1−𝑃_𝑆,(30) then the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (1) is locally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑆(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡)) be any solution of system (1). We define error 𝑒₁(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡),𝑒₂(𝑡)=𝑆(𝑡),𝑒₃(𝑡)=𝐼(𝑡)−𝐼^∗(𝑡),𝑒₄(𝑡)=𝑦_𝐽(𝑡)−𝑦^∗_𝐽(𝑡),𝑒₅(𝑡)=𝑦_𝑀(𝑡)−𝑦^∗_𝑀(𝑡). The linearized system of (1) at (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) is 𝑒^′₁(𝑡)=𝑥𝑟−2𝑟^∗(𝑡)𝐾𝑒₁(𝑡)−𝑃₁𝑥^∗(𝑡)𝑒₂(𝑡),𝑒^′₂(𝑡)=𝛽𝑃₁𝑥^∗(𝑡)−𝑔𝐼^∗(𝑡)−𝑃^′₂(0)𝑦^∗_𝑀(𝑡)−𝑑_𝑆𝑒₂(𝑡),𝑒^′₃(𝑡)=𝑔𝐼^∗(𝑡)𝑒₂(𝑡)−𝑑_𝐼𝑒₃(𝑡),𝑒^′₄(𝑡)=𝜆𝑃^′₂(0)𝑦^∗_𝑀(𝑡)𝑒₂(𝑡)−𝑑_𝐽+𝑚𝑒₄(𝑡),𝑒^′₅(𝑡)=𝑚𝑒₄(𝑡)−𝑑_𝑀𝑒₅(𝑡),𝑡≠(𝑛+𝜏−1)𝑇,𝑡≠𝑛𝑇,Δ𝑒₁(𝑡)=−𝛿𝑒₁(𝑡),Δ𝑒₂(𝑡)=−𝑃_𝑆𝑒₂(𝑡),Δ𝑒₃(𝑡)=−𝑃_𝐼𝑒₃(𝑡),Δ𝑒₄(𝑡)=−𝑃_𝐽𝑒₄(𝑡),Δ𝑒₅(𝑡)=−𝑃_𝑀𝑒₅(𝑡),𝑡=(𝑛+𝜏−1)𝑇,Δ𝑒₁(𝑡)=Δ𝑒₂(𝑡)=Δ𝑒₃(𝑡)=Δ𝑒₄(𝑡)=Δ𝑒₅(𝑡)=0,𝑡=𝑛𝑇.(31) Let Ψ(𝑡) be the fundamental matrix of (31), then Ψ(𝑡) satisfies 𝑑Ψ(𝑡)=𝑥𝑑𝑡𝑟−2𝑟^∗(𝑡)𝐾𝑃₁𝑥^∗(𝑡)0000𝛽𝑃₁𝑥^∗(𝑡)−𝑔𝐼^∗(𝑡)−𝑃^′₂(0)𝑦^∗_𝑀(𝑡)−𝑑_𝑆𝐼0000𝑔^∗(𝑡)−𝑑_𝐼000𝜆𝑃^′₂(0)𝑦^∗_𝑀(𝑡)0−𝑑_𝐽+𝑚0000𝑚−𝑑_𝑀Ψ(𝑡).(32)Then, a fundamental matrix Ψ(𝑡)(Ψ(0)=𝐼₄) of (31) isΨ(𝑡)=𝑒^{∫𝑡0(𝑟−2𝑟(𝑥∗(𝑠)/𝐾))𝑑𝑠}𝜙₁₂(𝑡)0000𝑒^{∫𝑡0(𝛽𝑃1(𝑥∗(𝑠))−𝑔(𝐼∗(𝑠))−𝑃′2(0)𝑦∗𝑀(𝑠)−𝑑𝑆)𝑑𝑠}0000𝜙₃₂(𝑡)𝑒^{−𝑑𝐼𝑡}000𝜙₄₂(𝑡)0𝑒^{−(𝑑𝐽+𝑚)𝑡}00𝜙₅₂(𝑡)0𝜙₅₄(𝑡)𝑒^{−𝑑𝑀𝑡},(33)where 𝜙₁₂(𝑡)=−𝑒^{−∫𝑡0(𝑟−2𝑟(𝑥∗(𝑠)/𝐾))𝑑𝑠}×^𝑡₀𝑃₁𝑥^∗(𝑠)𝑒^{∫𝑠0(𝛽𝑃1(𝑥∗(𝜉))−𝑔(𝐼∗(𝜉))−𝑃′2(0)𝑦∗𝑀(𝜉)−𝑑𝑆)𝑑𝜉}×𝑒^{∫𝑠0(𝑟−2𝑟(𝑥∗(𝜉)/𝐾))𝑑𝜉}𝜙𝑑𝑠,₃₂(𝑡)=𝑒^{−𝑑𝐼𝑡𝑡}₀𝑔𝐼^∗(𝑠)𝑒^{∫𝑠0(𝛽𝑃1(𝑥∗(𝜉))−𝑔(𝐼∗(𝜉))−𝑃′2(0)𝑦∗𝑀(𝜉)−𝑑𝑆)𝑑𝜉}×𝑒^𝑑𝐼𝑠𝜙𝑑𝑠,₄₂(𝑡)=𝑒^{−𝑑𝐽𝑡𝑡}₀𝜆𝑃^′₂(0)𝑦^∗_𝑀(𝑠)𝑒^{∫𝑠0(𝛽𝑃1(𝑥∗(𝜉))−𝑔(𝐼∗(𝜉))−𝑃′2(0)𝑦∗𝑀(𝜉)−𝑑𝑆)𝑑𝜉}×𝑒^𝑑𝐽𝑠𝜙𝑑𝑠,₅₂(𝑡)=𝑒^{−𝑑𝑀𝑡𝑡}₀𝑚𝜙₄₂(𝑠)𝑒^𝑑𝑀𝑠𝜙𝑑𝑠,₅₄(𝑡)=𝑚𝑒^{−(𝑑𝐽+𝑚)𝑡}−𝑒^{−𝑑𝑀𝑡}𝑑_𝑀−𝑑_𝐽.+𝑚(34) The resetting impulsive condition of (31) becomes 𝑒₁(𝑛+𝜏−1)𝑇⁺𝑒₂(𝑛+𝜏−1)𝑇⁺𝑒₃(𝑛+𝜏−1)𝑇⁺𝑒₄(𝑛+𝜏−1)𝑇⁺𝑒₅(𝑛+𝜏−1)𝑇⁺=1−𝛿000001−𝑃_𝑆000001−𝑃_𝐼000001−𝑃_𝐽000001−𝑃_𝑀×𝑒₁)𝑇)𝑒((𝑛+𝜏−1₂)𝑇)𝑒((𝑛+𝜏−1₃)𝑇)𝑒((𝑛+𝜏−1₄)𝑇)𝑒((𝑛+𝜏−1₅)𝑇),𝑒((𝑛+𝜏−1₁𝑛𝑇⁺𝑒₂𝑛𝑇⁺𝑒₃𝑛𝑇⁺𝑒₄𝑛𝑇⁺𝑒₅𝑛𝑇⁺=10000𝑒01000001000001000001¹()𝑒𝑛𝑇₂()𝑒𝑛𝑇₃()𝑒𝑛𝑇₄()𝑒𝑛𝑇₅().𝑛𝑇(35) Then, it is easy to obtain all eigenvalues of 𝑀=1−𝛿000001−𝑃_𝑆000001−𝑃_𝐼000001−𝑃_𝐽000001−𝑃_𝑀×Ψ(𝑇).1000001000001000001000001(36) We have 𝜆₁=(1−𝛿)𝑒^{∫𝑇0(𝑟−2𝑟(𝑥∗(𝑠)/𝐾))𝑑𝑠}, 𝜆₂=(1−𝑃_𝑆)𝑒^{∫𝑇0(𝛽𝑃1(𝑥∗(𝑠))−𝑔(𝐼∗(𝑠))−𝑃′2(0)𝑦∗𝑀(𝑠)−𝑑𝑆)𝑑𝑠}, 𝜆₃=(1−𝑃_𝐼)𝑒^{−𝑑𝐼𝑇}<1, 𝜆₄=(1−𝑃_𝐽)𝑒^{−(𝑑𝐽+𝑚)𝑇}<1 and 𝜆₅=(1−𝑃_𝑀)𝑒^{−𝑑𝑀𝑇}<1. Since 𝑥^∗(𝑡)>(𝐾/2), so 𝜆₁<1. By the condition (30), we have 𝜆₂<1. Therefore, according to Lemma 10, the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (1) is locally asymptotically stable. The proof is completed.

