Modelling and Analysis of a Pest-Control Pollution Model with Integrated Control Tactics

Abstract

A hybrid impulsive pest control model with stage structure for pest and Holling II functional response is proposed and investigated, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Sufficient conditions for global attractiveness of the pest-extinction periodic solution and permanence of the system are obtained, which show that there exists a globally asymptotically stable pest-extinction periodic solution when the number of natural enemies released is more than some critical value, whereas the system can be permanent when the number of natural enemies released is less than another critical value. Furthermore, numerical simulations are carried out to illustrate our theoretical results and facilitate their interpretation.

1. Introduction

Since the beginning of recorded history, outbreaks of pests have plagued humanity, coming in direct competition with people for life-sustaining food. Reportedly, an estimated 67,000 different pest species attack agricultural crops, and about 35% of the yearly agricultural crop production is lost to pests worldwide [1, 2]. That problem is one of how to control or suppress damaging populations of pests over widespread areas. As we know, the most effective strategy for controlling pests may be to combine methods in an approach known as integrated pest management (IPM) that emphasizes preventing pest damage. In IPM, information about pests and available pest-control methods (including biological, cultural, and chemical) is used to manage pest damage by the most economical means and with the least possible hazard to people, property, and environment [3–5].

Biological control of pests in agriculture is a method of controlling pests (including insects, mites, weeds, and plant diseases) that relies on predation, parasitism, herbivory, or other natural mechanisms. It can be an important component of integrated pest management (IPM) programs. It is defined as the reduction of pest populations by natural enemies and typically involves an active human role such as augmentation which involves the supplemental release of natural enemies. Biological control is not a "quick fix" for most pest problems. Natural enemies usually take longer to suppress a pest population than other forms of pest-control, and farmers often regard this as a disadvantage. Cultural controls are manipulations of the agroecosystem that make the cropping system less friendly to the establishment and proliferation of pest populations. Although they are designed to have positive effects on farm ecology and pest management, negative impacts may also result, due to variations in weather or changes in crop management [6]. Another important method for pest-control is chemical control. Chemical control is the approach of controlling pests through the spraying pesticide which is liable to reduce the pest populations considerably and which is indispensable when there are not enough natural enemies to decrease pest populations. In most cropping systems, insecticides are still the principal means of controlling pests once the economic threshold has been reached. They can be relatively cheap and are easy to apply, fast acting, and in most instances reliable in controling the pests [7]. Despite the advantages of conventional insecticides, the problems associated with their use have been well documented. These include the resurgence of pest populations after decimation of the natural enemies, development of insecticide-resistant populations, and negative impacts on nontarget organisms within and outside the crop system [8]. When considering these actions, in the process of effective control of pest, excessive use of a single control strategy is undesirable. Wherever possible, different pest-control techniques should work together rather than against each other. Even so, in many cases, the most effective release rate or spraying rate has not been identified as it will vary depending on crop type and target host density. Therefore, human beings have been forced to face the new challenge in the integrated pest management (IPM) program. One of the most important questions in IPM is how many natural enemies should be released and what fraction of the pest population should be killed to avoid economic damage and reduce the pesticide applications when the pest population reaches or exceeds the economic threshold level.

According to the idea of IPM, many mathematical models have been constructed and studied for understanding the range of possible ecological interactions between pest, natural enemy, and pesticides in the last decades. In order to consider the consequences of especially spraying pesticide and introducing additional predators into a natural pest-predator system, impulsive differential equations have been employed to describe such a system by many researchers [9–13], and the references cited therein. Impulsive differential equations are found in almost every domain of applied sciences [14, 15] and have been studied in many investigations [16–18]. They generally describe phenomena which are subject to steep or instantaneous changes. In IPM, impulsive reduction of the pest population is possible by trapping the pests and/or by poisoning them with chemicals. An impulsive increase of the natural enemy density can be achieved by releasing the natural enemy based on laboratory breeding into the field [5, 11]. Unfortunately, most of the pest-control models in the literature, which were modeled by impulsive differential equations, assume that at every impulsive spraying period, the pest population (including the natural enemies) may be killed immediately, and the instant killing rate of pesticide is a proportional constant. However, the actual situation is not always the same. Generally, pesticide appears in environment first, then it is absorbed by organism, and the individuals are affected, that is, the toxicity of pesticide does not act on the organism at once; in other words, it will last for some time before toxins are capable of decreasing the average growth rate of the species [19]. This fact urges us to consider the effect of pollution time delay on the extinction and permanence of population in a polluted environment. With this in mind, it is necessary to introduce the pollution model to model the process of pest-control problems and study its dynamics, and this is different from the previous pest-control model which assumed that pests were reduced proportionally by spraying pesticides.

As we know, since Hallam and his coworkers proposed a toxicant-population model in the early 1980s [20–23], mathematical models of single or multiple populations with toxicant effect have been constructed and studied extensively [19, 24–29]. However, the majority of these studies have been focused on the effects of toxicant emitted into the environment from industrial and household resources on biological species, and only a few attempts have been made to combine pollution model to study pest-control problems with pesticide (toxin) input. Recently, by using pollution model and impulsive delay differential equation, Liu et al. [19] constructed, and investigated, a pest-control model with age structure for pest by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moment. It is assumed in their model that each individual has the same dose response parameter to the organismal toxicant concentration regardless of the difference in many aspects between the immature and mature pest populations. However, in the natural world, there are many species whose individual members have a life story that takes them through two stages, immature and mature. Those species hatch from egg. Moreover, the immature and mature species express great differences in many aspects. One of the facts is that only the mature individuals are affected by the toxin (pesticide) and the immature individuals are not. For example, locust and salt-cedar leaf beetle, and so forth, are such species whose immature individuals (eggs) are protected by their eggshell and hardly injured by pesticides.

Based on all the above points, in this paper, we propose and investigate a pest-control model with a constant periodic pesticide input and natural enemies release at different fixed moment, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Moreover, we assume that the pest individuals have two life stages: immature (egg) and mature with a constant maturation time delay, pesticide (toxin) has no effect on the immature individuals, and the capacity of the environment is so large that the change of toxin in the environment that comes from uptake and egestion by the organisms can be ignored. On the other hand, it is well known that functional response is a basic modeling unit in community ecology [30]. So, we further assume that natural enemy (predator) only feeds on mature pest (prey), and the functional response of natural enemy (predator) to mature pest (prey) species takes the Holling type II form. Meanwhile, because we may artificially pick on the appropriate releasing time when there is the lowest chance of adversely affecting natural enemies; thus, we further assume that pesticide input has little influence on the natural enemies, that is, the effect of pesticide input on natural enemies can be ignored. We are interested in a theoretical study about the effects of our control tactics on dynamical behavior of populations and attempt to obtain a theoretical threshold value which determines extinction of pest species and permanence of the system.

The organization of this paper is as follows. In Section 2, we set up our model and introduce some notations, definitions, and lemmas. In Section 3, sufficient conditions for extinction of the pest species and permanence of the system are given, respectively. The numerical simulations are carried out to study the effects of the impulsive varying parameters on the system as well as to illustrate our theoretical results in Section 4. Finally, a brief discussion is given to conclude this work.

