Dynamics of a Beddington-DeAngelis Type Predator-Prey Model with Impulsive Effect

In view of the logical consistence, the model of a two-prey one-predator system with Beddington-DeAngelis functional response and impulsive control strategies is formulated and studied systematically. By using the Floquet theory of impulsive equation, small amplitude perturbation method, and comparison technique, we obtain the conditions which guarantee the global asymptotic stability of the two-prey eradication periodic solution. We also proved that the system is permanent under some conditions. Numerical simulations find that the system appears the phenomenon of competition exclusion.


Introduction
In an ecological system, understanding the dynamical relationships between predator and prey is one of the central goals.One important component of the predator-prey relationships is the predator's functional response.Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes.In 1965, Holling [1] gave three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic.The Beddington-DeAngelis functional response was introduced by Beddington [2] and DeAngelis et al. [3] in 1975.It is similar to the Holling type functional response but contains an extra term describing mutual interference by predators.Thus, a predator-prey model with Beddington-DeAngelis functional response is as the following form [2,3]: where () and () represent the population density of the prey and the predator at time , respectively.Usually,  is called the carrying capacity of the prey.The constant  is called intrinsic growth rate of the prey. is the conversion rate and  is the death rate of the predator.The term () measures the mutual interference between predators.In 2004, Fan and Kuang [4] explored the dynamics of the nonautonomous, spatially homogeneous, and continuous time predator-prey system with the Beddington-DeAngelis functional response.The explorations involved the permanence, extinction, and global asymptotic stability (general nonautonomous case); the existence, uniqueness, and stability of a positive (almost) periodic solution; and a boundary (almost) periodic solution for the periodic (almost periodic) case.
It is straightforward to generalize the above two-species model to the situation when two prey species competing for the same resource but the two prey species are predated by one predator species.This results in the following threespecies model: ( If  and  are identical species, then ( If model ( 2) is logically sound one, it should reduce itself to (3) According to the logical consistence proposed by Arditi and Michalski in 1996 [5], a logically consistent model of the two-prey one-predator interaction should take the form (5) There are a number of factors in the environment to be considered in predator-prey models.One of the important factors is an impulsive perturbation such as fire and flood; these are not suitable to be considered continuously.The impulsive perturbations bring sudden changes to the system.Let us think of prey as a pest and predator as a natural enemy of prey.There are many ways to beat agricultural pests, for example, biological control such as harvesting on prey and releasing natural enemies [6][7][8][9][10][11] and chemical control such as spreading pesticides.However, integrated pest management (IPM) is a more effective approach to control pests in farm, which has been proved by experiments (see [12][13][14]).Such tactics are discontinuous and periodical.With the idea of periodic forcing and impulsive perturbations, Baek [15] has investigated the predator-prey model with periodic constant impulsive immigration of the predator and periodic variation in the intrinsic growth rate of the prey.And an ecological model consisting of two preys and one predator with impulsive release of the predator is studied in [16].In this paper, we will consider an ecological model consisting of two preys and one predator with impulsive control strategy.We give some assumptions as follows.
(H1) Two prey species compete the same resource.
(H2) Functional response is effected for mutual interference by predators.
Thus, the model can be described by the following impulsive differential equations: where (), (), and () are the densities of the two preys and a predator at time , respectively. is the total carrying capacity of two prey species;   ( = 1, 2) are the intrinsic growth rate;   ( = 1, 2) are the catching rate;   ( = 1, 2) scale the impact of predators mutual interference;   ( = 1, 2) are saturation constants;   ( = 1, 2) denote the efficiency with which resources are converted to new consumers;  is the mortality rate of the predator. is the period of the impulsive effect;  ∈ ; is the set of all nonnegative integers;   ( = 1, 2, 3) represent the fractions of prey and predator which die due to pesticides;  is the release amount of the predator at  = .
The paper is arranged as follows.In Section 2, we give some notations and lemmas.In Section 3, we consider the local stability and global asymptotic stability of the twoprey eradication periodic solution by using Floquet theory of the impulsive equation, small amplitude perturbation, and techniques of comparison, and in Section 4 we show that the system is permanent.The paper ends with some interesting numerical simulations and conclusions.

Preliminaries
Firstly, we give some notations, definitions, and lemmas which will be useful for our main results. Let )  the map defined by the right hand of the former three equations in system (6) and denoted by  the set of all nonnegative integers.Let  :  + ×  3 + →  + ; then,  is said to belong to class  0 if (1)  is continuous in (, ( + 1)] ×  3  + and, for each (2)  is locally Lipschitzian in ; Definition 1.Let  ∈  0 ; then, for (, ) ∈ (, ( + 1)] ×  3  + , the upper right derivative of (, ) with respect to the impulsive differential system ( 6) is defined as The solution of system ( 6) is a piecewise continuous function  :  + →  3  + , () is continuous on (, (+1)],  ∈ , and ( + ) = lim  →  + () exists.The smoothness properties of  guarantee the global existence and uniqueness of solution of system (6) (see [17] for the details).Definition 2. System ( 6) is said to be permanent if there exists a compact Ω ⊂ Int 3 + such that every solution ((), (), ()) of system (6) will eventually enter and remain in the region Ω.
The following lemma is obvious.
We will use an important comparison theorem on impulsive differential equation.[17]).Suppose  ∈  0 .Assume that
To research the stability of the prey-free periodic solution of ( 6), we present the Floquet theory for the linear -periodic impulsive equations: Then, we introduce the following conditions.Let Φ() be a fundamental matrix of (10); then, there exists a unique nonsingular matrix  ∈  × such that For this equality, we call the constant matrix  is the monodromy matrix of (10) (corresponding to the fundamental matrix Φ()) All monodromy matrices of ( 10) are similar and have the same eigenvalues.The eigenvalues  1 , . . .,   of the monodromy matrices are called the Floquet multipliers of (10).
Therefore, we obtain the complete expression for the two-prey eradication periodic solution (0, 0,  * ()) of system (6).We will study the stability of the two-prey eradication periodic solution of system (6) in the next section.
The proof is completed.Now, we study the stability of the two-prey eradication periodic solution of system (6).Theorem 8. Let ((), (), ()) be any solution of system (6); then, (0, 0,  * ()) is said to be locally asymptotically stable if Proof.The local stability of periodic solution (0, 0,  * ()) may be determined by considering the behavior of small amplitude perturbation of the solution.Define Putting ( 18) into ( 6), the linearization of the system becomes Therefore, we have where Φ() satisfies Φ(0) = , the identity matrix, and ) . ( The stability of the periodic solution (0, 0,  * ()) is determined by the eigenvalues of If absolute values of all eigenvalues are less than one, then the periodic solution (0, 0,  * ()) is locally asymptotically stable.All eigenvalues of Θ are as follows: According to the Floquet theory of impulsive differential equation, (0, 0,  * ()) is locally asymptotically stable if | 1 | < 1 and | 2 | < 1; that is to say, This completes the proof.

Permanence
In this section, we investigate the permanence of the system (6).

Figure 1 :
Figure 1: Time series of a solution of model (6) with  = 3.9: (a) time series of the prey population (); (b) time series of the prey population (); (c) time series of the predator population ().

Figure 2 :
Figure 2: Time series of a solution of model (6) with  = 66: (a) time series of the prey population (); (b) time series of the prey population (); (c) time series of the predator population ().

Figure 4 :
Figure 4: Time series of a solution of model (6) with  = 60: (a) time series of the prey population (); (b) time series of the prey population (); (c) time series of the predator population ().