Theorem 12. If 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)𝑑𝑡−^𝑇₀𝑔𝐼^∗(𝑡)𝑑𝑡−min_{0≤𝜛≤𝑈𝑆}𝑃^′₂(𝜛)^𝑇₀𝑦^∗_𝑀(𝑡)𝑑𝑡−𝑑_𝑆1𝑇<ln1−𝑃_𝑆,(37) where 𝑈_𝑆 is an ultimate boundedness constant for S, then the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (1) is globally asymptotically stable.

Proof. Since 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)𝑑𝑡−^𝑇₀𝑔𝐼^∗(𝑡)𝑑𝑡−min_{0≤𝜛≤𝑈𝑆}𝑃^′₂(𝜛)^𝑇₀𝑦^∗_𝑀(𝑡)𝑑𝑡−𝑑_𝑆1𝑇<ln1−𝑃_𝑆,(38) we can choose an 𝜀 small enough such that 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)+𝜀𝑑𝑡−^𝑇₀𝑔𝐼^∗(𝑡)−𝜀𝑑𝑡−min_{0≤𝜛≤𝑈𝑆}𝑃^′₂(𝜛)^𝑇₀𝑦^∗_𝑀(𝑡)−𝜀𝑑𝑡−𝑑_𝑆𝑇1−ln1−𝑃_𝑆=𝜖<0,(39) where 𝑦^∗_𝑀(𝑡) is defined in the following. According to system (1), we have 𝑥^′(𝑡)(𝑡)𝑥(𝑡)≤𝑟𝑥1−𝐾,𝐼^′(𝑡)≥−𝑑_𝐼𝐼(𝑡),𝑦^′_𝐽(𝑡)≥−𝑑_𝐽𝑦_𝐽(𝑡)−𝑚𝑦_𝐽(𝑡),𝑦^′_𝑀(𝑡)=𝑚𝑦_𝐽(𝑡)−𝑑_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡),(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥Δ𝐼=−𝑃_𝐼𝐼(𝑡),Δ𝑦_𝐽(𝑡)=−𝑃_𝐽𝑦_𝐽(𝑡),Δ𝑦_𝑀(𝑡)=−𝑃_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡)𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,Δ𝐼=𝛿_𝐼,Δ𝑦_𝐽(𝑡)=𝛿_𝐽,Δ𝑦_𝑀(𝑡)=𝛿_𝑀,𝑡=𝑛𝑇,𝑛∈ℕ.(40) From the first equation of system (40), we obtain the following comparison system: 𝜈^′(𝑡)(𝑡)𝜈(𝑡)=𝑟𝜈1−𝐾()(𝑡)(𝑡)(),𝑡≠𝑛+𝜏−1𝑇,Δ𝜈=−𝛿𝜈,𝑡=𝑛+𝜏−1𝑇.(41) By Lemma 7, system (41) has a positive periodic solution 𝜈^∗(𝑡), and for any solution 𝜈(𝑡) of (41), 𝜈(𝑡)→𝜈^∗(𝑡) as 𝑡 large enough, where 𝜈^∗(𝑡)=𝑥^∗(𝑡). Then, according to Lemmas 4 and 7, there exists a positive constant 𝑛^∗ such that for all 𝑡≥𝑛^∗𝑇𝑥(𝑡)≤𝑥^∗(𝑡)+𝜀.(42) Let us define 𝑉(𝑡)=(𝑉₁(𝑡),𝑉₂(𝑡))^𝑇∈𝐶[ℝ₊×ℝ²,ℝ²₊] and 𝑉_𝑖(𝑡)∈𝑉₀,(𝑖=1,2), where 𝑉₁(𝑡)=𝐼(𝑡), 𝑉₂(𝑡)=𝑦_𝐽(𝑡). Then, we have 𝑉^′(𝑡)≥−𝑑_𝐼𝐼(𝑡)−𝑑_𝐽+𝑚𝑦_𝐽(𝑡)=−𝑑_𝐼00−𝑑_𝐽𝑉(𝑡),()𝑉()𝑇+𝑚𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,(43)𝑛+𝜏−1⁺=1−𝑃_𝐼001−𝑃_𝐽𝑉)𝑇),𝑉((𝑛+𝜏−1𝑛𝑇⁺=𝑉()+10𝛿01𝑛𝑇^𝐼𝛿_𝐽,𝑉0⁺=𝐼0⁺,𝑦_𝐽0⁺.(44) Then, the multicomparison system of (43) is 𝑤^′(𝑡)(𝑡)()𝑤𝑡=𝐴𝑤,𝑡≠𝑛+𝜏−1𝑇,𝑛𝑇,₊(𝑡)()𝑤𝑡=𝐵𝑤,𝑡=𝑛+𝜏−1𝑇,₊=𝐼₂𝑤(𝑡)𝑤0+𝐶,𝑡=𝑛𝑇,₊=𝑉0⁺,(45) where 𝐴=^−𝑑𝐼0_{0−(𝑑𝐽+𝑚)},𝐵=^{1−𝑃𝐼0}_{01−𝑃𝐽}, and 𝐶=𝛿_𝐼𝛿_𝐽.
By Lemma 9, it is easy to obtain a periodic solution (𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡))^𝑇 of system (45). Then, according to Lemmas 4 and 9, one may find 𝑛^∗₀(𝑛^∗₀>𝑛^∗) such that for all 𝑡≥𝑛^∗₀𝑇𝐼(𝑡)≥𝐼^∗(𝑡)−𝜀,𝑦_𝐽(𝑡)≥𝑦^∗_𝐽(𝑡)−𝜀.(46) From the fourth equation of system (40), we have 𝑦^′_𝑀(𝑡)≥𝑚(𝑦^∗_𝐽(𝑡)−𝜀)−𝑑_𝑀𝑦_𝑀(𝑡), by Lemmas 4 and 9, there exists 𝑛^∗₁(𝑛^∗₁>𝑛^∗₀) such that for all 𝑡≥𝑛^∗₁𝑇𝑦_𝑀(𝑡)≥𝑦^∗_𝑀(𝑡)−𝜀,(47) where 𝑦^∗_𝑀(𝑡)=𝑒^{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝑀(0⁺)+𝑚∫₀^{𝑡−(𝑛−1)𝑇}𝑦^∗_𝐽(𝑡)−𝜀𝑒^𝑑𝑀𝑠,)()𝑇],𝑒𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝑀(𝜏𝑇⁺)𝑒^{𝑑𝑀𝜏𝑇}+𝑚∫^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑦^∗_𝐽(𝑡)−𝜀𝑒^𝑑𝑀𝑠,)],𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇(48)𝑦^∗_𝑀(0⁺)=𝑚1−𝑃_𝑀0^𝜏𝑇𝑦^∗_𝐽(𝑡)−𝜀𝑒^𝑑𝑀𝑠+𝑑𝑠_𝑇𝜏𝑇𝑦^∗_𝐽(𝑡)−𝜀𝑒^𝑑𝑀𝑠𝑑𝑠𝑒^{−𝑑𝑀𝑇}+𝛿_𝑀×1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇−1},𝑦^∗_𝑀(𝜏𝑇⁺)=1−𝑃_𝑀𝑚₀^𝜏𝑇𝑦^∗_𝐽(𝑡)−𝜀𝑒^𝑑𝑀𝑠𝑑𝑠+𝑚𝑒^{−𝑑𝑀𝑇𝑇}_𝜏𝑇𝑦^∗_𝐽(𝑡)−𝜀𝑒^𝑑𝑀𝑠𝑑𝑠+𝛿_𝑀×𝑒^{−𝑑𝑀𝜏𝑇}×1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇−1}.(49) Therefore, 𝑆^′(𝑡)≤𝛽𝑃₁𝑥^∗(𝑡)+𝜀−𝑔𝐼^∗(𝑡)−𝜀−min_{0≤𝜛≤𝑈𝑆}𝑃^′₂(𝜛)𝑦^∗_𝑀(𝑡)−𝜀−𝑑_𝑆𝑆(𝑡),()(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝑃_𝑆𝑆(𝑡)()(𝑡),𝑡=𝑛+𝜏−1𝑇,Δ𝑆=0,𝑡=𝑛𝑇,(50) for 𝑡≥𝑛^∗₁𝑇. Let 𝑁∈ℕ and (𝑁+𝜏−1)≥𝑛^∗₁. Integrating (50) on ((𝑛+𝜏−1)𝑇,(𝑛+𝜏)𝑇],𝑛≥𝑁, we have 𝑆)𝑇))𝑇)((𝑛+𝜏≤𝑆((𝑛+𝜏−11−𝑃_𝑆×𝑒^{∫(𝑛+𝜏)𝑇(𝑛+𝜏−1)𝑇(𝛽𝑃1(}𝑥^∗(𝑡)_{−𝜀)−𝑔(}𝐼^∗(𝑡)_{+𝜀)−min𝑆0≤𝜛≤𝑈𝑃}^′_2(𝜛)(𝑦^∗_𝑀(𝑡)_{+𝜀)−𝑑𝑆)𝑑𝑡}=𝑆((𝑛+𝜏−1)𝑇)𝑒^𝜖.