Abstract

The application of pest management involves two thresholds when the chemical control and biological control are adopted, respectively. Our purpose is to provide an appropriate balance between the chemical control and biological control. Therefore, a Smith predator-prey system for integrated pest management is established in this paper. In this model, the intensity of implementation of biological control and chemical control depends linearly on the selected control level (threshold). Firstly, the existence and uniqueness of the order-one periodic solution (i.e., OOPS) are proved by means of the subsequent function method to confirm the feasibility of the biological and chemical control strategy of pest management. Secondly, the stability of system is proved by the limit method of the successor points’ sequences and the analogue of the Poincaré criterion. Moreover, an optimization strategy is formulated to reduce the total cost and obtain the best level of pest control. Finally, the numerical simulation of a specific model is performed.

1. Introduction

In the practical production, effective control of pests is a very important issue of the world, which catches attention of scholars for pest management method [1–7]. Integrated pest management (IPM), also known as integrated pest control (IPC), is an effective approach that integrates biological, chemical tactics, and physical methods for pests control [8–11]. Due to population dynamics and its related environment, IPM utilize effective methods and techniques comprehensively to reduce the level of economic harm caused by pests. The aim of IPM is to control the density of the insects under the economic threshold by integrated usage of less harmful pesticides and biological control methods for maximizing the protection of the ecosystem.

In mathematics, impulsive differential equations (IDES) is such a powerful tool to describe these phenomena that rapid changes in biological populations are caused by the variety of the pests control by artificial intervention [12–22]. In recent years, the theoretical studies on IDES have produced a lot of good research results [23–34]. Based on the theoretical research, some scholars have introduced impulsive differential equations in Lotka-Volterra system such as the regular release of predators [35–37]; the periodic release of infected pests [38–40]; the periodic release of predators together with regular spray of pesticides [41–43]; the periodic release of predators and infected pests together with regular spray of pesticides [39, 44]. In the practical application, the two control measures can be adopted at two different levels of pest density concerning this case. Nie et al. [45], Tian et al. [46], Zhao et al. [47], and Zhang et al. [48] studied the following predator-prey system and assumed that different control measures were adopted at different thresholds,where the intrinsic growth rate of prey is denoted by , the environment carrying capacity is denoted by , the predation rate by natural enemies is denoted by , and the transformation rate and the death rate of predator are denoted by and , respectively. The is a positive parameter, and the effect of pesticide to predator and prey species is denoted by and , respectively. The releasing quantity of natural enemy are denoted by and , respectively.

It is of great practical significance to adopt biological and chemical control strategies based on the different pest thresholds. But an important issue in this process should be pointed out, in which the biological control is carried out when the density of pest denoted by reaches the threshold , and when the density reaches the threshold , the integrated control strategy is adopted. But no strategy adopted for the density of pest denoted by , where , which is obviously unreasonable. In addition, from an economic and practical point of view, the control taken at threshold seems to be early and the amount of releasing predators will also be huge, while the control taken at threshold seems to be late and the intensity of chemical control will also be high. Considering the above problems, we should choose a pest control method between and .

An outline of this paper is as follows. In next section, a pest management Smith model is formulated. Then the existence, uniqueness, and the asymptotically orbit stability of order-one periodic solution (OOPS) of system (7) are proved in Section 3. In Section 4, an optimization problem is formulated and obtained the minimized total cost in pest control. The theoretical results are verified by numerical simulations in Section 5. Finally, a conclusion is drawn.

2. Model Formulation

In biological mathematics, Logistic model [10]is a classical mathematical model, where the predator and prey densities at time are denoted by and . denotes the intrinsic rate of growth and denotes the maximum environment carrying capacity, while system (2) is based on the assumption that the relative growth rate of the population size is linear function . In 1963, F.E.Smith found that the data about the population of Daphnia did not conform to the linear function [49]. Thus, Smith assumed that the relative growth rate of population density at time is proportional to the amount of remaining food; i.e.,where is the rate of food demand of the population at time ; is the rate of demand for food in a population saturated state. Smith assumed that the food required to keep the population is and the food required for the population to reproduce is . That is to say, Then Considering the demand for food of population reproduction, the Smith model uses the hyperbolic function instead of the linear function in the Logistic model. Thus, the Smith model is a further improvement of Logistic model. With the absence of predators, the per capita growth rate of the pest is assumed to be the Smith growth [49] model.

By the control strategy, the following predator-prey Smith system is investigated in this paper:where the releasing amount of the predator is denoted by and , , where . The strength of chemical control to the prey is and that to the predator is , where the parameters are continuous functions and satisfies , . A pest control level is between and . denotes the level of the predator at a lower density. By calculation we obtain , where are constants. When the density of predator is below , the chemical control is taken. Clearly, the control strategy of system (7) changes into the biological control strategy of system (1) when parameters , , and , of system (7), are chosen , , , , respectively. When parameters , , and , of system (7), are chosen , , , respectively, the control strategy of system (7) turns into the integrated control strategy of system (1). Therefore, system (7) is the further promotion of system (1).

