Abstract

We construct a pest control pollution model with stage-structure for pests and with epidemic in the predator by spraying pesticides and releasing susceptible predators together. We assume that only the pests and infective predators are affected by pesticide. We show that there exists a globally attractive pest-extinction periodic solution and we get the condition of global attractiveness of the pest-extinction periodic solution by applying comparison theorem of impulsive differential equation. Further, the condition for the permanence of the system is also given.

1. Introduction

With the rapid development of modern technology, industry, and agriculture, it is of great interest to consider the effects of toxicant on ecological communities from both an environmental and conservational point of view. Qualitatively estimating the effect of a toxicant on a population by mathematical models is a relatively new field that began only in the early 1980s [13]. Population toxicant coupling has been applied in several contexts including Lotka-Volterra and chemostat-like environments, resulting in ordinary, integrodifferential, and impulsive differential equation and stochastic models. So in this paper, we consider the above effects and 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 [46].

In the natural world, many species have a life history that takes their individual members through two stages: immature and mature; the authors of [7] studied an ecological model with stage-structure for predator. A general functional response was considered, and the authors analyzed the stability and the permanence of the system. The authors [8, 9] analyzed prey-predator models with age structure and impulsive control. The authors [10] investigated the dynamics of a pest control model with age structure for pests by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moments. they analyzed the conditions for the global attractivity of the pest-extinction periodic solution and the permanence of the system.

Modeling studies on disease-dominated ecological populations have addressed issues like disease-related mortality, reduction in reproduction, change in population sizes, and disease-induced oscillation of population states. Chattopadhyay and Arino [11] formulated a prey-predator model with prey infection and observed destabilization due to infection. Venturino analyzed prey-predator models with disease in the prey [12] and the predator [13].

Motivated by the above, in this paper, we construct a pest control model with epidemic in the predator by spraying pesticides and releasing susceptible predators at the same time. The pest is stage-structured, and the effects of spraying pesticides into the environment and into the organism are considered. So the pollution model provides a natural description of such a system and should be introduced to our model.

The organization of this paper is as follows: in Section 2, we introduce a pest control pollution model with stage-structure for pest and with epidemic in the predator by introducing a constant periodic pesticide input and releasing susceptible predators together. In Section 3, we will introduce some definitions and lemmas which will be used in the paper. In Section 4, sufficient conditions are obtained for the global attractiveness of pest-extinction periodic solution. In Section 5, sufficient conditions are obtained for the permanence of the system. We give a brief conclusion of our results in the last section.

2. Model Formulation

In this paper, we suppose that pesticides hardly have influence on the susceptible predators, and the susceptible predators only feed on mature pests. Now we consider the following impulsive differential equation:̇𝑥1(𝑡)=𝛼𝑥2(𝑡)𝑑1𝑥1(𝑡)𝛼𝑒𝑑1𝜏𝑥2(𝑡𝜏)𝑟1𝑐𝑜(𝑡)𝑥1(𝑡)̇𝑥2(𝑡)=𝛼𝑒𝑑1𝜏𝑥2(𝑡𝜏)𝑓𝑥22(𝑡)𝛽1𝑥2(𝑡)𝑆(𝑡)𝑟2𝑐𝑜(𝑡)𝑥2(̇𝛽𝑡)𝑆(𝑡)=1𝑐𝑥2(𝑡)𝑆(𝑡)1+𝛽1𝑥2𝛽(𝑡)2𝑆(𝑡)𝐼(𝑡)1+𝑘𝐼(𝑡)𝑑2̇𝛽𝑆(𝑡)𝐼(𝑡)=2𝑆(𝑡)𝐼(𝑡)1+𝑘𝐼(𝑡)𝑑3𝐼(𝑡)𝑟4𝑐𝑜(𝑡)𝐼(𝑡)̇𝑐𝑜(𝑡)=𝑘𝑐𝑒(𝑡)𝑔𝑐𝑜(𝑡)𝑚𝑐𝑜(𝑡)̇𝑐𝑒(𝑡)=𝑐𝑒𝑥(𝑡),𝑡𝑛𝑇,𝑛𝑁,1𝑡+=𝑥1(𝑡),𝑥2𝑡+=𝑥2𝑡(𝑡),𝑆+𝐼𝑡=𝑆(𝑡)+𝑝+=𝐼(𝑡),𝑐𝑜𝑡+=𝑐𝑜(𝑡),𝑐𝑒𝑡+=𝑐𝑒(𝑡)+𝑞,𝑡=𝑛𝑇,𝑛𝑁.(2.1)