(51) Then 𝑆(𝑡)≤𝑆((𝑛+𝜏)𝑇)𝑒^𝑘𝜖 for 𝑡∈((𝑛+𝜏+𝑘)𝑇,(𝑛+𝜏+𝑘+1)𝑇]. Since 𝜖<0, we can easily get 𝑆(𝑡)→0, as 𝑡→∞. In the following, we prove 𝑥(𝑡)→𝑥^∗(𝑡),𝐼(𝑡)→𝐼^∗(𝑡),𝑦_𝐽(𝑡)→𝑦^∗_𝐽(𝑡),𝑦_𝑀(𝑡)→𝑦^∗_𝑀(𝑡), as 𝑡→∞. Give 𝜀₀>0 small enough (𝜀₀<(𝑟/𝑃^′₁(0))), there must exist 𝑛^∗₂(𝑛^∗₂>𝑛^∗₁) such that 𝑆(𝑡)<𝜀₀, for 𝑡≥𝑛^∗₂𝑇. Then, we have 𝑥^′(𝑡)≥𝑟−𝑃^′₁(0)𝜀₀𝑥(𝑡(𝑡))1−𝑟𝑥𝐾𝑟−𝑃^′₁(0)𝜀₀,𝐼^′(𝑡)≤−𝑑_𝐼−𝑔^′(0)𝜀₀𝐼(𝑡),𝑦^′_𝐽(𝑡)≤𝜆𝑃₂𝜀₀𝑀−𝑑_𝐽+𝑚𝑦_𝐽(𝑡),𝑦^′_𝑀(𝑡)=𝑚𝑦_𝐽(𝑡)−𝑑_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡),(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥Δ𝐼=−𝑃_𝐼𝐼(𝑡),Δ𝑦_𝐽(𝑡)=−𝑃_𝐽𝑦_𝐽(𝑡),Δ𝑦_𝑀(𝑡)=−𝑃_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡)𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,Δ𝐼=𝛿_𝐼,Δ𝑦_𝐽(𝑡)=𝛿_𝐽,Δ𝑦_𝑀(𝑡)=𝛿_𝑀,𝑡=𝑛𝑇.(52) Analyzing (52) with similarity as (40), there exists 𝑛^∗₃(𝑛^∗₃>𝑛^∗₂) such that for all 𝑡≥𝑛^∗₃𝑇𝑥(𝑡)≥𝑥^∗(𝑡)−𝜀,𝐼(𝑡)≤𝐼^∗𝑦(𝑡)+𝜀,_𝐽(𝑡)≤𝑦^∗_𝐽(𝑡)+𝜀,𝑦_𝑀(𝑡)≤𝑦^∗_𝑀(𝑡)+𝜀,(53) where 𝑥^∗=𝐾(𝑡)𝑟−𝑃^′₁(0)𝜀₀1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝜀0)𝑇}×𝑟1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝜀0)𝑇}+𝛿𝑒^{−(𝑟−𝑃′1(0)𝜀0)(𝑇−𝜏𝑇)}𝑒^{−(𝑟−𝑃′1(0)𝜀0)(𝑡−(𝑛−1)𝑇)−1},)()𝑇],𝐾𝑡∈((𝑛−1𝑇,𝑛+𝜏−1𝑟−𝑃^′₁(0)𝜀₀1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝜀0)𝑇}𝑟1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝜀0)𝑇}+𝛿𝑒^{−(𝑟−𝑃′1(0)𝜀0)(𝑡−(𝑛+𝜏−1)𝑇)},)],𝐼𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗=𝛿(𝑡)_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝜀0)(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝜀0)𝑇},)()𝑇],𝛿𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_𝐼1−𝑃_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝜀0)(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝜀0)𝑇},)],𝑦𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗𝐽=𝑒(𝑡)_{−(𝑚+𝑑𝐽)(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝐽(0⁺)+𝜆𝑃₂𝜀₀𝑀∫₀^{𝑡−(𝑛−1)𝑇}𝑒^{(𝑚+𝑑𝐽)𝑠},)()𝑇],𝑒𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_{−(𝑚+𝑑𝐽)(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝐽𝜏𝑇⁺𝑒^{(𝑚+𝑑𝐽)𝜏𝑇}+𝜆𝑃₂𝜀₀𝑀∫^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠},)],𝑦𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗𝐽(0⁺)=𝜆𝑃₂𝜀₀𝑀1−𝑃_𝐽0^𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠+^𝑇_𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠×𝑒^{−(𝑚+𝑑𝐽)𝑇}+𝛿_𝐽1−(1−𝑃_𝐽)𝑒^{−(𝑚+𝑑𝐽)𝑇−1},𝑦^∗_𝐽(𝜏𝑇⁺)=1−𝑃_𝐽𝜆𝑃₂𝜀₀𝑀₀^𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠+𝑒^{−(𝑚+𝑑𝐽)𝑇}𝜆𝑃₂𝜀₀𝑀^𝑇_𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠+𝛿_𝐽×𝑒^{−(𝑚+𝑑𝐽)𝜏𝑇}1−(1−𝑃_𝐽)𝑒^{−(𝑚+𝑑𝐽)𝑇−1},𝑦^∗_𝑀=𝑒(𝑡)_{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝑀(0⁺)+𝑚∫₀^{𝑡−(𝑛−1)𝑇}𝑦^∗_𝐽(𝑡)+𝜀𝑒^𝑑𝑀𝑠,)()𝑇],𝑒𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝑀𝜏𝑇⁺𝑒^{𝑑𝑀𝜏𝑇}+𝑚∫^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀𝑒^𝑑𝑀𝑠,)],𝑦𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗𝑀(0⁺)=𝑚1−𝑃_𝑀0^𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀𝑒^𝑑𝑀𝑠𝑑𝑠+^𝑇_𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀𝑒^𝑑𝑀𝑠𝑑𝑠𝑒^{−𝑑𝑀𝑇}+𝛿_𝑀×1−(1−𝑃_𝑀)𝑒^{−𝑑𝑀𝑇−1},𝑦^∗_𝑀(𝜏𝑇⁺)=1−𝑃_𝑀×𝑚₀^𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀𝑒^𝑑𝑀𝑠𝑑𝑠+𝑚𝑒^{−𝑑𝑀𝑇}×^𝑇_𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀𝑒^𝑑𝑀𝑠𝑑𝑠+𝛿_𝑀𝑒^{−𝑑𝑀𝜏𝑇}×1−(1−𝑃_𝑀)𝑒^{−𝑑𝑀𝑇−1}.(54) Letting 𝜀,𝜀₀→0, we have 𝑦^∗_𝑀(𝑡)→𝑦^∗_𝑀(𝑡),𝑥^∗(𝑡)→𝑥^∗(𝑡),𝐼^∗(𝑡)→𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡)→𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)→𝑦^∗_𝑀(𝑡). Together with (42), (46), (47), and (53), we get 𝑥(𝑡)→𝑥^∗(𝑡),𝐼(𝑡)→𝐼^∗(𝑡),𝑦_𝐽(𝑡)→𝑦^∗_𝐽(𝑡),𝑦_𝑀(𝑡)→𝑦^∗_𝑀(𝑡) as 𝑡→∞. Therefore, the susceptible pest-eradication periodic solution (𝑥^∗(𝑡), 0, 𝐼^∗(𝑡), 𝑦^∗_𝐽(𝑡), 𝑦^∗_𝑀(𝑡)) is globally attractive. Since (37) implies (30), it follows from Theorem 11 that (𝑥^∗(𝑡), 0, 𝐼^∗(𝑡), 𝑦^∗_𝐽(𝑡)) is locally asymptotically stable. So, the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (1) is globally asymptotically stable. The proof is completed.