In our paper, , , and are assumed to have the following linear form [10]

3. Dynamical Analysis of System (7)

In this section, we dynamically analyze system (7) to study the existence, uniqueness and orbital asymptotical stability of the OOPS. For convenience, OOPS is used to represent the order-one periodic solution.

3.1. Qualitative Analysis of System (7)

We first study the following continuous system of system (7); i.e.,LetThen we get three equilibria , , and , where Let where . Thus, we get the following theorem.

Theorem 1. The positive equilibrium point is locally asymptotically stable, if holds.

Proof. At the point , the Jacobian matrix is then When (I) holds, then . Thus the point is locally asymptotically stable.

Theorem 2. If holds, then the point is globally asymptotically stable.

Proof. Let , then we have when , then .
By the method in [48], the point is globally asymptotically stable (see Figure 1.)

Figure 1 

The phase diagram of system (7) with , , , , , , and .

3.2. Existence and Uniqueness of the Periodic Orbit of System (7)

For convenience, let denote the first integral of system (7), where the implicit function is divided into upper and lower branches by isoclinic line denoted by and , where the starting point is . The impulsive set of system (7) is denoted by and the phase set is denoted by . Let and as the isoclinic line and , respectively. For any point , where and are denoted as the abscissa and ordinate of point , respectively. By the location of the threshold and positive equilibrium point , we get the following theorem.

Theorem 3. If holds, then a uniqueness OOPS exists in system (7).

Proof.  
Case I (). In view of Theorems 1 and 2, for any point where the trajectory has an intersection point with phase set . Thus, we discuss the trajectory tendency of the initial point on the phase set .
Assuming the intersection point of phase set and isoclinic line is point , where and . The trajectory intersects with pulse set at point , then the impulsive function can transfer the point into the point . Thus, we have define By the magnitudes between and , one has
(i) .
If , the subsequent function of point is , then the trajectory is an OOPS.
If , the point under the point , thus the subsequent function of point is The phase set intersects with x-axis at point , where . By the orbit tendency, intersects with the impulsive set at the point which jumps to the point . Obviously, the point is under the point . The subsequent function of the point is According to the continuity of subsequent function, there must be a point between point and , which makes (See Figures 2(a) and 2(b).)
(ii)
If , then the point must be above the point . Thus the subsequent function . The orbit will intersect with impulsive set at point , then hits phase set at point . Clearly, the point is under the point . Thus the point must be under the point . The subsequent function of point is So there must be a point , such that (see Figure 2(c)).
Now, the uniqueness of OOPS of system (7) is to be discussed.
Assuming that , then orbit and are OOPS, where .Assume then whereandSo , ; that is to say, is a decreasing function. For , then .
Thus which is a contradiction.
When , then there exists a point such that . Thus, for any , the subsequent function of point is According to the proof above, we have . Thus, if , then .
If , then . Thus the uniqueness of OOPS of system (7) in case of is proved. The proof is completed.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 2 

The existence of the OOPS of system (7). (a) Discussion in Case (i) if . (b) Discussion in Case (i) if . (c) Discussion in Case (ii).

Theorem 4. If holds, then we have two cases. If holds, then system (7) has no OOPS. If holds, then system (7) has a unique OOPS, where .

Proof.  
Case II ). Two cases are discussed according the magnitude of and .
(i) If , the trajectory tending to point is based on the global asymptotical stability of . For any , the trajectory tends to . (See Figure 3.)
(ii) If , the trajectory must intersect with the impulsive set at the point , which jumps to the point . The subsequent function of point is . Similar to the proof of Case I, when , an OOPS exists in system (7). If , we also can prove that an OOPS exists in system (7) by same method of Case I. (See Figure 4.)

Figure 3 

The existence of the OOPS of system (2) if in Case (II).

(a)

(b)

(c)

(a)
(b)
(c)

Figure 4 

The existence of the OOPS of system (7) if in Case (II).

4. The Orbital Asymptotical Stability of OOPS of System (7)

According to the discussion above, a unique OOPS exists in system (7), denoted by . Then we get the following theorem.

Theorem 5. If , then the OOPS of system (7) is orbitally asymptotically stable and globally attractive to the point .

Proof. We choose arbitrary point on the phase set . If , then after several pulse effects the trajectory will jump to the segment . Thus we assume that ; the trajectory will hit the impulsive set at point , which jumps to the point . The trajectory will intersect with impulsive set at point and then jumps to the point . Repeat the process above; we get a point sequences , where such that The sequence is a monotonic decreasing sequence with lower bound . According to the monotonic bounded theorem, there must exist a limit such that , which means that Since , if and only if , then . That is to say .
Similarly, we can use the above method to get an increasing point sequences such that There must exist a limit such that , which means that Since , if and only if , then . That is to say, . By the arbitrariness of the point and , one has Thus the OOPS of system (7) is orbitally asymptotically stable and globally attractive (see Figure 5).