Here 𝑥1=𝑥1(𝑡) and 𝑥2=𝑥2(𝑡) represent the density of the immature and mature pest (the prey) at time 𝑡, respectively; 𝑆=𝑆(𝑡) and 𝐼=𝐼(𝑡) represent the density of susceptible predator and infective predator at time 𝑡, respectively; 𝑐𝑒(𝑡) represents the concentration of pesticide in the environment at time 𝑡; 𝑐𝑜(𝑡) represents the concentration of pesticide in the organism at time 𝑡; we use a special functional response, that is, when the number of the prey captured is less, the digestive capacity of the predator will increase with the density of the prey. Here 𝛼 is the birth rate of the immature pest; 𝑑1 is the death rate of the mature pest; 𝑑2 and 𝑑3 are the death rates of the susceptible predator and the infective predator, respectively; 𝜏 is the mean length of the juvenile period; 𝛽1 represents the capturing rate of the susceptible predator; represents digestive time of the susceptible predator; 𝑐 is the transformation rate of the susceptible predator; 𝑓 represents the intraspecific competition coefficient of mature pest; 𝛽2𝑆(𝑡)𝐼(𝑡)/(1+𝑘𝐼(𝑡)) represents saturation incidence rate; 𝑇 is the period of the impulsive effect; 𝑝 is the releasing amount of the susceptible predator at 𝑡=𝑛𝑇; 𝑞 is the amount of the pesticides spraying at every impulsive period 𝑛𝑇; 𝑟1,𝑟2,and𝑟4 represent the decreasing rate of the intrinsic growth rate associated with the uptake of the pesticide in the organism for the immature pest, mature pest, susceptible predator, and infective predator, respectively; 𝑘𝑐𝑒(𝑡) represents an organism’s net uptake of toxin from the environment; 𝑔𝑐𝑜(𝑡) and 𝑚𝑐𝑜(𝑡) represent the digestion and depuration rates of pesticide in an organism, respectively; 𝑐𝑒(𝑡) represents the loss of pesticide in the environment due to natural degradation. All the coefficients are positive constants.

The initial conditions of system (2.1) are𝑥1(𝑡),𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒[](𝑡)𝐶𝜏,0,𝑅6+,𝑥1(0)>0,𝑥2(0)>0,𝑆(0)>0,𝐼(0)>0,𝑐𝑜(0)>0,𝑐𝑒(0)>0,(2.2) where 𝑅6+={(𝑥1,𝑥2,𝑆,𝐼,𝑐𝑜,𝑐𝑒)𝑥10,𝑥20,𝑆0,𝐼0,𝑐𝑜0,𝑐𝑒0}. To assure the continuity of the initial values, we assume that 𝑥1(0)=0𝜏𝛼𝑒𝛾𝜃𝑥2(𝜃)𝑑𝜃. This suggests that if we know the properties of 𝑥2(𝑡), then the properties of 𝑥1(𝑡) can be obtained.

Note that the variable 𝑥1(𝑡) does not appear in the second, third, forth, fifth, and sixth equations of system (2.1), hence we only need to consider the subsystem of (2.1) as follows:̇𝑥2(𝑡)=𝛼𝑒𝑑1𝜏𝑥2(𝑡𝜏)𝑓𝑥22(𝑡)𝛽1𝑥2(𝑡)𝑆(𝑡)𝑟2𝑐𝑜(𝑡)𝑥2̇𝛽(𝑡)𝑆(𝑡)=1𝑐𝑥2(𝑡)𝑆(𝑡)1+𝛽1𝑥2𝛽(𝑡)2𝑆(𝑡)𝐼(𝑡)1+𝑘𝐼(𝑡)𝑑2̇𝛽𝑆(𝑡)𝐼(𝑡)=2𝑆(𝑡)𝐼(𝑡)1+𝑘𝐼(𝑡)𝑑3𝐼(𝑡)𝑟4𝑐𝑜(𝑡)𝐼(𝑡)̇𝑐𝑜(𝑡)=𝑘𝑐𝑒(𝑡)𝑔𝑐𝑜(𝑡)𝑚𝑐𝑜(𝑡)̇𝑐𝑒(𝑡)=𝑐𝑒(𝑥𝑡),𝑡𝑛𝑇,𝑛𝑁,2𝑡+=𝑥2𝑡(𝑡),𝑆+𝑡=𝑆(𝑡)+𝑝,𝐼+𝑐=𝐼(𝑡)𝑜𝑡+=𝑐𝑜(𝑡),𝑐𝑒𝑡+=𝑐𝑒(𝑡)+𝑞,𝑡=𝑛𝑇,𝑛𝑁.(2.3)

The initial conditions for system (2.3) are𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒[](𝑡)𝐶𝜏,0,𝑅5+,𝑥2(0)>0,𝑆(0)>0,𝐼(0)>0,𝑐𝑜(0)>0,𝑐𝑒(0)>0,(2.4) where 𝑅5+={(𝑥2,𝑆,𝐼,𝑐𝑜,𝑐𝑒)𝑥20,𝑆0,𝐼0,𝑐𝑜0,𝑐𝑒0}.