4.2. Permanence

Next, we will discuss the permanence of system (1). In order to facilitate discussion, we give one lemma.

Lemma 13. There exists a constant 𝑚₄>0, such that 𝑥(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡)≥𝑚₄  for each solution of (1) with 𝑡 large enough.

Proof . First, we discuss 𝑥(𝑡). Since 𝑆(𝑡)≤𝑀, by the first equation of system (1), we have 𝑥^′(𝑡)≥𝑟−𝑃^′₁(0)𝑀𝑥(𝑡(𝑡))1−𝑟𝑥𝐾𝑟−𝑃^′₁(0)𝑀,()(𝑡)(𝑡)()(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥,𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,𝑡=𝑛𝑇.(55) Then, we obtain the following comparison system: 𝜒^′(𝑡)=𝑟−𝑃^′₁(0)𝑀𝜒(𝑡(𝑡))1−𝑟𝜒𝐾𝑟−𝑃^′₁(0)𝑀,()(𝑡)(𝑡)()(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝜒=−𝛿𝜒,𝑡=𝑛+𝜏−1𝑇,Δ𝜒=0,𝑡=𝑛𝑇.(56) Letting 𝑟>𝑃^′₁(0)𝑀 and (1−𝛿)𝑒^{(𝑟−𝑃′1(0)𝑀)𝑇}>1, by Lemma 7, the system (56) has a positive periodic solution 𝜒^∗(𝑡), and for any solution 𝜒(𝑡) of (56), 𝜒(𝑡)→𝜒^∗(𝑡) as 𝑡 large enough, where 𝜒^∗(𝑡)=𝐾𝑟−𝑃^′₁(0)𝑀1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑀)𝑇}×𝑟1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑀)𝑇}+𝛿𝑒^{−(𝑟−𝑃′1(0)𝑀)(𝑇−𝜏𝑇)}×𝑒^{−(𝑟−𝑃′1(0)𝑀)(𝑡−(𝑛−1)𝑇)−1},)()𝑇],𝐾𝑡∈((𝑛−1𝑇,𝑛+𝜏−1𝑟−𝑃^′₁(0)𝑀1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑀)𝑇}×𝑟1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑀)𝑇}+𝛿𝑒^{−(𝑟−𝑃′1(0)𝑀)(𝑡−(𝑛+𝜏−1)𝑇)−1},𝑡∈((𝑛+𝜏−1)𝑇,𝑛𝑇].(57) According to Lemmas 4 and 7, one may find 𝑛^∗₄∈ℕ such that 𝑥(𝑡)≥𝜒^∗(𝑡)−𝜀 for 𝑡≥𝑛^∗₄𝑇. Since 𝜒^∗(𝑡)−𝜀≥(𝐾(𝑟−𝑃^′₁(0)𝑀)(1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑀)𝑇})/𝑟(1−𝑒^{−(𝑟−𝑃′1(0)𝑀)𝑇}))−𝜀=𝑚₀>0, so 𝑥(𝑡)≥𝑚₀ for 𝑡≥𝑛^∗₄𝑇. Next, we will discuss the rest of parts.
From (46) and (47), we know that there exists 𝑛^∗₅(𝑛^∗₅>max{𝑛^∗₁,𝑛^∗₄}) such that 𝐼(𝑡)≥𝐼^∗(𝑡)−𝜀, 𝑦_𝐽(𝑡)≥𝑦^∗_𝐽(𝑡)−𝜀, and 𝑦_𝑀(𝑡)≥𝑦^∗_𝑀(𝑡)−𝜀 for all 𝑡≥𝑛^∗₅𝑇. By (22), (23), and (48), we have 𝐼(𝑡)≥(𝛿_𝐼(1−𝑃_𝐼)𝑒^{−𝑑𝐼𝑇}/(1−(1−𝑃_𝐼)𝑒^{−𝑑𝐼𝑇}))−𝜀=𝑚₁>0, 𝑦_𝐽(𝑡)≥(𝛿_𝐽(1−𝑃_𝐽)𝑒^{−(𝑚+𝑑𝐽)𝑇}/(1−(1−𝑃_𝐽)𝑒^{−(𝑚+𝑑𝐽)𝑇}))−𝜀=𝑚₂>0, and 𝑦_𝑀(𝑡)≥𝑚_𝑀−𝜀=𝑚₃>0, where 𝑚_𝑀=min_{0<𝑡≤𝑇}𝑦^∗_𝑀(𝑡). Let 𝑚₄=min{𝑚₀,𝑚₁,𝑚₂,𝑚₃}, then 𝑥(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡)≥𝑚₄ for 𝑡≥𝑛^∗₅𝑇. The proof is completed.