2. Model and Preliminaries

According to the above analysis and assumption, we construct a pest-control pollution model with stage structure for the pest and Holling II functional response concerning integrated control tactics. The model takes the following form: 𝑑𝑥_𝑗(𝑡)(𝑡)𝑑𝑡=𝑎𝑥−𝑏₁𝑥_𝑗(𝑡)−𝑎𝑒^−𝑏1𝜏𝑥(),(𝑡)𝑡−𝜏𝑑𝑥𝑑𝑡=𝑎𝑒^−𝑏1𝜏𝑥(𝑡−𝜏)−𝑓𝑥²(𝑡)−𝛽𝑥(𝑡)𝑦(𝑡)𝜂+𝑥(𝑡)−𝑟𝑐_𝑜(𝑡)𝑥(𝑡),𝑑𝑦(𝑡)(𝑡)𝑦(𝑡)𝑑𝑡=𝜆𝛽𝑥𝜂+𝑥(𝑡)−𝑏₂𝑦(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,𝑑𝑐_𝑜(𝑡)𝑑𝑡=𝑘𝑐_𝑒(𝑡)−𝑔𝑐_𝑜(𝑡)−𝑚𝑐_𝑜(𝑡),𝑑𝑐_𝑒(𝑡)𝑑𝑡=−ℎ𝑐_𝑒(𝑡),Δ𝑥_𝑗(𝑡)=0,Δ𝑥(𝑡)=0,Δ𝑦(𝑡)=𝜇₁,Δ𝑐_𝑜(𝑡)=0,Δ𝑐_𝑒(𝑡)=0,𝑡=(𝑛+𝑙−1)𝑇,Δ𝑥_𝑗(𝑡)=0,Δ𝑥(𝑡)=0,Δ𝑦(𝑡)=0,Δ𝑐_𝑜(𝑡)=0,Δ𝑐_𝑒(𝑡)=𝜇₂,𝑡=𝑛𝑇. (2.1) The initial conditions are 𝑥_𝑗(𝑡),𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡)=𝜙₁(𝑡),𝜙₂(𝑡),𝜙₃(𝑡),𝜙₄(𝑡),𝜙₅(𝑡)∈𝐶⁺₅,𝐶⁺₅=𝐶[−𝜏,0],𝑅⁵₊,𝑡∈[−𝜏,0],𝜙_𝑖(0)>0,𝑖=1,2,3,4,5, (2.2) where 𝑥_𝑗(𝑡),  𝑥(𝑡), and  𝑦(𝑡) represent the density of the immature pest (egg), mature pest and natural enemy at time 𝑡, respectively, 𝑐_𝑒(𝑡) represents the concentration of pesticide in the environment at time 𝑡; 𝑐₀(𝑡) represents the concentration of pesticide in the organism for the mature pest at time 𝑡, 𝑎 is the growth rate of the immature pest; 𝑏₁ and 𝑏₂ show the death rate of the immature pest and natural enemy, respectively, 𝜏 represents a constant time to maturity, 𝑓 represents the intraspecific competition coefficient of mature species, expression 𝛽𝑥(𝑡)/(𝜂+𝑥(𝑡)) is Holling II functional response function, 𝛽>0, 𝜂>0, 𝜆 represents the rate of conversion of consumed mature pest to natural enemy, 𝑟 represents the decreasing rate of the intrinsic growth rate associated with the uptake of pesticide in the organism for the mature pest, 𝑘𝑐_𝑒(𝑡) represents the organism's net uptake pesticide from the environment, 𝑔𝑐₀(𝑡) and 𝑚𝑐₀(𝑡) represent the egestion and depuration rates of pesticide in the organism for the mature pest, respectively, −ℎ𝑐_𝑒(𝑡) represents the loss of pesticide in the environment due to natural degradation, Δ𝑥_𝑗(𝑡)=𝑥_𝑗(𝑡⁺)−𝑥_𝑗(𝑡), Δ𝑐₀(𝑡)=𝑐₀(𝑡⁺)−𝑐₀(𝑡), Δ𝑐_𝑒(𝑡)=𝑐_𝑒(𝑡⁺)−𝑐_𝑒(𝑡); 0≤𝑙≤1, 𝑇 is the period of impulsive effect, 𝑛∈𝑍₊={1,2,…}, 𝜇₁ is the releasing amount of the natural enemy at time𝑡=(𝑛+𝑙−1)𝑇, and 𝜇₂ is the amount of pesticide input at time 𝑡=𝑛𝑇.

Obviously, the first equation of system (2.1) can be written as 𝑥_𝑗(𝑡)=^𝑡_𝑡−𝜏𝑎𝑒^{−𝑏1(𝑡−𝑠)}𝑥(𝑠)𝑥𝑑𝑠,_𝑗(0)=⁰_−𝜏𝑎𝑒^𝑏1𝑠𝑥(𝑠)𝑑𝑠 (2.3)(2.4) which means that the property of 𝑥_𝑗(𝑡) can be investigated by 𝑥(𝑡). Moreover, the condition (2.4) presents the total surviving immature population from the observed birth on −𝜏≤𝑡≤0. On the other hand, because the immature pest (egg) does little harm to the crops and it cannot breed, we just need to consider the control of the mature pest. Meanwhile, note that the variable 𝑥_𝑗(𝑡) does not appear in the second, third, fourth, and fifth equations of system (2.1); hence, we only need to consider the subsystem of (2.1) as follows: 𝑑𝑥(𝑡)𝑑𝑡=𝑎𝑒^−𝑏1𝜏𝑥(𝑡−𝜏)−𝑓𝑥²(𝑡)−𝛽𝑥(𝑡)𝑦(𝑡)𝜂+𝑥(𝑡)−𝑟𝑐_𝑜(𝑡)𝑥(𝑡),𝑑𝑦(𝑡)(𝑡)𝑦(𝑡)𝑑𝑡=𝜆𝛽𝑥𝜂+𝑥(𝑡)−𝑏₂𝑦(𝑡),𝑑𝑐_𝑜(𝑡)𝑑𝑡=𝑘𝑐_𝑒(𝑡)−𝑔𝑐_𝑜(𝑡)−𝑚𝑐_𝑜(𝑡),𝑑𝑐_𝑒(𝑡)𝑑𝑡=−ℎ𝑐_𝑒(𝑡),()(𝑡)(𝑡)𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,Δ𝑥=0,Δ𝑦=𝜇₁,Δ𝑐_𝑜(𝑡)=0,Δ𝑐_𝑒(𝑡)()(𝑡)(𝑡)=0,𝑡=𝑛+𝑙−1𝑇,Δ𝑥=0,Δ𝑦=0,Δ𝑐_𝑜(𝑡)=0,Δ𝑐_𝑒(𝑡)=𝜇₂,𝑡=𝑛𝑇. (2.5) The initial conditions for system (2.5) are 𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡)=𝜙₂(𝑡),𝜙₃(𝑡),𝜙₄(𝑡),𝜙₅(𝑡)∈𝐶⁺₄,𝐶⁺₄=𝐶[−𝜏,0],𝑅⁴₊,𝑡∈[−𝜏,0],𝜙_𝑖(0)>0,𝑖=2,3,4,5. (2.6) Furthermore, since 𝑐₀(𝑡) and 𝑐_𝑒(𝑡) are the concentration of toxicant, to ensure 0≤𝑐₀(𝑡)≤1 and 0≤𝑐_𝑒(𝑡)≤1, we assume that condition 𝑔≤𝑘≤𝑔+𝑚, 𝜇₂≤1−𝑒^−ℎ𝑇 holds in this paper. Meanwhile, considering the biological meaning, we assume that 𝑘<ℎ.

In Sections 3 and 4, we mainly consider the global stability of pest-extinction solution and the uniform permanence of system (2.1); before introducing our main results, we give some preliminaries needed in next sections.