3. Definitions and Lemmas

Let 𝑅+=[0,],𝑅5+={𝑋=(𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒(𝑡))𝑥20,𝑆0,𝐼0,𝑐𝑜0,𝑐𝑒0}, and 𝑁 be the set of all nonnegative integers, let 𝑓=(𝑓1,𝑓2,𝑓3,𝑓4,𝑓5)𝑇 the map defined by the right of system (2.3). The solution of system (2.3), denoted by 𝑋=(𝑥2(𝑡),,𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒(𝑡))𝑅+𝑅5+, is continuously differentiable on (𝑛𝑇,(𝑛+1)𝑇). Let 𝑉𝑅+×𝑅5+𝑅+, then V is said to belong to class 𝑉0 if;

(1) 𝑉 is continuous on (𝑛𝑇,(𝑛+1)𝑇]×𝑅5+, and for each 𝑋(𝑡)𝑅5+,𝑛𝑁 and

lim(𝑡,𝑦)(𝑛𝑇+,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑛𝑇+,𝑥) exist;

(2) 𝑉 is locally Lipschitzian in𝑋.

Definition 3.1. Let 𝑉𝑉0, then for (𝑡,𝑋)(𝑛𝑇,(𝑛+1)𝑇]×𝑅5+, the upper right derivative of 𝑉(𝑡,𝑋) with respect to impulsive differential system (2.3) is defined as 𝐷+𝑉(𝑡,𝑋)=lim0+1sup[].𝑉(𝑡+,𝑋+𝑓(𝑡,𝑋))𝑉(𝑡,𝑋)(3.1)

Definition 3.2. System (2.3) is said to be permanent if there are constants 𝑚,𝑀>0(independent of initial value) and a finite time 𝑇0 such that for every positive solution (𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒(𝑡))𝑅5+ with initial conditions of system (2.3) satisfies 𝑚𝑥2(𝑡)𝑀, 𝑚𝑆(𝑡)𝑀, 𝑚𝐼(𝑡)𝑀, 𝑚𝑐𝑜(𝑡)𝑀,𝑚𝑐𝑒(𝑡)𝑀 for all 𝑡𝑇0, here 𝑇0 may depend on the initial condition of system (2.3).

Lemma 3.3 (the comparison theorem of impulsive differential equation [14]). Let 𝑉𝑉0. Assume that 𝐷+𝑉𝑡𝑉(𝑡,𝑥)𝑔(𝑡,𝑉(𝑡,𝑥)),𝑡𝑛𝑇,𝑡,𝑋+𝜑𝑛(𝑉(𝑡,𝑥)),𝑡=𝑛𝑇,(3.2)where 𝑔𝑅+×𝑅+𝑅 is continuous in (𝑛𝑇,(𝑛+1)𝑇]×𝑅+, and for each 𝑢𝑅+,𝑛𝑁,lim(𝑡,𝑣)(𝑛𝑇+,𝑢)𝑔(𝑡,𝑣)=𝑔(𝑛𝑇+,𝑢) exists and is finite; 𝜑𝑛𝑅+𝑅+ is nondecreasing.
Let 𝑟(𝑡) be the maximal solution of the scalar impulsive differential equation defined on [0,+), then𝑢𝑡̇𝑢(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑡𝑛𝑇,+=𝜑𝑛𝑢0(𝑢(𝑡)),𝑡=𝑛𝑇,+=𝑢0.(3.3) So 𝑉(0+,𝑋0)𝑢0 implies that 𝑉(𝑡,𝑋(𝑡))𝑟(𝑡),𝑡0,(3.4) where 𝑋(𝑡) is any solution of system (2.3).

Lemma 3.4 (see [15]). Consider the following equation: 𝑑𝑥𝑑𝑡=𝑎𝑥(𝑡𝜏)𝑏𝑥(𝑡),(3.5) where 𝑎,𝑏,𝜏>0, 𝑥(𝑡)>0 for 𝑡[𝜏,0], one has(i)if 𝑎<𝑏, then lim𝑡𝑥(𝑡)=0,(ii)if 𝑎>𝑏, then lim𝑡𝑥(𝑡)=+.

Remark 3.5 (see [16]). 𝑐𝑜(𝑡),𝑐𝑒(𝑡) are the concentration of toxicant. To assure 0𝑐𝑜(𝑡)1,0𝑐𝑒(𝑡)1, it is necessary that 𝑔𝑘𝑔+𝑚.

Remark 3.6 (see [16]). From the point of the biological meaning, we assume that 𝑘<.

