Abstract

A three-species Lotka-Volterra system with impulsive control strategies containing the biological control (the constant impulse) and the chemical control (the proportional impulse) with the same period, but not simultaneously, is investigated. By applying the Floquet theory of impulsive differential equation and small amplitude perturbation techniques to the system, we find conditions for local and global stabilities of a lower-level prey and top-predator free periodic solution of the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. We also give a numerical example that seems to indicate the existence of chaotic behavior.

1. Introduction

The mathematical study of a predator-prey system in population dynamics has a long history starting with the work of Lotka and Volterra. The principles of Lotka-Volterra models have remained valid until today and many theoretical ecologists adhere to their principles [1–9]. Thus, we need to consider a Lotka-Volterra-type food chain model which can be described by the following differential equations:where , , and are the densities of the lowest-level prey, mid-level predator, and top predator at time , respectively; is called intrinsic growth rate of the prey; is the coefficient of intraspecific competition; and are the per-capita rate of predation of the predator; and denote the product of the per-capita rate of predation and the conversion rate; and denote the death rate of the predators.

Now, we regard as a pest to establish a new system dealing with impulsive pest control strategies from system (1.1). There are many ways to control pest population. One of the most important methods for pest control is chemical control. A principal substance in chemical control is pesticide. Pesticides are often useful because they quickly kill a significant portion of pest population. However, there are many deleterious effects associated with the use of chemicals that need to be reduced or eliminated. These include human illness associated with pesticide applications, insect resistance to insecticides, contamination of soil and water, and diminution of biodiversity. As a result, we should combine pesticide efficacy tests with other ways of control like biological control. Biological control is another important strategy to control pest population. It is defined as the reduction of the pest population by natural enemies and typically involves an active human role. Natural enemies of insect pests, also known as biological control agents, include predators, parasites, and pathogens. Virtually, all pests have some natural enemies, and the key to successful pest control is to identify the pest and its natural enemies and release them at fixed time for pest control. Such different pest control tactics should work together rather than against each other to accomplish successful pest population control [10–12]. Thus, in this paper, we consider the following Lotka-Volterra-type food chain model with periodic constant releasing natural enemies (mid-level predator) and spraying pesticide at different fixed time:where , is the period of the impulsive immigration or stock of the mid-level predator, present the fraction of the prey and the predator which die due to the harvesting or pesticides, and is the size of immigration or stock of the predator. Such system is an impulsive differential equation whose theories and applications were greatly developed by the efforts of Bainov and Simeonov [13] and Lakshmikantham et al. [14]. Also, Nieto and O'Regan. [18] presented a new approach to obtain the existence of solutions to some impulsive problems. Moreover [15–17], the theory of impulsive differential equations is being recognized to be not only richer than the corresponding theory of differential equations without impulses, but also to represent a more natural framework for mathematical modeling of real-world phenomena [18–22].

In recent years, many authors have studied two-dimensional predator-prey systems with impulsive perturbations [23–29]. Moreover, three-species food chain systems with sudden perturbations have been intensively researched, such as those of Holling-type [30, 31] and Beddington-type [32, 33]. However, most researches about food chain systems mentioned above have just dealt with biological control and have only given conditions for extinction of the lowest-level prey and top predator by observing the local stability of lower-level prey and top-predator free periodic solution. For this reason, the main purpose of this paper is to investigate the conditions for the extinction and the permanence of system (1.2).

The organization of the paper is as follows. In the next section, we introduce some notations and lemmas which are used in this paper. In Section 3, we find conditions for local and global stabilities of a lower-level prey and top-predator free periodic solution by applying the Floquet theory and for permanence of system (1.2) by using the comparison theorem. In Section 4, we give some numerical examples including chaotic phase portrait. Finally, we have a conclusion in Section 5.

2. Preliminaries

Now, we will introduce a few notations and definitions together with a few auxiliary results relating to comparison theorem, which will be useful for our main results.