Theorem 14. If 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)𝑑𝑡−^𝑇₀𝑔𝐼^∗(𝑡)𝑑𝑡−𝑃^′₂(0)×^𝑇₀𝑦^∗_𝑀(𝑡)𝑑𝑡−𝑑_𝑆1𝑇>ln1−𝑃_𝑆,(58) then system (1) is permanent.

Proof. By Lemmas 6 and 13, we have already known that there exist two constants 𝑚₄,𝑀>0, such that 𝑥(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡)≥𝑚₄ and 𝑥(𝑡),𝑆(𝑡),𝐼(𝑡),𝑦_𝐽(𝑡),𝑦_𝑀(𝑡)≤𝑀 for 𝑡 large enough. Thus, we only need to find 𝑚^∗>0 such that 𝑆(𝑡)≥𝑚^∗ for 𝑡 large enough. We will do this in the following two steps.
Step  1. Let 𝑚₅>0 and 𝜀₁>0 small enough, so that 𝑚₅<min{(𝑟/𝑃^′₁(0)),(𝑑_𝐼/𝑔^′(0)),𝑀} and 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)−𝜀₁𝑑𝑡−^𝑇₀𝑔𝐼^∗(𝑡)+𝜀₁𝑑𝑡−𝑃^′₂(0)×^𝑇₀𝑦^∗_𝑀(𝑡)+𝜀₁𝑑𝑡−𝑑_𝑆1𝑇−ln1−𝑃_𝑆=𝜂>0,(59) where 𝑥^∗(𝑡)=𝐾𝑟−𝑃^′₁(0)𝑚₅1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑚5)𝑇}×𝑟1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑚5)𝑇}+𝛿𝑒^{−(𝑟−𝑃′1(0)𝑚5)(𝑇−𝜏𝑇)}×𝑒^{−(𝑟−𝑃′1(0)𝑚5)(𝑡−(𝑛−1)𝑇)−1},)()𝑇],𝐾𝑡∈((𝑛−1𝑇,𝑛+𝜏−1𝑟−𝑃^′₁(0)𝑚₅1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑚5)𝑇}×𝑟1−𝛿−𝑒^{−(𝑟−𝑃′1(0)𝑚5)𝑇}+𝛿𝑒^{−(𝑟−𝑃′1(0)𝑚5)(𝑡−(𝑛+𝜏−1)𝑇)−1},)],𝐼𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗(𝑡)=𝛿_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝑚5)(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝑚5)𝑇},)()𝑇],𝛿𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_𝐼1−𝑃_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝑚5)(𝑡−(𝑛−1)𝑇)}1−1−𝑃_𝐼𝑒^{−(𝑑𝐼−𝑔′(0)𝑚5)𝑇},)],𝑦𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗𝐽(𝑡)=𝑒^{−(𝑚+𝑑𝐽)(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝐽(0⁺)+𝜆𝑃₂𝑚₅𝑀∫₀^{𝑡−(𝑛−1)𝑇}𝑒^{(𝑚+𝑑𝐽)𝑠},)()𝑇],𝑒𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_{−(𝑚+𝑑𝐽)(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝐽𝜏𝑇⁺𝑒^{(𝑚+𝑑𝐽)𝜏𝑇}+𝜆𝑃₂𝑚₅𝑀∫^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠},)],𝑦𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗𝐽(0⁺)=𝜆𝑃₂𝑚₅𝑀1−𝑃_𝐽0^𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠+^𝑇_𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠×𝑒^{−(𝑚+𝑑𝐽)𝑇}+𝛿_𝐽1−(1−𝑃_𝐽)𝑒^{−(𝑚+𝑑𝐽)𝑇−1},𝑦^∗_𝐽𝜏𝑇⁺=1−𝑃_𝐽×𝜆𝑃₂𝑚₅𝑀₀^𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠+𝑒^{−(𝑚+𝑑𝐽)𝑇}×𝜆𝑃₂𝑚₅𝑀^𝑇_𝜏𝑇𝑒^{(𝑚+𝑑𝐽)𝑠}𝑑𝑠+𝛿_𝐽×𝑒^{−(𝑚+𝑑𝐽)𝜏𝑇}1−(1−𝑃_𝐽)𝑒^{−(𝑚+𝑑𝐽)𝑇−1},𝑦^∗_𝑀(𝑡)=𝑒^{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝑀(0⁺)+𝑚∫₀^{𝑡−(𝑛−1)𝑇}×𝑦^∗_𝐽(𝑡)+𝜀₁𝑒^𝑑𝑀𝑠,)()𝑇],𝑒𝑑𝑠𝑡∈((𝑛−1𝑇,𝑛+𝜏−1_{−𝑑𝑀(𝑡−(𝑛−1)𝑇)}×𝑦^∗_𝑀(𝜏𝑇⁺)𝑒^{𝑑𝑀𝜏𝑇}+𝑚∫^{𝑡−(𝑛−1)𝑇}_𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀₁𝑒^𝑑𝑀𝑠,)],𝑦𝑑𝑠𝑡∈((𝑛+𝜏−1𝑇,𝑛𝑇_∗𝑀(0⁺)=𝑚1−𝑃_𝑀0^𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀₁𝑒^𝑑𝑀𝑠+𝑑𝑠_𝑇𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀₁𝑒^𝑑𝑀𝑠𝑑𝑠×𝑒^{−𝑑𝑀𝑇}+𝛿_𝑀1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇−1},𝑦^∗_𝑀𝜏𝑇⁺=1−𝑃_𝑀×𝑚₀^𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀₁𝑒^𝑑𝑀𝑠𝑑𝑠+𝑚𝑒^{−𝑑𝑀𝑇𝑇}_𝜏𝑇𝑦^∗_𝐽(𝑡)+𝜀₁𝑒^𝑑𝑀𝑠𝑑𝑠+𝛿_𝑀×𝑒^{−𝑑𝑀𝜏𝑇}1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇−1}.(60) We shall prove that one cannot have 𝑆(𝑡)<𝑚₅ for all 𝑡>0, otherwise 𝑥^′(𝑡)≥𝑟−𝑃^′₁(0)𝑚₅𝑥(𝑡(𝑡))1−𝑟𝑥𝐾𝑟−𝑃^′₁(0)𝑚₅,𝐼^′(𝑡)≤−𝑑_𝐼−𝑔^′(0)𝑚₅𝐼(𝑡),𝑦^′_𝐽(𝑡)≤𝜆𝑃₂𝑚₅𝑀−𝑑_𝐽+𝑚𝑦_𝐽(𝑡),𝑦^′_𝑀(𝑡)=𝑚𝑦_𝐽(𝑡)−𝑑_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡),(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥Δ𝐼=−𝑃_𝐼𝐼(𝑡),Δ𝑦_𝐽(𝑡)=−𝑃_𝐽𝑦_𝐽(𝑡),Δ𝑦_𝑀(𝑡)=−𝑃_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡)𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,Δ𝐼=𝛿_𝐼,Δ𝑦_𝐽(𝑡)=𝛿_𝐽,Δ𝑦_𝑀(𝑡)=𝛿_𝑀,𝑡=𝑛𝑇.