Lemma 3.7. Consider the following subsystem of system (2.3) ̇𝑐𝑜(𝑡)=𝑘𝑐𝑒(𝑡)𝑔𝑐𝑜(𝑡)𝑚𝑐𝑜(𝑡)̇𝑐𝑒(𝑡)=𝑐𝑒𝑐(𝑡),𝑡𝑛𝑇,𝑛𝑁,𝑜𝑡+=𝑐𝑜𝑐(𝑡)𝑒𝑡+=𝑐𝑒(𝑡)+𝑞,𝑡=𝑛𝑇,𝑛𝑁.(3.6) Then, system (3.6) has a unique positive T-periodic solution (𝑐𝑜(𝑡),𝑐𝑒(𝑡)) and for each solution (𝑐𝑜(𝑡),𝑐𝑒(𝑡)) of system (3.6), 𝑐𝑜(𝑡)𝑐𝑜(𝑡) and 𝑐𝑒(𝑡)𝑐𝑒(𝑡) as 𝑡+. Where 𝑐𝑜(𝑡)=𝑐𝑜(0)𝑒(𝑔+𝑚)(𝑡𝑛𝑇)+𝑒𝑘𝑞(𝑔+𝑚)(𝑡𝑛𝑇)𝑒(𝑡𝑛𝑇)(𝑔𝑚)1𝑒𝑇,𝑐𝑒(𝑡)=𝑞𝑒(𝑡𝑛𝑇)1𝑒𝑇,𝑐𝑜𝑒(0)=𝑘𝑞(𝑔+𝑚)𝑇𝑒𝑇(𝑔𝑚)1𝑒𝑇1𝑒(𝑔+𝑚)𝑇,𝑐𝑒𝑞(0)=1𝑒𝑇,(3.7) for 𝑡(𝑛𝑇,(𝑛+1)𝑇] and 𝑛𝑁.

Lemma 3.8. There exists a constant 𝑀>0 such that 𝑥1(𝑡)𝑀,𝑥2(𝑡)𝑀,𝑆(𝑡)𝑀,𝐼(𝑡)𝑀,𝑐𝑜(𝑡)𝑀,𝑐𝑒(𝑡)𝑀 for each solution 𝑋(𝑡)=(𝑥1(𝑡),𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒(𝑡)) of system (2.1) with all 𝑡 large enough.

Proof. Define 𝑉(𝑡)=𝑐(𝑥1+𝑥2)+𝑆+𝐼+𝑐𝑜+𝑐𝑒. Choose 0<𝑙<min{𝑑1,𝑑2,𝑑3,𝑔+𝑚,𝑘}, we have 𝑉𝑉0 and 𝐷+𝑉(𝑡)+𝑙𝑉(𝑡)𝑐𝑓𝑥22+𝑐(𝑙+𝛼)𝑥2+𝑙𝑑2𝑆+𝑙𝑑3𝐼+(𝑙𝑔𝑚)𝑐𝑜+(𝑙+𝑘)𝑐𝑒+𝑐𝑙𝑑1𝑥1,𝑡𝑛𝑇,𝑛𝑁,𝑐𝑓𝑥22+𝑐(𝑙+𝛼)𝑥2,𝑉𝑛𝑇+=𝑉(𝑛𝑇)+𝑝+𝑞,𝑡=𝑛𝑇,𝑛𝑁.(3.8) Hence there exists a positive constant 𝐾 such that 𝐷+𝑉𝑉(𝑡)𝐾𝑙𝑉(𝑡),𝑡𝑛𝑇,𝑛𝑁,𝑛𝑇+=𝑉(𝑛𝑇)+𝑝+𝑞,𝑡=𝑛𝑇,𝑛𝑁,(3.9) by Lemma 3.3, for 𝑡0, we have 𝑉𝑉0(𝑡)+𝐾𝑙𝑒𝑙𝑡+(𝑝+𝑞)1𝑒𝑛𝑙𝑇𝑒𝑙(𝑡𝑛𝑇)1𝑒𝑙𝑇+𝐾𝑙],𝑡(𝑛𝑇,(𝑛+1)𝑇,𝑛𝑁.(3.10) Then lim𝑡𝑉(𝑡)(𝐾/𝑙)+(𝑝+𝑞)𝑒𝑙𝑇/(𝑒𝑙𝑇1).
So 𝑉(𝑡) is uniformly ultimately bounded. By the definition of 𝑉(𝑡), there exists a constant 𝑀>0 such that 𝑥1(𝑡)𝑀,𝑥2(𝑡)𝑀,𝑆(𝑡)𝑀,𝐼(𝑡)𝑀,𝑐𝑜(𝑡)𝑀,𝑐𝑒(𝑡)𝑀 for 𝑡 large enough. The proof is complete.

4. The Global Attractivity of Periodic Solution

In this section, the sufficient conditions are obtained for the global attractivity of the pest-extinction periodic solution.

We first demonstrate the expression of the pest-extinction solution of system (2.3), in which the pest individual and infective predator individual are entirely absent from the model, that is, 𝑥2(𝑡)=0,𝐼(𝑡)=0 for all 𝑡0.