Let and . Denote as the set of all nonnegative integers, , and as the right-hand side of the first three equations in (1.2). We first give the definition of a solution for (1.2).

Definition 2.1 (see [14]). Let be an open set and let . A function , , is said to be a solution of (1.2) if(1) and for ,(2) is continuously differentiable and satisfies the first three equations in (1.2) for , , and ,(3); then is left continous at and , and

Now, we introduce another definition to formulate the comparison result. Let , then is said to be in a class if(1) is continuous on , and exists;(2) is locally Lipschitzian in .

Definition 2.2 (see [14]). For , one defines the upper-right Dini derivative of with respect to the impulsive differential system (1.2) at by

Remark 2.3. The smoothness properties of guarantee the global existence and uniqueness of solutions of system (1.2). (See [13, 14] for details.)

We will use a comparison result of impulsive differential inequalities. We suppose that satisfies the following hypothesis.

(H) is continuous on and the limit exists and is finite for and .

Lemma 2.4 (see [14]). Suppose and where satisfies and are nondecreasing for all . Let be the maximal solution for the impulsive Cauchy problemdefined on . Then implies that , where is any solution of (2.3).

We now indicate a special case of Lemma 2.4 which provides estimations for the solution of a system of differential inequalities. For this, we let denote the class of real piecewise continuous (real piecewise continuously differentiable) functions defined on .

Lemma 2.5 (see [14]). Let the function satisfy the inequalitieswhere and , are constants, and is a strictly increasing sequence of positive real numbers. Then, for ,

Similar results can be obtained when all conditions of the inequalities in Lemmas 2.4 and 2.5 are reversed. Using Lemma 2.5, it is possible to prove that the solutions of the Cauchy problem (2.4) with strictly positive initial value remain strictly positive.

Lemma 2.6. The positive octant is an invariant region for system (1.2).

Proof. Let be a solution of system (1.2) with a strictly positive initial value . By Lemma 2.5, we can obtain that, for ,where , , and . Thus, , and remain strictly positive on .

Lemma 2.7. If , then as for any solution of the following impulsive differential equation:

Proof. It is easy to see that for a given initial condition ,for a solution of (2.8). It follows from and (2.9) thatThus, we know that the sequence is monotonically decreasing and bounded from below by 0, and so it converges to some . From (2.10), we obtain thatSince , we get . It is from (2.9) that, for ,Therefore, we have .

Now, we give the basic properties of another impulsive differential equation as follows: System (2.13) is a periodically forced linear system. It is easy to obtain that, and is a positive periodic solution of (2.13). Moreover, we can obtain thatis a solution of (2.13). From (2.14) and (2.15), we get easily the following result.

Lemma 2.8. All solutions of (2.13) tend to . That is, as .

It is from Lemma 2.8 that the general solution of (2.13) can be synchronized with the positive periodic solution of (2.13).

3. Extinction and Permanence

Firstly, we show that all solutions of (1.2) are uniformly ultimately bounded.

Theorem 3.1. There is an such that and for all large enough, where is a solution of system (1.2).

Proof. Let be a solution of (1.2) with and let for . Then, if , and , then we obtain thatChoosing , we getAs the right-hand side of (3.2) is bounded from above by , it follows thatIf , then and if , then , where . From Lemma 2.5, we get thatSince the limit of the right-hand side of (3.4) as isit easily follows that is bounded for sufficiently large . Therefore, and are bounded by a constant for sufficiently large . Hence, there is an such that and for a solution with all large enough.

Theorem 3.2. The periodic solution is locally asymptotically stable ifwhere

Proof. The local stability of the periodic solution of system (1.2) may be determined by considering the behavior of small amplitude perturbations of the solution. Let be any solution of system (1.2). Define . Then they may be written aswhere satisfiesand , the identity matrix. So the fundamental solution matrix isThe resetting impulsive conditions of system (1.2) becomeNote that all eigenvalues ofare , and . Sincethe conditions and are equivalent to the equation (3.7). By Floquet theory [13, Chapter 2], we obtain that is locally asymptotically stable.