Let 𝑅₊=[0,∞) and 𝑅⁵₊={𝑋∈𝑅⁵∶𝑋>0}. Denote 𝐹=(𝑓₁,𝑓₂,𝑓₃,𝑓₄,𝑓₅) as the map defined by the right hand of system (2.1). The solution of (2.1), denoted by 𝑋(𝑡)=(𝑥_𝑗(𝑡),𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡))∶𝑅₊→𝑅⁵₊, is continuous on ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇] and ((𝑛+𝑙−1)𝑇,𝑛𝑇]. 𝑋((𝑛+𝑙−1)𝑇⁺)=lim_{𝑡→(𝑛+𝑙−1)𝑇}⁺𝑋(𝑡)and 𝑋(𝑛𝑇⁺)=lim_{𝑡→𝑛𝑇}⁺𝑋(𝑡) exist. Obviously, the global existence and uniqueness of solutions of (2.1) is guaranteed by the smoothness properties of 𝐹 (see [15]). Furthermore, the following lemma is easily obtained.

Lemma 2.1. If 𝑋(𝑡) is a solution of system (2.1) with (𝜙₁(𝑡),𝜙₂(𝑡),𝜙₃(𝑡),𝜙₄(𝑡),𝜙₅(𝑡))>0(−𝜏≤𝑡≤0), then 𝑋(𝑡)>0 for all 𝑡≥0.

Consider the following system𝑑𝑦(𝑡)𝑑𝑡=−𝑏₂𝑦(𝑡)()(𝑡),𝑡≠𝑛+𝑙−1𝑇,Δ𝑦=𝜇₁,𝑡=(𝑛+𝑙−1)𝑇. (2.7)

Lemma 2.2 (see [19]). System (2.7) has a unique positive periodic solution given by𝑦^∗(𝑡)=𝜇₁𝑒^{−𝑏2[𝑡−(𝑛+𝑙−1)𝑇]}1−𝑒^−𝑏2𝑇,for(𝑛+𝑙−1)𝑇<𝑡≤(𝑛+𝑙)𝑇 (2.8) which is globally asymptotically stable.

Consider the following system𝑑𝑣(𝑡)(𝑡)()(𝑡)()𝑑𝑡=𝑝−𝑞𝑣,𝑡≠𝑛+𝑙−1𝑇,Δ𝑣=𝜇,𝑡=𝑛+𝑙−1𝑇. (2.9)

Lemma 2.3 (see [19]). System (2.9) has a unique positive periodic solution given by𝜐^∗(𝑡)=𝑝𝑞+𝜇𝑒^{−𝑞[𝑡−(𝑛+𝑙−1)𝑇]}1−𝑒^−𝑞𝑇,for(𝑛+𝑙−1)𝑇<𝑡≤(𝑛+𝑙)𝑇 (2.10) which is globally asymptotically stable.

Now we consider some basic properties of the following subsystem of system (2.5)𝑑𝑐_𝑜(𝑡)𝑑𝑡=𝑘𝑐_𝑒(𝑡)−𝑔𝑐_𝑜(𝑡)−𝑚𝑐_𝑜(𝑡),𝑑𝑐_𝑒(𝑡)𝑑𝑡=−ℎ𝑐_𝑒(𝑡),𝑡≠𝑛𝑇Δ𝑐_𝑜(𝑡)=0,Δ𝑐_𝑒(𝑡)=𝜇₂,𝑡=𝑛𝑇,0≤𝑐_𝑜(0)≤1,0≤𝑐_𝑒(0)≤1. (2.11)

Lemma 2.4 (see [19]). System (2.11) has a unique positive 𝑇-periodic solution given by𝑐^∗₀(𝑡)=𝑐^∗₀(0)𝑒^{−(𝑔+𝑚)(𝑡−𝑛𝑇)}+𝑘𝑐^∗_𝑒(0)𝑒^{−(𝑔+𝑚)(𝑡−𝑛𝑇)}−𝑒^{−ℎ(𝑡−𝑛𝑇)},𝑐ℎ−𝑔−𝑚_∗𝑒(𝑡)=𝑐^∗_𝑒(0)𝑒^{−ℎ(𝑡−𝑛𝑇)},𝑐^∗₀(0)=𝑘𝑐^∗_𝑒(0)𝑒^{−(𝑔+𝑚)𝑇}−𝑒^−ℎ𝑇ℎ−𝑔−𝑚1−𝑒^{−(𝑔+𝑚)𝑇},𝑐^∗_𝑒(0)=𝜇₂1−𝑒^−ℎ𝑇, (2.12) for 𝑛𝑇<𝑡≤(𝑛+1)𝑇, which is globally asymptotically stable.

Lemma 2.5 (see [19]). Considering the following equation𝑑𝑥(𝑡)()(𝑡)𝑑𝑡=𝑎𝑥𝑡−𝜏−𝑏𝑥−𝑐𝑥²(𝑡), (2.13) where 𝑎, 𝑏,𝑐, and 𝜏 are all positive constants, 𝑥(𝑡)>0 for −𝜏≤𝑡≤0, one has(1)if 𝑎<𝑏, then lim_𝑡→+∞𝑥(𝑡)=0;(2)if 𝑎>𝑏, thenlim_𝑡→+∞𝑥(𝑡)=(𝑎−𝑏)/𝑐.

Definition 2.6. System (2.5) is said to be permanent if there are constants 𝑀₁,𝑀₂>0 (independent of initial value) and a finite time 𝑇₀ such that for every positive solution (𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡))∈𝑅⁺₄ with initial conditions, (2.6) satisfies 𝑀₁≤𝑥(𝑡)≤𝑀₂, 𝑀₁≤𝑦(𝑡)≤𝑀₂, 𝑀₁≤𝑐_𝑜(𝑡)≤𝑀₂, 𝑀₁≤𝑐_𝑒(𝑡)≤𝑀₂ for all 𝑡≥𝑇₀. Hence, 𝑇₀ may depend on the initial conditions (2.6).

3. Extinction and Permanence

Firstly, we show that all solutions of system (2.1) are uniformly ultimately bounded.

Theorem 3.1. There exists a constant 𝐿>0 such that 𝑥_𝑗(𝑡)≤𝐿/𝜆, 𝑥(𝑡)≤𝐿/𝜆, 𝑦(𝑡)≤𝐿, 𝑐_𝑜(𝑡)≤𝐿, 𝑐_𝑒(𝑡)≤𝐿 for each solution (𝑥_𝑗(𝑡),𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡)) of system (2.1) with large enough 𝑡.

Obviously, when the pest individuals are entirely absent from the model, that is, 𝑥(𝑡)=0 for 𝑡≥0, 𝑦(𝑡) satisfies the system (2.7). Accordingly, by Lemmas 2.2 and 2.4, we can get that system (2.1) exists for an immature and mature pest-extinction periodic solution (0,0,𝑦^∗(𝑡),𝑐^∗_𝑜(𝑡),𝑐^∗_𝑒(𝑡)), whose global attractiveness is equivalent to global attractiveness of the mature pest-extinction periodic solution (0,𝑦^∗(𝑡),𝑐^∗_𝑜(𝑡),𝑐^∗_𝑒(𝑡)) of system (2.5). In the following, we give the sufficient conditions for global attractiveness of solution (0,𝑦^∗(𝑡),𝑐^∗_𝑜(𝑡),𝑐^∗_𝑒(𝑡)).