(61) Analyzing (61) with similarity as (40), it is easy to obtain that there exists a positive constant 𝑛^∗₆, such that 𝑥(𝑡)≥𝑥^∗(𝑡)−𝜀₁,𝐼(𝑡)≤𝐼^∗(𝑡)+𝜀₁,𝑦_𝐽(𝑡)≤𝑦^∗_𝐽(𝑡)+𝜀₁,𝑦_𝑀(𝑡)≤𝑦^∗_𝑀(𝑡)+𝜀₁ for 𝑡≥𝑛^∗₆𝑇. Therefore, 𝑆^′(𝑡)≥𝛽𝑃₁𝑥^∗(𝑡)−𝜀₁𝑆(𝑡)−𝑔𝐼^∗(𝑡)+𝜀₁𝑆(𝑡)−𝑃^′₂(0)𝑦^∗_𝑀(𝑡)+𝜀₁𝑆(𝑡)−𝑑_𝑆𝑆(𝑡),()(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝑃_𝑆𝑆(𝑡)()(𝑡),𝑡=𝑛+𝜏−1𝑇,Δ𝑆=0,𝑡=𝑛𝑇,(62) for 𝑡≥𝑛^∗₆𝑇. Let 𝑁₀∈ℕ and (𝑁₀+𝜏−1)≥𝑛^∗₆. Integrating (62) on ((𝑛+𝜏−1)𝑇,(𝑛+𝜏)𝑇],𝑛≥𝑁₀, we have 𝑆)𝑇))𝑇)((𝑛+𝜏≥𝑆((𝑛+𝜏−11−𝑃_𝑆×𝑒^{∫(𝑛+𝜏)𝑇(𝑛+𝜏−1)𝑇(𝛽𝑃1(}𝑥^∗(𝑡)_{−𝜀1)−𝑔(}𝐼^∗(𝑡)_{+𝜀1)−𝑃}^′₂₍₀₎₍𝑦^∗_𝑀(𝑡)_{+𝜀1)−𝑑𝑆)𝑑𝑡}=𝑆((𝑛+𝜏−1)𝑇)𝑒^𝜂.(63) Then 𝑆((𝑁₀+𝜏+𝑘)𝑇)≥𝑆((𝑁₀+𝜏)𝑇)𝑒^𝑘𝜂→∞ as 𝑘→∞, which is a contradiction. So there exists a 𝑡₁(𝑡₁>𝑛^∗₆𝑇) such that 𝑆(𝑡₁)≥𝑚₅.
Step  2. If 𝑆(𝑡)≥𝑚₅ for all 𝑡≥𝑡₁, then Our purpose is obtained. If not, let 𝑡₂=inf{𝑡>𝑡₁∣𝑆(𝑡)<𝑚₅}. Then 𝑆(𝑡)≥𝑚₅ for 𝑡∈[𝑡₁,𝑡₂) and 𝑆(𝑡₂)=𝑚₅. In this step, we consider two possible cases for 𝑡₂.
Case  1. 𝑡₂=(𝑛₁+𝜏−1)𝑇,𝑛₁∈ℕ. Then 𝑆(𝑡⁺₂)=(1−𝑃_𝑆)𝑆(𝑡₂)<𝑚₅. Select 𝑛₂,𝑛₃∈ℕ such that (𝑛₂−1)≥𝑛^∗₆ and (1−𝑃_𝑆)^𝑛2𝑒^{𝑛3𝜂+𝑛2𝜎𝑇}>(1−𝑃_𝑆)^𝑛2𝑒^{𝑛3𝜂+(𝑛2+1)𝜎𝑇}>1, where 𝜎=𝛽𝑃₁(𝑚₄)−𝑔(𝑀)−𝑃^′₂(0)𝑀−𝑑_𝑆<0. Let 𝑇=(𝑛₂+𝑛₃)𝑇, then we have the claim: there exists 𝑡₃∈(𝑡₂,𝑡₂+𝑇] such that 𝑆(𝑡₃)≥𝑚₅. If the claim is false, we will get a contradiction in the following.
According to Step 1, we have 𝑥(𝑡)≥𝑥^∗(𝑡)−𝜀₁,𝐼(𝑡)≤𝐼^∗(𝑡)+𝜀₁,𝑦_𝐽(𝑡)≤𝑦^∗_𝐽(𝑡)+𝜀₁,𝑦_𝑀(𝑡)≤𝑦^∗_𝑀(𝑡)+𝜀₁ for 𝑡≥(𝑛₁+𝑛₂−1)𝑇. Then, we have 𝑆^′(𝑡)≥𝛽𝑃₁𝑥^∗(𝑡)−𝜀₁𝑆(𝑡)−𝑔𝐼^∗(𝑡)+𝜀₁𝑆(𝑡)−𝑃^′₂(0)𝑦^∗_𝑀(𝑡)+𝜀₁𝑆(𝑡)−𝑑_𝑆𝑆(𝑡),()(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝑃_𝑆𝑆(𝑡)()(𝑡),𝑡=𝑛+𝜏−1𝑇,Δ𝑆=0,𝑡=𝑛𝑇,(64) for 𝑡∈[𝑡₂+𝑛₂𝑇,𝑡₂+𝑇]. As in Step 1, we have 𝑆𝑡₂+𝑇≥𝑆𝑡₂+𝑛₂𝑇𝑒^𝑛3𝜂.(65)
Since 𝑥(𝑡)≥𝑚₄,𝐼(𝑡)≤𝑀,𝑦_𝑀(𝑡)≤𝑀 and 𝑃₂(𝑆(𝑡))<𝑃^′₂(0)𝑆(𝑡), we have 𝑆^′(𝑡)≥𝛽𝑃₁𝑚₄−𝑔(𝑀)−𝑃^′₂(0)𝑀−𝑑_𝑆𝑆(𝑡)(𝑡),()(𝑡)=𝜎𝑆𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝑃_𝑆𝑆(𝑡)()(𝑡),𝑡=𝑛+𝜏−1𝑇,Δ𝑆=0,𝑡=𝑛𝑇,(66) for 𝑡∈[𝑡₂,𝑡₂+𝑛₂𝑇]. Integrating (66) on [𝑡₂,𝑡₂+𝑛₂𝑇], we have 𝑆𝑡₂+𝑛₂𝑇≥𝑆𝑡⁺₂𝑒^{𝑛2𝜎𝑇}=1−𝑃_𝑆𝑚₅𝑒^{𝑛2𝜎𝑇}≥1−𝑃_𝑆^𝑛2𝑚₅𝑒^{𝑛2𝜎𝑇}.(67) Thus, by (65) and (67), we have 𝑆(𝑡₂+𝑇)≥(1−𝑃_𝑆)^𝑛2𝑚₅𝑒^{𝑛3𝜂+𝑛2𝜎𝑇}>𝑚₅, which is a contradiction. Let 𝑡₄=inf{𝑡>𝑡₂∣𝑆(𝑡)≥𝑚₅}, then for 𝑡∈[𝑡₂,𝑡₄),𝑆(𝑡)<𝑚₅ and 𝑆(𝑡₄)=𝑚₅. So, 𝑆(𝑡)≥𝑆(𝑡⁺₂)𝑒^{𝜎(𝑡−𝑡2)}=(1−𝑃_𝑆)𝑚₅𝑒^{𝜎(𝑡−𝑡2)}≥(1−𝑃_𝑆)^𝑛2+𝑛3𝑚₅𝑒^{𝜎(𝑛2+𝑛3)𝑇}=𝑚₁ for 𝑡∈[𝑡₂,𝑡₄).
Case  2 (𝑡₂≠(𝑛₁+𝜏−1)𝑇,𝑛₁∈ℕ). Suppose that 𝑡₂∈((𝑛^′₁+𝜏−1)𝑇,(𝑛^′₁+𝜏)𝑇),𝑛^′₁∈ℕ.  𝑆(𝑡)≥𝑚₅ for 𝑡∈[𝑡₁,𝑡₂) and 𝑆(𝑡₂)=𝑚₅. There are two possible cases for 𝑡∈(𝑡₂,(𝑛^′₁+𝜏)𝑇).
Case  2a. If 𝑆(𝑡)≤𝑚₅ for all 𝑡∈(𝑡₂,(𝑛^′₁+𝜏)𝑇), similar to Case 1, we can prove there exists a 𝑡^′₃∈((𝑛^′₁+𝜏)𝑇,(𝑛^′₁+𝜏)𝑇+𝑇] such that 𝑆(𝑡^′₃)≥𝑚₅. Let 𝑡^′₄=inf{𝑡>𝑡₂∣𝑆(𝑡)≥𝑚₅}, then for 𝑡∈[𝑡₂,𝑡^′₄),𝑆(𝑡)<𝑚₅ and 𝑆(𝑡^′₄)=𝑚₅. So, 𝑆(𝑡)≥𝑆(𝑡₂)𝑒^{𝜎(𝑡−𝑡2)}=𝑚₅𝑒^{𝜎(𝑡−𝑡2)}≥(1−𝑃_𝑆)^𝑛2+𝑛3𝑚₅𝑒^{𝜎(𝑛2+𝑛3+1)𝑇}=𝑚^∗<𝑚₁ for all 𝑡∈[𝑡₂,𝑡^′₄).
Case  2b. If there exists a 𝑡∈(𝑡₂,(𝑛^′₁+𝜏)𝑇) such that 𝑆(𝑡)≥𝑚₅. Let 𝑡^′₄=inf{𝑡>𝑡₂∣𝑆(𝑡)≥𝑚₅}, then for 𝑡∈[𝑡₂,𝑡^′₄),𝑆(𝑡)<𝑚₅ and 𝑆(𝑡^′₄)=𝑚₅. So, 𝑆(𝑡)≥𝑆(𝑡₂)𝑒^{𝜎(𝑡−𝑡2)}=𝑚₅𝑒^{𝜎(𝑡−𝑡2)}≥𝑚₅𝑒^𝜎𝑇>𝑚^∗ for all 𝑡∈[𝑡₂,𝑡^′₄).
Since 𝑆(𝑡)≥𝑚₅ for some 𝑡≥𝑡₁, in both cases a similar discussion can be continued. The proof is completed.