When 𝑥2(𝑡)=0 and 𝐼(𝑡)=0, 𝑆(𝑡) satisfies the following system:̇𝑆(𝑡)=𝑑2𝑆𝑡𝑆,𝑡𝑛𝑇,+=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇,(4.1)

𝑆(0+)=𝑆0. Clearly, we can obtain the unique positive periodic solution with the form𝑆(𝑡)=𝑝𝑒𝑑2(𝑡𝑛𝑇)1𝑒𝑑2𝑇],𝑡(𝑛𝑇,(𝑛+1)𝑇,𝑛𝑁.(4.2)

Therefore, 𝑆(𝑡)=(𝑆(0+)𝑝/1𝑒𝑑2𝑇)𝑒𝑑2𝑡+𝑆(𝑡),𝑡(𝑛𝑇,(𝑛+1)𝑇], 𝑛𝑁 is the solution of system (4.1) with initial value 𝑆0.

Denote 𝜂1=𝑒(𝛽2/𝑘+𝑑2)𝑇/(1𝑒(𝛽2/𝑘+𝑑2)𝑇),𝜂2=𝑘(𝑒(𝑔+𝑚)𝑇𝑒𝑇)𝑒(𝑔+𝑚)𝑇/(𝑔𝑚)(1𝑒(𝑔+𝑚)𝑇)(1𝑒𝑇).

Theorem 4.1. Let 𝑋(𝑡)=(𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡)𝑐𝑜(𝑡),𝑐𝑒(𝑡)) be any solution of system (2.3) with positive initial values, if 𝛼𝑒𝑑1𝜏𝛽1𝜂1𝑝+𝑟2𝑞𝜂2𝛽<1,2𝑝𝑒𝑑2𝑇𝑑31𝑒𝑑2𝑇<1,(4.3) then (0,𝑆(𝑡),0,𝑐𝑜(𝑡),𝑐𝑒(𝑡)) is globally attractive.