If we denote 𝐴=𝑒^−𝑏2𝑇/(1−𝑒^−𝑏2𝑇), 𝐵=𝑘[𝑒^{−(𝑔+𝑚)𝑇}−𝑒^−ℎ𝑇]𝑒^{−(𝑔+𝑚)𝑇}/(ℎ−𝑔−𝑚)(1−𝑒^−ℎ𝑇)[1−𝑒^{−(𝑔+𝑚)𝑇}], 𝑏=min{𝑏₁,𝑏₂,ℎ−𝑘,𝑔+𝑚}, 𝜎₁=𝑒^𝑏𝑙𝑇/(𝑒^𝑏𝑇−1), 𝜎₂=𝑒^𝑏𝑇/(𝑒^𝑏𝑇−1), 𝑀₀=𝜆(𝑎+𝑏)²/4𝑓, then we have the following.

Theorem 3.2. The mature pest-extinction periodic solution (0,𝑦^∗(𝑡),𝑐^∗_𝑜(𝑡),𝑐^∗_𝑒(𝑡)) of system (2.5) is globally attractive provided that

𝜇₁>𝑎𝑒^−𝑏1𝜏𝑏𝜆𝜂+𝑀₀−𝑏𝜆𝜂𝑟𝐵−𝜎₂𝑏𝑎𝑒^−𝑏1𝜏+𝑀₀𝑟𝐵𝜇₂−𝜎₂𝑟𝑏𝐵𝜇²₂𝑏𝛽𝜆𝐴−𝜎₁𝑎𝑒^−𝑏1𝜏+𝜎₁𝑟𝐵𝜇₂ (3.1) holds true.

Remark 3.3. The pest-extinction periodic solution (0,0,y^∗(t),c^∗_o(t),c^∗_e(t)) of system (2.1) is also globally attractive if the condition (3.1) holds.

Now, we give the sufficient conditions for permanence of system (2.5). If we denote 𝜎=1/(1−𝑒^−𝑏2𝑇), 𝛿=𝑘[𝑒^{−(𝑔+𝑚)𝑇}−𝑒^−ℎ𝑇]/(ℎ−𝑔−𝑚)(1−𝑒^−ℎ𝑇)[1−𝑒^{−(𝑔+𝑚)𝑇}]+𝑘/|ℎ−𝑔−𝑚|(1−𝑒^−ℎ𝑇), and then we have the following.

Theorem 3.4. System (2.5) is permanent provided that

𝜇₁<𝜂𝑎𝑒^−𝑏1𝜏−𝑟𝛿𝜇₂𝛽𝜎 (3.2) holds true.

Remark 3.5. System (2.1) is also permanent if the condition (3.2) holds.

For convenience, the proofs of Theorems 3.1, 3.2, and 3.4 are given in Appendices A–C, respectively.

The above results show that many factors including maturation time delay, functional response of the predator, the organism's net uptake pesticide from the environment, the egestion and depuration rates of pesticide in an organism, the loss of pesticide in the environment due to natural degradation, the natural enemy releasing amount, the pesticide spraying amount, and the releasing and spraying period, can induce variation in the characteristics of populations. Meanwhile, the results imply that the modelling methods described can help in the design of appropriate control strategies and assist management decision-making. In fact, the conditions (3.1) and (3.2) imply that there exist two theoretical criteria values in system (2.1), which can be, respectively, denoted as follows: 𝜇^∗₁=𝑎𝑒^−𝑏1𝜏𝑏𝜆𝜂+𝑀₀−𝑏𝜆𝜂𝑟𝐵−𝜎₂𝑏𝑎𝑒^−𝑏1𝜏+𝑀₀𝑟𝐵𝜇₂−𝜎₂𝑟𝑏𝐵𝜇²₂𝑏𝛽𝜆𝐴−𝜎₁𝑎𝑒^−𝑏1𝜏+𝜎₁𝑟𝐵𝜇₂,𝜇₁^∗∗=𝜂𝑎𝑒^−𝑏1𝜏−𝑟𝛿𝜇₂.𝛽𝜎 (3.3) Moreover, if 𝜇₁>𝜇^∗₁, the pest-extinction periodic solution is globally asymptotically stable; if 𝜇₁<𝜇₁^∗∗, the insect pests and the natural enemies can coexist, that is, system (2.1) that we consider permanent. It is well known that, in a definitive ecological environment, the appropriate artificial release of natural enemies and spraying of pesticides play an important role in the success of pest-control. Due to the antagonism between chemical and biological methods, we should reduce the pesticide application to avoid antagonism and especially negative impacts on nontarget organisms. Theorems 3.2 and 3.4 indicate that we can choose the appropriate impulsive parameters to reduce pests to tolerable levels with little economical cost and minimal effect on the environment. Therefore, our impulsive strategy is more effective than the classical one if the chemical control is adopted rationally. To confirm our mathematical findings and facilitate their interpretation, we proceed to investigate further by using numerical simulations in the following section.

4. Numerical Simulations

In this section, numerical simulations are carried out to investigate effects of impulsive varying parameters on dynamical behaviors of system (2.1) as well as to illustrate our theoretical results. Owing to the lack of biologically realistic parametric values, the solution of the system with initial conditions in the first octant is obtained numerically for biologically feasible ranges of parametric values dominated by Theorems 3.2 and 3.4. For convenience, we assume that some parametric values of system (2.1) are kept as 𝑎=0.8,𝑏₁=0.9,𝜏=0.7,𝑓=0.2,𝛽=0.9,𝜂=1,𝑟=0.7,𝑏₂=0.3,𝜆=0.9,𝑘=1,𝑔=0.5,𝑚=0.7,ℎ=2,𝑙=0.1,𝑇=2,𝜇₂=0.1. (4.1)

Firstly, we give numerical results of the system, in which there are no impulsive perturbations (including natural enemy releasing and pesticide spraying), in other words, that is the unforced system of (2.1). The model takes the following form: 𝑑𝑥_𝑗(𝑡)(𝑡)𝑑𝑡=𝑎𝑥−𝑏₁𝑥_𝑗(𝑡)−𝑎𝑒^−𝑏1𝜏𝑥(),(𝑡)𝑡−𝜏𝑑𝑥𝑑𝑡=𝑎𝑒^−𝑏1𝜏𝑥(𝑡−𝜏)−𝑓𝑥²(𝑡)−𝛽𝑥(𝑡)𝑦(𝑡)(𝑡),(𝑡)𝜂+𝑥𝑑𝑦(𝑡)𝑦(𝑡)𝑑𝑡=𝜆𝛽𝑥𝜂+𝑥(𝑡)−𝑏₂𝑦(𝑡), (4.2) where the value of parameters for model (4.2) can be seen in (4.1). We can easily plot the time series of every population and phase portrait of the system and find that the solution of (4.2) with initial values 𝑥_𝑗(0)=0.2, 𝑥(0)=0.1,  and 𝑦(0)=0.1 would tend to a positive equilibrium solution (see Figures 1(a) and 1(b) in details). From the following discussion, we can observe that the solution of the unforced system would become unstable via impulsive perturbation. Further, it indicates that the system is impulsively controllable.

(a)

(b)

Figure 1 

Dynamical behavior of system (4.2) with 𝑎=0.8, 𝑏1=0.9, 𝜏=0.7, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑏2=0.3, 𝜆=0.9, initial values 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1. (a) Time series of system (4.2). (b) Phase portrait of system (4.2).

From theoretical criteria values formula (3.3) and the above parameter hypothesis (4.1), by a straightforward calculation, we can obtain that two theoretical criteria values of system (2.1) are 𝜇^∗₁=5.5247 and  𝜇₁^∗∗=0.1654, respectively.