5. Numerical Simulations and Conclusions

In this section, we will give an example and its simulations to show the efficiency of the criteria derived in Section 4.

In system (1), let 𝑃₁(𝑥(𝑡))=𝑎𝑥(𝑡),  𝑔(𝐼(𝑡))=𝑏𝐼(𝑡), and 𝑃₂(𝑆(𝑡))=ℎ(1−𝑒^{−𝑐𝑆(𝑡)}), 𝑎,𝑏,𝑐,ℎ>0. Namely, 𝑃₁(𝑥(𝑡)) describes an Holling type-I functional response of the pest, 𝑃₂(𝑆(𝑡)) describes a Ivlev-type functional response of the pest's natural predator. Therefore, we consider the pest management model with impulsive releasing and harvesting at two different fixed moments: 𝑥^′(𝑡)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾(𝑡)𝑆(𝑡),𝑆−𝑎𝑥_′(𝑡)=𝛽𝑎𝑥(𝑡)𝑆(𝑡)−𝑏𝐼(𝑡)𝑆(𝑡)−ℎ1−𝑒^{−𝑐𝑆(𝑡)}𝑦_𝑀(𝑡)−𝑑_𝑆𝑆(𝑡),𝐼^′(𝑡)=𝑏𝐼(𝑡)𝑆(𝑡)−𝑑_𝐼𝐼(𝑡),𝑦^′_𝐽(𝑡)=𝜆ℎ1−𝑒^{−𝑐𝑆(𝑡)}𝑦_𝑀(𝑡)−𝑑_𝐽𝑦_𝐽(𝑡)−𝑚𝑦_𝐽(𝑡),𝑦^′_𝑀(𝑡)=𝑚𝑦_𝐽(𝑡)−𝑑_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡),(𝑡)𝑡≠𝑛+𝜏−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=−𝛿𝑥Δ𝑆=−𝑃_𝑆𝑆(𝑡),Δ𝐼(𝑡)=−𝑃_𝐼𝐼(𝑡),Δ𝑦_𝐽(𝑡)=−𝑃_𝐽𝑦_𝐽(𝑡),Δ𝑦_𝑀(𝑡)=−𝑃_𝑀𝑦_𝑀(𝑡),()(𝑡)(𝑡)(𝑡)𝑡=𝑛+𝜏−1𝑇,Δ𝑥=0,Δ𝑆=0,Δ𝐼=𝛿_𝐼,Δ𝑦_𝐽(𝑡)=𝛿_𝐽,Δ𝑦_𝑀(𝑡)=𝛿_𝑀,𝑡=𝑛𝑇.(68) So, by (22), (23), and (25), we have 𝛽^𝑇₀𝑃₁𝑥^∗(𝑡)𝑑𝑡=𝛽𝑎𝐾𝑟(𝑟𝑇+ln(1−𝛿))=𝜃₁,_𝑇0𝑔𝐼^∗(𝑡=)𝑑𝑡𝑏𝛿_𝐼1−𝑒^{−𝑑𝐼𝜏𝑇}+1−𝑃_𝐼𝑒^{−𝑑𝐼𝜏𝑇}−𝑒^{−𝑑𝐼𝑇}𝑑_𝐼1−1−𝑃_𝐼𝑒^{−𝑑𝐼𝑇}=𝜃₂,_𝑇0𝑦^∗_𝑀(𝑡)=𝑦𝑑𝑡_∗𝑀0⁺𝑑_𝑀1−𝑒^{−𝑑𝑀𝜏𝑇}+𝑚𝛿_𝐽1−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑑_𝑀−𝑚+𝑑_𝐽×1−𝑒^{−(𝑚+𝑑𝐽)𝜏𝑇}𝑚+𝑑_𝐽−1−𝑒^{−𝑑𝑀𝜏𝑇}𝑑_𝑀+𝑦^∗_𝑀𝜏𝑇⁺𝑒^{𝑑𝑀𝜏𝑇}𝑑_𝑀𝑒^{−𝑑𝑀𝜏𝑇}−𝑒^𝑑𝑀𝑇+𝑚𝛿_𝐽1−𝑃_𝐽1−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑑_𝑀−𝑚+𝑑_𝐽×𝑒^{−(𝑚+𝑑𝐽)𝜏𝑇}−𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑚+𝑑_𝐽−𝑒^{−𝑑𝑀𝜏𝑇}−𝑒^{−𝑑𝑀𝑇}𝑑_𝑀𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝜏𝑇}=𝜃₃,(69) where 𝑦^∗_𝑀0⁺=1−𝑃_𝑀𝐴(𝜏𝑇)+𝐵(𝑇)𝑒^{−𝑑𝑀𝑇}+𝛿_𝑀1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇},𝑦^∗_𝑀𝜏𝑇⁺=1−𝑃_𝑀𝐴(𝜏𝑇)+𝑒^{−𝑑𝑀𝑇}𝐵(𝑇)+𝛿_𝑀𝑒^{−𝑑𝑀𝜏𝑇}1−1−𝑃_𝑀𝑒^{−𝑑𝑀𝑇},𝐴(𝜏𝑇)=𝑚𝛿_𝐽𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝜏𝑇}−11−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑑_𝑀−𝑚+𝑑_𝐽,𝐵(𝑇)=𝑚𝛿_𝐽1−𝑃_𝐽𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝑇}−𝑒^{(𝑑𝑀−(𝑚+𝑑𝐽))𝜏𝑇}1−1−𝑃_𝐽𝑒^{−(𝑚+𝑑𝐽)𝑇}𝑑_𝑀−𝑚+𝑑_𝐽.(5) Then, by Theorems 11 and 14, we have the following. (𝑇1) If 𝜃₁−𝜃₂−𝜃₃−𝑑_𝑆𝑇<ln(1/(1−𝑃_𝑆)), then the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (68) is locally asymptotically stable. (𝑇2) If 𝜃₁−𝜃₂−ℎ𝑐𝑒^−𝑐𝑀𝜃₃−𝑑_𝑆𝑇<ln(1/(1−𝑃_𝑆)), then the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (68) is globally asymptotically stable, where 𝑀 is defined in Lemma 6. (𝑇3) If 𝜃₁−𝜃₂−ℎ𝑐𝜃₃−𝑑_𝑆𝑇>ln(1/(1−𝑃_𝑆), then system (68) is permanent.