Proof. From system (2.3), we have ̇𝑑𝑆(𝑡)2+𝛽2𝑘𝑆𝑡𝑆,𝑡𝑛𝑇,+=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇.(4.4) Consider the following comparison system 𝑑̇𝑢(𝑡)=2+𝛽2𝑘𝑢𝑡𝑢,𝑡𝑛𝑇,+𝑢0=𝑢(𝑡)+𝑝,𝑡=𝑛𝑇,+0=𝑆+.(4.5) Obviously, system (4.5) has a positive periodic solution 𝑢(𝑡)=𝑝𝑒(𝑑2𝛽2/𝑘)(𝑡𝑛𝑇)1𝑒(𝑑2𝛽2/𝑘)𝑇],,𝑡(𝑛𝑇,(𝑛+1)𝑇(4.6) which is globally asymptotically stable. By Lemma 3.3, we conclude that for an arbitrary positive constant 𝜀1 small enough, there exists an 𝑁1𝑍 such that 𝑆(𝑡)𝑍(𝑡)>𝑍(𝑡)𝜀1𝑁,𝑡1𝑁𝑇,1𝑇+1.(4.7) From which, we get 𝑆(𝑡)>𝑝𝑒(𝑑2𝛽2/𝑘)𝑇1𝑒(𝑑2𝛽2/𝑘)𝑇𝜀1𝑁,𝑡1𝑁𝑇,1𝑇.+1(4.8) That is, 𝑆(𝑡)>𝑝𝜂1𝜀1𝑁,𝑡1𝑁𝑇,1𝑇+1.(4.9) By Lemma 3.7, we conclude that for a sufficiently small 𝜀2>0, there exists an 𝑁2𝑍 such that 𝑐0(𝑡)>𝑐0(𝑡)𝜀2𝑁,𝑡2𝑁𝑇,2𝑇+1.(4.10) that is, 𝑐0(𝑡)>𝑞𝜂2𝜀2𝑁,𝑡2𝑁𝑇,2𝑇+1.(4.11) Let 𝑇=max{𝑁1𝑇,𝑁2𝑇}, from the first equation of systems (2.3), (4.9), and (4.11). We have ̇𝑥2(𝑡)<𝛼𝑒𝑑1𝜏𝑥2(𝑡𝜏)𝑓𝑥22(𝑡)𝛽1𝑥2(𝑡)𝑝𝜂1𝜀1𝑟2𝑥2(𝑡)𝑞𝜂2𝜀2,𝑡>𝑇+𝜏.(4.12) Now consider the following comparison equation: ̇𝑃(𝑡)=𝛼𝑒𝑑1𝜏𝑃(𝑡𝜏)𝑓𝑃2𝛽(𝑡)1𝑝𝜂1𝜀1+𝑟2𝑞𝜂2𝜀2𝑃(𝑡).(4.13) Since the first condition of the theorem holds, we can choose the above 𝜀1,𝜀2 small enough such that 𝛼𝑒𝑑1𝜏<𝛽1𝑝𝜂1𝜀1+𝑟2𝑞𝜂2𝜀2.(4.14) By (4.14) and Lemma 3.4, we have lim𝑡𝑃(𝑡)=0.
By Lemma 3.3, we get lim𝑡𝑥2(𝑡)lim𝑡𝑃(𝑡)=0.(4.15) Incorporating the positivity of 𝑥2(𝑡), we get lim𝑡𝑥2(𝑡)=0.
Then for a sufficiently small 𝜀3(0,𝑑2) and 𝑡 large enough, we have 0<𝑥2(𝑡)<𝜀3/𝑐𝛽1, without loss of generality, we may assume 0<𝑥2(𝑡)<𝜀3/𝑐𝛽1 as 𝑡0.
From the second equation of system (2.3), we havė𝑆𝜀(𝑡)3𝑑2𝑆𝑡𝑆,𝑡𝑛𝑇,+=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇.(4.16) Consider the following system: ̇𝐺𝜀(𝑡)=3𝑑2𝐺𝐺𝑡(𝑡),𝑡𝑛𝑇,+𝐺0=𝐺(𝑡)+𝑝,𝑡=𝑛𝑇,+0=𝑆+.(4.17) Obviously, system (4.17) has a positive periodic solution 𝐺(𝑡)=𝑝𝑒(𝑑2+𝜀3)(𝑡𝑛𝑇)1𝑒(𝑑2+𝜀3)𝑇,𝑛𝑇<𝑡(𝑛+1)𝑇,(4.18) which is globally asymptotically stable. Thus, for a sufficiently small 𝜀4>0, when 𝑡 is large enough, we have 𝑆(𝑡)𝐺(𝑡)<𝑉(𝑡)+𝜀4.(4.19) By the second condition of the theorem, when 𝑡 is large enough, we have 𝛽2𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇+𝜀4𝑑3<0.(4.20) By (4.19), when 𝑡 is large enough, 𝑆(𝑡)<𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇+𝜀4.(4.21) Combining the third equation of system (2.3) with (4.21), we obtain ̇𝛽𝐼(𝑡)2𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇+𝜀4𝑑3𝐼(𝑡).(4.22) By (4.20) and 𝐼0, we have 𝐼(𝑡)0 as 𝑡.
Further, since 𝐼(𝑡)0, for an arbitrary positive constant 𝜀5 small enough, we have 𝐼𝜀5 as 𝑡 is large enough. Theṅ𝑆𝑑(𝑡)2+𝛽2𝜀5𝑆𝑡𝑆,𝑡𝑛𝑇,+=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇.(4.23) Consider the following system: ̇𝑈𝑑(𝑡)=2+𝛽2𝜀5𝑈𝑡𝑈,𝑡𝑛𝑇,+𝑈0=𝑈(𝑡)+𝑝,𝑡=𝑛𝑇,+0=𝑆+.(4.24) So system (4.24) has a positive periodic solution 𝑈(𝑡)=𝑝𝑒(𝑑2+𝛽2𝜀5)(𝑡𝑛𝑇)1𝑒(𝑑2+𝛽2𝜀5)𝑇,𝑛𝑇<𝑡(𝑛+1)𝑇,(4.25) which is globally asymptotically stable. Therefore, for an arbitrary positive constant 𝜀6 small enough, when 𝑡 is large enough, we have 𝑆(𝑡)𝑈(𝑡)>𝑈(𝑡)𝜀6.(4.26) Combining (4.26) with (4.19), we obtain 𝑈(𝑡)𝜀6<𝑆(𝑡)<𝐺(𝑡)+𝜀4, since 𝜀4,𝜀5,𝜀6 are all sufficient small constants, we know lim𝑡𝑆(𝑡)=𝑆(𝑡).(4.27) By Lemma 3.7, we get lim𝑡𝑐0(𝑡)=𝑐0(𝑡),lim𝑡𝑐𝑒(𝑡)=𝑐𝑒(𝑡).(4.28) The proof is complete.

Remark 4.2. Obviously, we know that the global attractiveness of pest-eradication periodic solution (0,0,𝑆(𝑡),0,𝑐𝑜(𝑡),𝑐𝑒(𝑡)) of system (2.2) is equivalent to the global attractiveness of mature pest-eradication periodic solution (0,𝑆(𝑡),0,𝑐𝑜(𝑡),𝑐𝑒(𝑡)) of system (2.3).

5. Permanence

Theorem 5.1. System (2.3) is permanent provided that 𝛼𝑒𝑑1𝜏𝛽1𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇𝑟2𝑒𝑘𝑞(𝑔+𝑚)𝑇𝑒𝑇(𝑔𝑚)1𝑒𝑇1𝑒(𝑔+𝑚)𝑇𝛽>0,2𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇𝑑3>1.(5.1)