Let 𝜇₁>𝜇^∗₁=5.5247, that is, the condition (3.1) holds true; we know that the pest-extinction periodic solution is globally asymptotically stable from Theorem 3.2. that is, if we let 𝜇₁=5.6>𝜇^∗₁, a typical pest-eradication periodic solution of system (2.1) with initial values 𝑥_𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐_𝑜(𝑡)=0.01, and 𝑐_𝑒(𝑡)=0.01 is shown in Figure 2(a), where we observe how the predator (natural enemy) 𝑦(𝑡) and the concentration and 𝑐_𝑒(𝑡) of pesticide in the environment and the concentration 𝑐₀(𝑡) of pesticide in the organism periodically oscillate; in contrast, both the immature pest 𝑥_𝑗(𝑡) and mature pest 𝑥(𝑡) rapidly decrease to zero. If we continue to increase 𝜇₁ and let 𝜇₁=6.5>𝜇^∗₁ and 𝜇₁=7.5>𝜇^∗₁, from Figures 2(b) and 2(c), the same phenomenon as above can be observed, respectively. This illustrates that the pest-extinction periodic solution of system (2.1) is globally asymptotically stable.

(a)

(b)

(c)

Figure 2 

Dynamical behavior (extinction) of system (2.1) with 𝑎=0.8, 𝑏1=0.9, 𝜏=0.7, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑟=0.7, 𝑏2=0.3, 𝜆=0.9, 𝑘=1, 𝑔=0.5, 𝑚=0.7, ℎ=2, 𝑙=0.1, 𝑇=2, 𝜇2=0.1, initial values 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐𝑜(𝑡)=0.01, 𝑐𝑒(𝑡)=0.01 (∗corresponding theoretical criteria value: 𝜇∗1=5.5247. (a) Time-series of system (2.1) with 𝜇1=5.6. (b) Time-series of system (2.1) with  𝜇1=6.5. (c) Time-series of system (2.1) with 𝜇1=7.5.

Let 𝜇₁<𝜇₁^∗∗=0.1654, that is, the condition (3.2) holds true; we know that system (2.1) that we consider is permanent from Theorem 3.4. That is, if we let 𝜇₁=0.16<𝜇₁^∗∗, a positive periodic solution of system (2.1) with initial values 𝑥_𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐_𝑜(𝑡)=0.0, 𝑐_𝑒(𝑡)=0.01 is shown in Figure 3, where we observe that each population of system (2.1) can coexist on a stable limit cycle. If we continue to decrease 𝜇₁, and let 𝜇₁=0.12<𝜇₁^∗∗ and 𝜇₁=0.08<𝜇₁^∗∗, from Figures 4 and 5, the same phenomenon as above can be observed, respectively. This illustrates that system (2.1) is permanent.

(a)

(b)

Figure 3 

Dynamical behavior (permanence) of system (2.1) with 𝑎=0.8, 𝑏1=0.9, 𝜏=0.7, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑟=0.7, 𝑏2=0.3, 𝜆=0.9, 𝑘=1, 𝑔=0.5, 𝑚=0.7, ℎ=2, 𝑙=0.1, and 𝑇=2, and 𝜇2=0.1; initial value 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐𝑜(𝑡)=0.01, and 𝑐𝑒(𝑡)=0.01 (∗corresponding theoretical criteria value: 𝜇1∗∗=0.1654). (a) Time-series of system (2.1) with 𝜇1=0.16. (b) Solution 𝑥𝑗(𝑡), 𝑥(𝑡), 𝑦(𝑡) with 𝜇1=0.16 will finally tend to a 𝑇-periodic solution.

(a)

(b)

Figure 4 

Dynamical behavior (permanence) of system (2.1) with 𝑎=0.8, 𝑏1=0.9, 𝜏=0.7, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑟=0.7, 𝑏2=0.3, 𝜆=0.9, 𝑘=1, 𝑔=0.5, 𝑚=0.7, ℎ=2, 𝑙=0.1, 𝑇=2, and 𝜇2=0.1, initial values 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐𝑜(𝑡)=0.01, and 𝑐𝑒(𝑡)=0.01 (∗corresponding theoretical criteria value:  𝜇1∗∗=0.1654. (a) Time-series of system (2.1) with 𝜇1=0.12. (b) Solution 𝑥𝑗(𝑡),𝑥(𝑡),𝑦(𝑡) with 𝜇1=0.12 will finally tend to a 𝑇-periodic solution.

(a)

(b)

Figure 5 

Dynamical behavior (permanence) of system (2.1) with 𝑎=0.8, 𝑏1=0.9, 𝜏=0.7, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑟=0.7, 𝑏2=0.3, 𝜆=0.9, 𝑘=1, 𝑔=0.5, 𝑚=0.7, ℎ=2, 𝑙=0.1, 𝑇=2, 𝜇2=0.1, initial values 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐𝑜(𝑡)=0.01, 𝑐𝑒(𝑡)=0.01 (∗corresponding theoretical criteria value: 𝜇1∗∗=0.1654. (a) Time series of system (2.1) with 𝜇1=0.08. (b) Solution 𝑥𝑗(𝑡),𝑥(𝑡),𝑦(𝑡) with 𝜇1=0.08 will finally tend to a 𝑇-periodic solution.

We must emphasize here that condition (3.1) and condition (3.2) are the only sufficient conditions which, respectively, assure global attractiveness of the pest-extinction periodic solution of system (2.1) and permanence of the populations. Accordingly, 𝜇^∗₁=5.5247 and 𝜇₁^∗∗=0.1654 are only two theoretical criteria values, not the threshold. Concerning the mathematical formula of theoretical threshold, we leave this for future work. We only give here an approximate threshold which can be obtained by numerical simulations. Indeed, by plotting the bifurcation diagram, we may observe that the theoretical threshold of parameter 𝜇₁ is approximately equal to 0.29 (see Figure 6 in details). That is to say, when 𝜇₁>0.29, the pest-extinction periodic solution of system (2.1) is globally asymptotically stable; reversely, when 𝜇₁<0.29, system (2.1) that we consider is permanent.

(a)

(b)

Figure 6 

Bifurcation diagrams of system (2.1), showing the effect of 𝜇1 with 𝑎=0.8, 𝑏1=0.9, 𝜏=0.7, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑟=0.7, 𝑏2=0.3, 𝜆=0.9, 𝑘=1, 𝑔=0.5, 𝑚=0.7, ℎ=2, 𝑙=0.1, 𝑇=2, and 𝜇2=0.1, initial values 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐𝑜(𝑡)=0.01, and 𝑐𝑒(𝑡)=0.01. (a) 𝑥(𝑡) is plotted for 𝜇1 over [0.05,6]. (b) 𝑦(𝑡) is plotted for 𝜇1 over [0.05,6].

According to the bifurcation theory, the properties of a dynamic system depend on certain parameter, and dynamic system with different parameters may have different dynamic behaviors. The above numerical results that we have investigated depend on parameter 𝜇₁, that is, 𝜇₁ is control parameter. In fact, from condition (3.1) and condition (3.2), the control parameter may also choose the other parameter as 𝑇, 𝜇₂, 𝑙, or 𝜏, and then the same argument as above can be continued. We only give here two numerical examples. Figure 7(a) is plotted by changing the parameter 𝜏=0.7 of Figure 3 to 𝜏=1.5. Figure 7(b) is plotted by changing the parameter 𝑇=2 of Figure 3 to 𝑇=1. As against Figure 3, Figure 7 implies that long maturation time delay and short impulsive period may induce variation in the characteristics of populations and cause pests eradication.