In the following, we analyze the locally asymptotical stability of the susceptible pest-eradication periodic solution and permanence of system (68).

Assume that 𝑥(0)=20, 𝑆(0)=2, 𝐼(0)=2, 𝑦_𝐽(0)=0.5, 𝑦_𝑀(0)=0.5, 𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑_𝑆=0.2, 𝑑_𝐼=0.5, 𝑑_𝐽=0.4, 𝑑_𝑀=0.2, 𝛿=0.4, 𝑃_𝑆=𝑃_𝐼=𝑃_𝐽=𝑃_𝑀=0.2, 𝛿_𝐼=0.2, 𝛿_𝐽=0.3, 𝛿_𝑀=0.5, 𝜏=0.3, 𝑇=0.5. Obviously, the condition of (T1) is satisfied, then the susceptible pest-eradication periodic solution of system (68) is locally asymptotically stable, which can be seen from the numerical simulation in Figures 1 and 2.

(a)

(b)

(c)

(d)

(e)

Figure 1 

Time series of the system (68) with 𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑𝑆=0.2, 𝑑𝐼=0.5, 𝑑𝐽=0.4, 𝑑𝑀=0.2, 𝛿=0.4, 𝑃𝑆=𝑃𝐼=𝑃𝐽=𝑃𝑀=0.2, 𝛿𝐼=0.2, 𝛿𝐽=0.3, 𝛿𝑀=0.5, 𝜏=0.3, 𝑇=0.5, 𝑥(0)=20, 𝑆(0)=2, 𝐼(0)=2, 𝑦𝐽(0)=0.5, 𝑦𝑀(0)=0.5.

(a)

(b)

Figure 2 

Phase portrait of the system (68) with 𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑𝑆=0.2, 𝑑𝐼=0.5, 𝑑𝐽=0.4, 𝑑𝑀=0.2, 𝛿=0.4, 𝑃𝑆=𝑃𝐼=𝑃𝐽=𝑃𝑀=0.2, 𝛿𝐼=0.2, 𝛿𝐽=0.3, 𝛿𝑀=0.5, 𝜏=0.3, 𝑇=0.5, 𝑥(0)=20, 𝑆(0)=2, 𝐼(0)=2, 𝑦𝐽(0)=0.5, 𝑦𝑀(0)=0.5.

Assume that 𝑥(0)=20, 𝑆(0)=0.2, 𝐼(0)=2, 𝑦_𝐽(0)=2, 𝑦_𝑀(0)=2, 𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑_𝑆=0.2, 𝑑_𝐼=0.5, 𝑑_𝐽=0.4, 𝑑_𝑀=0.2, 𝛿=0.4, 𝑃_𝑆=𝑃_𝐼=𝑃_𝐽=𝑃_𝑀=0.2, 𝛿_𝐼=0.2, 𝛿_𝐽=0.3, 𝛿_𝑀=0.5, 𝜏=0.3, 𝑇=1. Obviously, the condition of (T3) is satisfied. Then, system (68) is permanent, which can also be seen from Figures 3 and 4.

(a)

(b)

(c)

(d)

(e)

Figure 3 

Time series of the system (68) with 𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑𝑆=0.2, 𝑑𝐼=0.5, 𝑑𝐽=0.4, 𝑑𝑀=0.2, 𝛿=0.4, 𝑃𝑆=𝑃𝐼=𝑃𝐽=𝑃𝑀=0.2, 𝛿𝐼=0.2, 𝛿𝐽=0.3, 𝛿𝑀=0.5, 𝜏=0.3, 𝑇=1, 𝑥(0)=20, 𝑆(0)=0.2, 𝐼(0)=2, 𝑦𝐽(0)=2, 𝑦𝑀(0)=2.

(a)

(b)

(c)

Figure 4 

Phase portrait of the system (68) with 𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑𝑆=0.2, 𝑑𝐼=0.5, 𝑑𝐽=0.4, 𝑑𝑀=0.2, 𝛿=0.4, 𝑃𝑆=𝑃𝐼=𝑃𝐽=𝑃𝑀=0.2, 𝛿𝐼=0.2, 𝛿𝐽=0.3, 𝛿𝑀=0.5, 𝜏=0.3, 𝑇=1, 𝑥(0)=20, 𝑆(0)=0.2, 𝐼(0)=2, 𝑦𝐽(0)=2, 𝑦𝑀(0)=2.

From results of the numerical simulation, we know that there exists an impulsive harvesting(or releasing) periodic threshold 𝑇^∗, which satisfies 0.5<𝑇^∗<1. If 𝑇<𝑇^∗ and the other parameters are fixed (𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑_𝑆=0.2, 𝑑_𝐼=0.5, 𝑑_𝐽=0.4, 𝑑_𝑀=0.2, 𝛿=0.4, 𝑃_𝑆=𝑃_𝐼=𝑃_𝐽=𝑃_𝑀=0.2, 𝛿_𝐼=0.2, 𝛿_𝐽=0.3, 𝛿_𝑀=0.5, 𝜏=0.3, 𝑇=0.5.), then the susceptible pest-eradication periodic solution (𝑥^∗(𝑡),0,𝐼^∗(𝑡),𝑦^∗_𝐽(𝑡),𝑦^∗_𝑀(𝑡)) of system (68) is locally asymptotically stable. If 𝑇>𝑇^∗ and the other parameters are fixed (𝑟=8, 𝐾=10, 𝑎=0.8, 𝛽=0.5, 𝑏=0.3, ℎ=8, 𝑐=0.2, 𝜆=0.6, 𝑚=2, 𝑑_𝑆=0.2, 𝑑_𝐼=0.5, 𝑑_𝐽=0.4, 𝑑_𝑀=0.2, 𝛿=0.4, 𝑃_𝑆=𝑃_𝐼=𝑃_𝐽=𝑃_𝑀=0.2, 𝛿_𝐼=0.2, 𝛿_𝐽=0.3, 𝛿_𝑀=0.5, 𝜏=0.3, 𝑇=0.5), then system (68) is permanent. The same discussion can be applied to other parameters.

In this paper, we proposed a pest management model with impulsive releasing (periodic infective pests, immature and mature natural enemies releasing) and harvesting (periodic crops harvesting) at two different fixed moments. By means of Floquet theory and multicomparison results for impulsive differential equations, two sufficient conditions ensuring the locally and globally asymptotical stability of the susceptible pest-eradication period solution and permanence of the system are derived.

Acknowledgments

This work is supported partially by National Natural Science Foundation of China under Grant no. 10971240 and no. 61004042, Key Project of Chinese Education Ministry under Grant no. 212138, Natural Science Foundation of Chongqing under Grant CQ CSTC 2011BB0117, and Foundation of Science and Technology Project of Chongqing Education Commission under Grant KJ120630.