Proof. By Lemma 3.8, we know that there exists an 𝑀>0,and 𝑀>(𝛼𝑒𝑑1𝜏)/𝑓 such that 𝑥2(𝑡)𝑀, 𝑆(𝑡)𝑀, 𝐼(𝑡)𝑀, 𝑐𝑜(𝑡)𝑀,𝑐𝑒(𝑡)𝑀 for all 𝑡>0, we will prove the theorem through the following five steps.Step 1. From (4.9), we know that there exists an 𝑚2=𝑝𝜂1𝜀1 such that 𝑆(𝑡)𝑚2 for 𝑡 large enough.Step 2. From (4.11), we know that there exists an 𝑚3=𝑞𝜂2𝜀2 such that 𝑐0(𝑡)𝑚3 for 𝑡 large enough.Step 3. From Lemma 3.7, for an arbitrary positive constant 𝜀0 small enough, when 𝑡 is large enough, 𝑐𝑒(𝑡)𝑞𝑒𝑇/(1𝑒𝑇)𝜀0𝑚4.Step 4. We will prove that there exists an 𝑚1>0 such that 𝑥2(𝑡)𝑚1 for 𝑡 large enough, we will do it in the following two steps.
(i) By the condition of the theorem, we can select positive constants 𝜀7 and 𝑚5 small enough such that𝑚5<𝛼𝑒𝑑1𝜏𝑓,𝛿=𝛽1𝑐𝑚5<𝑑2,𝛼𝑒𝑑1𝜏𝑓𝑚5𝛽1𝑝𝑒(𝑑2+𝛿)𝑇1𝑒(𝑑2+𝛿)𝑇𝛽1𝜀7𝑟2𝜂𝑟2𝜀2>0.(5.2) Now we will prove that 𝑥2(𝑡)<𝑚5 cannot hold for all 𝑡0, otherwise, ̇𝑆(𝑡)𝑑2+𝛿𝑆.(5.3) Let ̇𝑅(𝑡)=𝑑2𝑅𝑡+𝛿𝑅,𝑡𝑛𝑇,+𝑅0=𝑅(𝑡)+𝑝,𝑡=𝑛𝑇,+0=𝑆+.(5.4)
By Lemma 3.3, 𝑆(𝑡)𝑅(𝑡) and 𝑅(𝑡)𝑅(𝑡),𝑡, where 𝑅(𝑡)=𝑝𝑒(𝑑2+𝛿)(𝑡𝑛𝑇)/(1𝑒(𝑑2+𝛿)𝑇),𝑡(𝑛𝑇,(𝑛+1)𝑇]. So there exists a 𝑇1>0 such that𝑆(𝑡)𝑅(𝑡)<𝑅(𝑡)+𝜀7,(5.5) for 𝑡>𝑇1. From Lemma 3.7, we know that when 𝑡 is large enough, 𝑐0(𝑡)<𝑐𝑜(0)𝑒(𝑔+𝑚)𝑇+𝑒𝑘𝑞(𝑔+𝑚)𝑇𝑒𝑇(𝑔𝑚)1𝑒𝑇+𝜀2𝜂+𝜀2,(5.6) for a sufficiently small 𝜀2>0, then ̇𝑥2(𝑡)𝛼𝑒𝑑1𝜏𝑥2(𝑡𝜏)𝑓𝑚5+𝛽1𝑝𝑒(𝑑2+𝛿)𝑇1𝑒(𝑑2+𝛿)𝑇+𝜀7+𝑟2𝜂+𝜀2𝑥2(𝑡),(5.7) by Lemma 3.4, 𝑥2(𝑡), as 𝑡. This is a contradiction to the boundedness of 𝑥2(𝑡). So there exists a 𝑡1>0 such that 𝑥2(𝑡1)𝑚5.
(ii) If 𝑥2(𝑡)𝑚5 for all 𝑡𝑡1, our aim is obtained. Otherwise, we consider that 𝑥2(𝑡) is oscillating about 𝑚5.
Let 𝑡2=inf𝑡𝑡1{𝑥2(𝑡)<𝑚5}, then 𝑥2(𝑡)𝑚5 for 𝑡[𝑡1,𝑡2), since 𝑥2(𝑡) is continuous, 𝑥2(𝑡2)=𝑚5; and because 𝑥2(𝑡) is oscillating about 𝑚5, we know that there exists a 𝑡3=inf𝑡𝑡2{𝑥2(𝑡)>𝑚5}, then 𝑥2(𝑡)𝑚5 for 𝑡[𝑡2,𝑡3), from the continuity of 𝑥2(𝑡), we get 𝑥2(𝑡3)=𝑚5; and continue we can obtain the time sequence 𝑡1𝑡2<𝑡3<<𝑡2𝑘<𝑡2𝑘+1<, which satisfies
(a)when 𝑖=2,3,4,,𝑥2(𝑡𝑖)=𝑚5,(b)when 𝑡(𝑡2𝑘,𝑡2𝑘+1),𝑘=1,2,,𝑥2(𝑡𝑖)<𝑚5,(c)when 𝑡(𝑡2𝑘+1,𝑡2𝑘+2),𝑘=1,2,,𝑥2(𝑡𝑖)>𝑚5.
We claim that there exists a 𝑇0=sup{𝑡2𝑘+1𝑡2𝑘, 𝑘𝑁}, otherwise, there exists a subsequence {𝑇𝑗𝑇𝑗=𝑡2𝑘𝑗+1𝑡2𝑘𝑗, 𝑗𝑁} such that lim𝑗𝑇𝑗=+; obviously, (5.7) holds when 𝑡(𝑡2𝑘𝑗,𝑡2𝑘𝑗+1).
By Lemma 3.4, we get lim𝑗𝑥2(𝑡2𝑘𝑗+1)=+; this is a contradiction to 𝑥2(𝑡2𝑘𝑗+1)=𝑚5.
By the boundedness of the system, we have 𝑆(𝑡)<𝑀,𝑐0(𝑡)<𝑀,(5.8) for 𝑡𝑡1. It is clear that ̇𝑥2(𝑡)𝑓𝑚5+𝛽1𝑀+𝑟2𝑀𝑥2(𝑡),𝑥2𝑡2𝑘=𝑚5,(5.9) for 𝑡(𝑡2𝑘,𝑡2𝑘+1), 𝑘=1,2, Let 𝑚1=𝑚5𝑒(𝑓𝑚5+𝛽1𝑀+𝑟2𝑀)𝑇0, then we have 𝑥2(𝑡)𝑚1 for all 𝑡𝑡1.
Step 5. We will prove that there exists an 𝑚6>0 such that 𝐼(𝑡)𝑚6 for 𝑡 large enough, we will do it in the following two steps.
(i) From the condition of the theorem, let 𝑚7>0 and 𝜀8>0 be small enough such that𝛽21+𝑘𝑚7𝑝𝑒(𝛽2𝑚7+𝑑2)𝑇1𝑒(𝛽2𝑚7+𝑑2)𝑇𝜀8𝑑3𝑟4𝑀>0.(5.10) We will prove 𝐼(𝑡)<𝑚7 cannot hold for all 𝑡0. Otherwise, we have ̇𝑆𝛽(𝑡)2𝑚7+𝑑2𝑆(𝑡).(5.11) Consider the following system: ̇𝑄𝑑(𝑡)=2+𝛽2𝑚7𝑄𝑡𝑄,𝑡𝑛𝑇,+𝑄0=𝑄(𝑡)+𝑝,𝑡=𝑛𝑇,+0=𝑆+.(5.12) By Lemma 3.3, we have 𝑆(𝑡)𝑄(𝑡) and 𝑄(𝑡)𝑄(𝑡), as 𝑡, where 𝑄(𝑡)=𝑝𝑒(𝑑2+𝛽2𝑚7)(𝑡𝑛𝑇)/(1𝑒(𝑑2+𝛽2𝑚7)𝑇),𝑡(𝑛𝑇,(𝑛+1)𝑇].
So there exists a 𝑇2>0, when 𝑡>𝑇2𝑆(𝑡)𝑄(𝑡)>𝑄(𝑡)𝜀8.
Hence, ̇𝛽𝐼(𝑡)21+𝑘𝑚7𝑝𝑒(𝛽2𝑚7+𝑑2)𝑇1𝑒(𝛽2𝑚7+𝑑2)𝑇𝜀8𝑑3𝑟4𝑀𝐼(𝑡),(5.13) from (5.10), we get 𝐼(𝑡), as 𝑡, this is a contradiction to the boundedness of 𝐼(𝑡).So there exists a 𝑡0>0 such that 𝐼(𝑡0)𝑚7.
(ii) Similar to the method of step (ii) of Step 4, we can find an 𝑚6=𝑚7𝑒𝑑3𝑇0 such that 𝐼(𝑡)𝑚6 for all 𝑡𝑡0.