(a)

(b)

Figure 7 

Dynamical behavior (extinction) of system (2.1) with 𝑎=0.8, 𝑏1=0.9, 𝑓=0.2, 𝛽=0.9, 𝜂=1, 𝑟=0.7,𝑏2=0.3, 𝜆=0.9, 𝑘=1, 𝑔=0.5, 𝑚=0.7, ℎ=2, 𝑙=0.1,  and 𝜇2=0.1, initial values 𝑥𝑗(0)=0.2, 𝑥(0)=0.1, 𝑦(0)=0.1, 𝑐𝑜(𝑡)=0.01,   and 𝑐𝑒(𝑡)=0.01. (a) Time series of system (2.1) with 𝜏=1.5,𝑇=2, 𝜇1=0.16. (b) Time series of system (2.1) with 𝜏=0.7, 𝑇=1, and 𝜇1=0.16.

5. Conclusion

In this paper, in order to investigate the consequences of periodically spraying pesticides and releasing natural enemies at different fixed moment in pest-natural enemy system, a hybrid impulsive pest-control model with stage structure for pest and Holling II functional response is proposed, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Sufficient conditions for global attractiveness of the pest-extinction periodic solution and permanence of the system have been obtained, which shows that there exists a globally asymptotically stable pest-eradication periodic solution when the number 𝜇₁ of natural enemies released is more than some critical value 𝜇^∗₁ (see Figure 2), whereas the system can be permanent when the number 𝜇₁ of natural enemies released is less than another critical value 𝜇₁^∗∗ (see Figures 3, 4, and 5). Meanwhile, numerical simulation results for biologically feasible ranges of parametric values can confirm our mathematical findings and facilitate their interpretation. We also note that the conditions for the extinction or permanence in system (2.1) are quite different from the corresponding system (4.2) without impulse. For example, the system (4.2) has a positive equilibrium which is orbitally asymptotically stable (see Figure 1); however, this properties are changed via additional impulsive perturbation (see Figures 2–7). Furthermore, by plotting the bifurcation diagram (see Figure 6), we obtained the theoretical threshold of control parameter 𝜇₁, which is crucial for extinction or permanence of the population if the other parameters of system (2.1) are fixed. Finally, the numerical results, which show that long maturation time delay and short impulsive period may cause pests eradicat, have been given (see Figure 7). Obviously, these results indicate that the models proposed in this paper can help us understand pest-natural enemy interactions, to design appropriate control strategies and to make management decisions in insect pest-control. We would like to mention here that an interesting but challenging problem associated with the studies of system (2.1) should be how to optimize the number of periodically releasing natural enemy and the dosage of spraying pesticides to reduce pests to tolerable levels with little economical cost and minimal effect on the environment. We leave this for future work.

Appendices

A. Proof of Theorem 3.1

Define 𝑉(𝑡)=𝜆𝑥_𝑗(𝑡)+𝜆𝑥(𝑡)+𝑦(𝑡)+𝑐_𝑜(𝑡)+𝑐_𝑒(𝑡),  𝑏=min{𝑏₁,𝑏₂,ℎ−𝑘,𝑔+𝑚}. When 𝑡≠(𝑛+𝑙−1)𝑇 and 𝑡≠𝑛𝑇, we have 𝑑𝑉(𝑡)(𝑡)()𝑥(𝑡)𝑑𝑡+𝑏𝑉≤𝜆𝑎+𝑏−𝑓𝜆𝑥²(𝑡)≤𝑀₀, (A.1) where 𝑀₀=𝜆(𝑎+𝑏)²/(4𝑓). In addition, 𝑉((𝑛+𝑙−1)𝑇⁺)=𝑉((𝑛+𝑙−1)𝑇)+𝜇₁, 𝑉(𝑛𝑇⁺)=𝑉(𝑛𝑇)+𝜇₂.

By a straightforward calculation, when 0<(𝑛+𝑙−1)𝑇<𝑛𝑇<𝑡<(𝑛+1+𝑙−1)𝑇, we have 𝑉(𝑡)≤𝑉(0)𝑒^−𝑏𝑡+^𝑡₀𝑀₀𝑒^{−𝑏(𝑡−𝑠)}𝑑𝑠+^𝑛_𝑖=1𝜇₁𝑒^{−𝑏[𝑡−(𝑖+𝑙−1)𝑇]}+𝜇₂𝑒^{−𝑏(𝑡−𝑖𝑇)}, (A.2) and when 0<𝑛𝑇<(𝑛+1+𝑙−1)𝑇<𝑡<(𝑛+1)𝑇, we have 𝑉(𝑡)≤𝑉(0)𝑒^−𝑏𝑡+^𝑡₀𝑀₀𝑒^{−𝑏(𝑡−𝑠)}𝑑𝑠+^𝑛+1_𝑖=1𝜇₁𝑒^{−𝑏[𝑡−(𝑖+𝑙−1)𝑇]}+^𝑛_𝑖=1𝜇₂𝑒^{−𝑏(𝑡−𝑖𝑇)}. (A.3) Accordingly, we have𝑉(𝑡)≤𝑀₀𝑏+𝜇₁𝑒^𝑏𝑙𝑇+𝜇₂𝑒^𝑏𝑇𝑒^𝑏𝑇−1^Δ=𝐿as𝑡→∞. (A.4)

So 𝑉(𝑡) is uniformly ultimately bounded. By the definition of 𝑉(𝑡), we have 𝑥_𝑗(𝑡)≤𝐿/𝜆, 𝑥(𝑡)≤𝐿/𝜆, 𝑦(𝑡)≤𝐿, 𝑐_𝑜(𝑡)≤𝐿, 𝑐_𝑒(𝑡)≤𝐿 for large enough 𝑡. The proof is completed.

B. Proof of Theorem 3.2

Suppose that (𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡)) is any solution of system (2.5) with initial conditions (2.6). From system (2.5), we have 𝑑𝑦(𝑡)𝑑𝑡≥−𝑏₂𝑦(𝑡)()(𝑡),𝑡≠𝑛+𝑙−1𝑇,Δ𝑦=𝜇₁,𝑡=(𝑛+𝑙−1)𝑇. (B.1)

By Lemma (2.2), we know that 𝑢^∗(𝑡)=𝜇₁𝑒^{−𝑏2[𝑡−(𝑛+𝑙−1)𝑇]}/(1−𝑒^−𝑏2𝑇) is the unique positive periodic solution of impulsive differential equation as follows:𝑑𝑢(𝑡)𝑑𝑡=−𝑏₂𝑢(𝑡),()(𝑡)𝑡≠𝑛+𝑙−1𝑇,Δ𝑢=𝜇₁,𝑡=(𝑛+𝑙−1)𝑇. (B.2)

By comparison theorem of impulsive equation [15], for any small enough 𝜀₁>0, there exists an integer 𝑁₁ such that𝑦(𝑡)>𝑢^∗(𝑡)−𝜀₁,for𝑁₁+𝑙−1𝑇<𝑡≤𝑁₁+𝑙𝑇. (B.3) Accordingly, we obtain𝑦(𝑡)>𝜇₁𝐴−𝜀₁,for𝑁₁+𝑙−1𝑇<𝑡≤𝑁₁+𝑙𝑇. (B.4) On the other hand, from Lemma 2.4, we can easily obtain that for any small enough 𝜀₂>0, there exists an integer 𝑁₂ such that 𝑐_𝑜(𝑡)>𝜇₂𝐵−𝜀₂,for𝑁₂𝑇<𝑡≤𝑁₂+1𝑇. (B.5) Leting 𝑇=max{(𝑁₁+𝑙−1)𝑇,𝑁₂𝑇}, from the first equation of system (2.5), (B.4), (B.5), and Theorem 3.1, we have