Therefore, 𝑚𝑥1(𝑡),𝑥2(𝑡),𝑆(𝑡),𝐼(𝑡),𝑐𝑜(𝑡),𝑐𝑒(𝑡)𝑀 for 𝑡 large enough, where 𝑚=min{𝑚1,𝑚2,𝑚3,𝑚4,𝑚6}. The proof is complete.

6. Conclusion

In this paper, we propose and analyze a pest control model with age structure for pest and pulse spraying pesticides and pulse releasing infective predators. By Lemma 3.8, we know that any solution of system (2.1) is bounded for 𝑡 large enough and get the specific form of the upper boundedness. From Theorem 4.1, we get the sufficient condition of global attractiveness of the pest-extinction periodic solution:𝛼𝑒𝑑1𝜏𝛽1𝜂1𝑝+𝑟2𝑞𝜂2𝛽<1,2𝑝𝑒𝑑2𝑇𝑑31𝑒𝑑2𝑇<1.(6.1)

By Theorem 5.1, we get the sufficient condition for the permanence of the system:𝛼𝑒𝑑1𝜏𝛽1𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇𝑟2𝑒𝑘𝑞(𝑔+𝑚)𝑇)𝑒𝑇(𝑔𝑚)1𝑒𝑇1𝑒(𝑔+𝑚)𝑇𝛽>0,2𝑝𝑒𝑑2𝑇1𝑒𝑑2𝑇𝑑3>1.(6.2)