   𝑑𝑥(𝑡)𝑑𝑡<𝑎𝑒^−𝑏1𝜏𝑥()−𝛽𝜇𝑡−𝜏¹𝐴−𝜀₁𝜇𝜂+𝐿/𝜆+𝑟₂𝐵−𝜀₂𝑥(𝑡)−𝑓𝑥²(𝑡),for𝑡>𝑇+𝜏, (B.6) where 𝐿=𝑀₀/𝑏+(𝜇₁𝑒^𝑏𝑙𝑇+𝜇₂𝑒^𝑏𝑇)/(𝑒^𝑏𝑇−1)=𝑀₀/𝑏+𝜇₁𝜎₁+𝜇₂𝜎₂ is obtained from Proof of Theorem 3.1. In the following we consider the comparison equation 𝑑𝑧(𝑡)𝑑𝑡=𝑎𝑒^−𝑏1𝜏𝑧()−𝛽𝜇𝑡−𝜏¹𝐴−𝜀₁𝜇𝜂+𝐿/𝜆+𝑟₂𝐵−𝜀₂𝑧(𝑡)−𝑓𝑧²(𝑡). (B.7) Because the condition (3.1) is equivalent to the condition 𝑎𝑒^−𝑏1𝜏<𝛽𝐴𝜇₁/(𝜂+𝐿/𝜆)+𝑟𝐵𝜇₂, therefore, we can choose 𝜀₁>0 and 𝜀₂>0 small enough such that 𝑎𝑒^−𝑏1𝜏<𝛽𝜇₁𝐴−𝜀₁𝜇𝜂+𝐿/𝜆+𝑟₂𝐵−𝜀₂. (B.8) For any solution 𝑧(𝑡) of (B.7), by Lemma 2.5 and (B.8), we can get lim_𝑡→+∞𝑧(𝑡)=0. Thus by the comparison theorem in delay differential equation and Lemma 2.1, we obtain that lim_𝑡→+∞𝑥(𝑡)≤lim_𝑡→+∞𝑧(𝑡)=0, and lim_𝑡→+∞𝑥(𝑡)=0.

Further, for any small enough  𝜀₃>0 and large enough 𝑡, we have 0<𝑥(𝑡)<𝜀₃. Without loss of generality, we may assume 0<𝑥(𝑡)<𝜀₃ for 𝑡≥0. And then from system (2.5), we obtain 𝑑𝑦(𝑡)≤𝑑𝑡𝜆𝛽𝐿𝜀₃𝜂+𝜀₃−𝑏₂𝑦(𝑡)()(𝑡),𝑡≠𝑛+𝑙−1𝑇,Δ𝑦=𝜇₁,𝑡=(𝑛+𝑙−1)𝑇. (B.9) By Lemma 2.3 and the comparison theorem in impulsive differential equation [15], for any 𝜀₄>0 is small enough, when large enough 𝑡, we have 𝑦(𝑡)<𝑌^∗(𝑡)+𝜀₄, (B.10) where 𝑌^∗(𝑡)=𝜆𝛽𝐿𝜀₃/𝑏₂(𝜂+𝜀₃)+𝜇₁𝑒^{−𝑏2[𝑡−(𝑛+𝑙−1)𝑇]}/(1−𝑒^−𝑏2𝑇) for (𝑛+𝑙−1)𝑇<𝑡≤(𝑛+𝑙)𝑇 is the unique positive periodic solution of impulsive differential equation as follows 𝑑𝑌(𝑡)=𝑑𝑡𝜆𝛽𝐿𝜀₃𝜂+𝜀₃−𝑏₂𝑌(𝑡)()(𝑡),𝑡≠𝑛+𝑙−1𝑇,Δ𝑌=𝜇₁,𝑡=(𝑛+𝑙−1)𝑇. (B.11)

Combining (B.3) with (B.10), when 𝑡 is large enough, we obtain 𝑢^∗(𝑡)−𝜀₁<𝑦(𝑡)<𝑌^∗(𝑡)+𝜀₄ (B.12) which implies lim_𝑡→+∞𝑦(𝑡)=𝑦^∗(𝑡) since 𝜀₁,𝜀₃,𝜀₄ are all sufficiently small positive constants. Moreover, by Lemma 2.4, when 𝑡→+∞, we have 𝑐_𝑜(𝑡)→𝑐^∗_𝑜(𝑡),𝑐_𝑒(𝑡)→𝑐^∗_𝑒(𝑡). Thus the proof is completed.

C. Proof of Theorem 3.4

Suppose that (𝑥(𝑡),𝑦(𝑡),𝑐_𝑜(𝑡),𝑐_𝑒(𝑡)) is any solution of system (2.5) with initial conditions (2.6). By Theorem 3.1, we have proved that there exists a constant 𝐿=𝑀₀/𝑏+(𝜇₁𝑒^𝑏𝑙𝑇+𝜇₂𝑒^𝑏𝑇)/(𝑒^𝑏𝑇−1)>0 such that 𝑥(𝑡)≤𝐿/𝜆, 𝑦(𝑡)≤𝐿, 𝑐_𝑜(𝑡)≤𝐿, 𝑐_𝑒(𝑡)≤𝐿 for large enough 𝑡. From Proof of Theorem 3.2, we know that 𝑦(𝑡)>𝜇₁𝐴−𝜀₁, 𝑐_𝑜(𝑡)>𝜇₂𝐵−𝜀₂ for large enough 𝑡 (see (B.4), and (B.5)). By Lemma 2.4, we easily obtain that 𝑐_𝑒(𝑡)>𝜇₂𝑒^−ℎ𝑇/(1−𝑒^−ℎ𝑇)−𝜀₂ for large enough 𝑡. Thus, from Definition 2.6, we only need to find a constant 𝑀₁>0 such that 𝑥(𝑡)≥𝑀₁ for 𝑡 large enough. We will do it in the following two steps.

(1) we prove that there exists a constant 𝑚₁>0 such that 𝑥(𝑡)<𝑚₁ cannot hold for all 𝑡≥𝑡₀. Otherwise, there is a constant 𝑡₀>0 such that 𝑥(𝑡)<𝑚₁ for all 𝑡≥𝑡₀. Thus, from system (2.5), when 𝑡≥𝑡₀, we have𝑑𝑦(𝑡)<𝑑𝑡𝜆𝛽𝑚₁𝐿𝜂+𝑚₁−𝑏₂𝑦(𝑡)()(𝑡),𝑡≠𝑛+𝑙−1𝑇,Δ𝑦=𝜇₁,𝑡=(𝑛+𝑙−1)𝑇. (C.1)

By Lemma (2.4) and comparison theorem of impulsive equation [15], for any 𝜀>0 small enough, there exists a 𝑇₁≥𝑡₀ such that 𝑦(𝑡)<𝜆𝛽𝑚₁𝐿𝑏₂𝜂+𝑚₁+𝜇₁1−𝑒^−𝑏2𝑇+𝜀=𝜆𝛽𝑚₁𝐿𝑏₂𝜂+𝑚₁+𝜇₁𝜎+𝜀. (C.2) For the above 𝜀, by Lemma 2.4, there exists a 𝑇₂≥𝑡₀ such that 𝑐_𝑜(𝑡)<𝜇₂𝛿+𝜀,for𝑡>𝑇₂. (C.3) Because the first equation of (2.5) can be rewritten as 𝑑𝑥(𝑡)=𝑑𝑡𝑎𝑒^−𝑏1𝜏(𝑡)−(𝑡)−𝑓𝑥𝛽𝑦𝜂+𝑥(𝑡)−𝑟𝑐_𝑜(𝑡)𝑥(𝑡)−𝑎𝑒^−𝑏1𝜏𝑑𝑑𝑡^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠. (C.4)

Now, we define 𝑉(𝑡)=𝑥(𝑡)+𝑎𝑒^{−𝑏1𝜏𝑡}_𝑡−𝜏𝑥(𝑠)𝑑𝑠. (C.5)

By calculating the derivative of 𝑉(𝑡) along system (2.5), we have 𝑑𝑉(𝑡)=𝑑𝑡𝑎𝑒^−𝑏1𝜏(𝑡)−(𝑡)−𝑓𝑥𝛽𝑦𝜂+𝑥(𝑡)−𝑟𝑐_𝑜(𝑡)𝑥(𝑡). (C.6) Let 𝑇=max{𝑇₁,𝑇₂}, then for 𝑡>𝑇, combining (C.2), (C.3), and (C.6), we have 𝑑𝑉(𝑡)>𝑑𝑡𝑎𝑒^−𝑏1𝜏−𝑓𝑚₁−𝛽𝜂𝜆𝛽𝑚₁𝐿𝑏₂𝜂+𝑚₁+𝜇₁𝜎+𝜀−𝑟𝜇₂𝛿+𝜀𝑥(𝑡). (C.7) Since condition (3.2) holds, we can choose 𝑚₁ and 𝜀 to be small enough such that 𝑎𝑒^−𝑏1𝜏>𝑓𝑚₁+𝛽𝜂𝜆𝛽𝑚₁𝐿𝑏₂𝜂+𝑚₁+𝜇₁𝜎+𝜀+𝑟𝜇₂𝛿+𝜀. (C.8) Leting 𝑚₂=min_𝑡∈[𝑇_,𝑇_+𝜏]𝑥(𝑡), we show that 𝑥(𝑡)≥𝑚₂ for 𝑡>𝑇. Otherwise, there is a nonnegative constant 𝑇₃ such that 𝑥(𝑡)≥𝑚₂ for 𝑡∈[𝑇,𝑇+𝜏+𝑇₃], 𝑥(𝑇+𝜏+𝑇₃)=𝑚₂, and 𝑥′(𝑇+𝜏+𝑇₃)<0. Further, from the first equation of (2.5), we obtain that 𝑑𝑥𝑇+𝜏+𝑇₃𝑑𝑡=𝑎𝑒^−𝑏1𝜏𝑥𝑇+𝑇₃−𝑓𝑥²𝑇+𝜏+𝑇₃−𝛽𝑥𝑇+𝜏+𝑇₃𝑦𝑇+𝜏+𝑇₃𝜂+𝑥𝑇+𝜏+𝑇₃−𝑟𝑐₀𝑇+𝜏+𝑇₃𝑥𝑇+𝜏+𝑇₃≥𝑚₂𝑎𝑒^−𝑏1𝜏−𝑓𝑚₁−𝛽𝜂𝜆𝛽𝑚₁𝐿𝑏₂𝜂+𝑚₁+𝜇₁𝜎+𝜀−𝑟𝜇₂𝛿+𝜀>0. (C.9)

This is a contradiction. So, we obtain that 𝑥(𝑡)≥𝑚₂ for 𝑡>𝑇. Combining (C.7) and (C.8), we have 𝑑𝑉(𝑡)>𝑑𝑡𝑎𝑒^−𝑏1𝜏−𝑓𝑚₁−𝛽𝜂𝜆𝛽𝑚₁𝐿𝑏₂𝜂+𝑚₁+𝜇₁𝜎+𝜀−𝑟𝜇₂𝛿+𝜀𝑚₂>0, (C.10)

for𝑡>𝑇.It implies that as 𝑡→+∞,𝑉(𝑡)→+∞. Meanwhile, by the definition of 𝑉(𝑡), we easily obtained that 𝑉(𝑡)≤𝐿(1+𝜏𝑎𝑒^−𝑏1𝜏)/𝜆. This is contradiction. Hence, for any constant 𝑡₀>0, 𝑥(𝑡)<𝑚₁ cannot hold for all 𝑡≥𝑡₀.

(2) If 𝑥(𝑡)≥𝑚₁ holds true for all large enough 𝑡, then our aim is obtained. Otherwise, 𝑥(𝑡) is oscillatory about 𝑚₁. Thus there exist two positive constant 𝑡,𝜃 such that x(𝑡)=𝑥(𝑡+𝜃)=𝑚₁ and 𝑥(𝑡)≤𝑚₁ for 𝑡<𝑡<𝑡+𝜃. Let 𝑚₃=min{𝑚₁/2,𝑚₁𝑒^{−(𝑓/𝜆+𝛽/𝜂+𝑟)𝜏𝐿}}. In the following, we firstly show that 𝑥(𝑡)≥𝑚₃ for 𝑡≤𝑡≤𝑡+𝜃 and then address that 𝑥(𝑡)≥𝑚₃ for 𝑡 large enough.

From system (2.5), we know that 𝑥(𝑡) is continuous and bounded. So, there exists a constant 𝑇₄ (0<𝑇₄<𝜏 and independent of the choice of 𝑡) such that 𝑥(𝑡)≥𝑚₁/2 for all 𝑡≤𝑡≤𝑡+𝑇₄. Moreover, when 𝑡 is large enough, by Theorem 3.1 and the first equation of (2.5), we have 𝑑𝑥(𝑡)𝑓𝑑𝑡≥−𝜆+𝛽𝜂+𝑟𝐿𝑥(𝑡),for𝑡≤𝑡≤𝑡+𝜃. (C.11)Accordingly, if 𝜃≤𝑇₄, our aim is obtained; if 𝑇₄<𝜃≤𝜏, from (C.11), we have 𝑥(𝑡)≥𝑚₁𝑒^{−(𝑓/𝜆+𝛽/𝜂+𝑟)𝜏𝐿}for𝑡<𝑡≤𝑡+𝜃≤𝑡+𝜏. (C.12)

It is obvious that 𝑥(𝑡)≥𝑚₃ for 𝑡≤𝑡≤𝑡+𝜃; if 𝜃>𝜏, from (C.11), we can obtain that 𝑥(𝑡)≥𝑚₃ for 𝑡<𝑡≤𝑡+𝜏. The same argument can be continued, so we can obtain that 𝑥(𝑡)≥𝑚₃ for 𝑡+𝜏<𝑡≤𝑡+𝜃. Since two positive constants, 𝑡, 𝜃, are arbitrarily chosen, we only assure 𝑡 to be large enough, and then we get that 𝑥(𝑡)≥𝑚₃ for 𝑡 large enough.

According to the above analysis, we can find a constant 𝑀₁>0 such that 𝑥(𝑡)≥𝑀₁ for large enough 𝑡. Thus the proof is completed.

Acknowledgments

This work is supported by Science and Research Project Foundation of Educational Department of Hubei Province (D20101902), the Key Project of Chinese Ministry of Education (210134), Youth-group Innovation Project for Colleges and Universities in Hubei (T200804), and the NFS of Hubei Province (2008CDB068). The authors would like to thank Editor Professor Z. Jin and the referees for helpful remarks that improved the paper.