Qualitative analysis of Beddington–DeAngelis type impulsive predator–prey models

Abstract

In the paper, the impulsive predator–prey models with Beddington–DeAngelis functional response are studied. Conditions for the existence and stability of a prey-free solution and for the existence of a nontrivial periodic solution have been established. Also, we find a sufficient condition that the model is permanent and show that the model has complex dynamical behaviors via bifurcation diagrams.

Introduction

Understanding the dynamical relationships between predator and prey is one of central goals in an ecological system. One of the most important components of the predator–prey relationship are so-called functional responses. Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. The Beddington–DeAngelis functional response was introduced by Beddington [1] and DeAngelis et al. [2]. It is similar to the well-known Holling type functional response but contains an extra term describing mutual interference by predators. Actually, there is much significant evidence to suggest that functional responses with predator interference occur quite frequently in laboratory and natural systems [3]. Thus, we can establish a predator–prey model with Beddington–DeAngelis functional response as the following form [1], [4], [5]:

$$x^{\prime }\left(t\right)=ax\left(t\right)(1-\frac{x\left(t\right)}{K})-\frac{ex\left(t\right)y\left(t\right)}{by\left(t\right)+x\left(t\right)+c}\text{,}$$$$y^{\prime }\left(t\right)=-Dy\left(t\right)+\frac{mx\left(t\right)y\left(t\right)}{by\left(t\right)+x\left(t\right)+c}\text{,}$$$$(x\left(0^{+}\right),y\left(0^{+}\right))=(x_0,y_0)=x_0\text{,}$$

where $x\left(t\right),y\left(t\right)$ represent the population density of the prey and the predator at time $t$, respectively. Usually, $K$ is called the carrying capacity of the prey. The constant $a$ is called intrinsic growth rate of the prey. The constants $m,D$ are the conversion rate and the death rate of the predator, respectively. The term $by$ measures the mutual interference between predators.

As Cushing [6] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons and so on). Such perturbations were often treated continually. However, there are still some other perturbations such as fire, flood, etc, that are not suitable to be considered continually. These impulsive perturbations bring sudden change to the system. Let’s think of prey as a pest and predator as a natural enemy of prey. There are many ways to beat agricultural pests. For examples, harvesting on prey, spreading pesticides, releasing natural enemies and so on. Such tactics are discontinuous and periodical. With the idea of periodic forcing and impulsive perturbations, in this paper, we consider the following predator–prey model with periodic constant impulsive immigration of the predator and periodic variation in the intrinsic growth rate of the prey.

$$\{\begin{array}{c}\begin{array}{c}{x}^{\prime }\left(t\right)=ax\left(t\right)(1-\frac{x\left(t\right)}{K})-\frac{ex\left(t\right)y\left(t\right)}{by\left(t\right)+x\left(t\right)+c}\text{,}\hfill \\ y^{\prime }\left(t\right)=-Dy\left(t\right)+\frac{mx\left(t\right)y\left(t\right)}{by\left(t\right)+x\left(t\right)+c}\text{,}\hfill \end{array}\}t\ne nT\text{,}\\ \begin{array}{c}x\left(t^{+}\right)=(1-p_1)x\left(t\right)\text{,}\hfill \\ y\left(t^{+}\right)=(1-p_2)y\left(t\right)+q\text{,}\hfill \end{array}\}t=nT\text{,}\\ (x\left(0^{+}\right),y\left(0^{+}\right))=(x_0,y_0)=x_0\text{,}\end{array}$$

where all parameters are positive constants, $T$ is the period of the impulsive immigration or stock of the predator, $0\le p_1,p_2<1$ present the fraction of prey and predator which die due to harvesting or pesticides etc, $q$ is the size of immigration or stock of the predator. Such a model is an impulsive differential equation whose theory and applications were greatly developed by the efforts of Bainov and Lakshmikantham et al. [7], [8] and, moreover, 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 represents a more natural framework for the mathematical modeling of real world phenomenons. In recent years, models with sudden perturbations have been intensively researched, such as Lotka–Volterra [9], Holling-type [10], [11], [12], [13], Ivlev-type [10], [14] and Watt-type [15], [16]. Most of the models mentioned above have dealt with impulsive harvesting and the immigration of predators at different fixed times. On the contrary, we consider the impulsive harvesting and immigration at the same time in our model which has not been studied well until now.

The organization of this paper is as follows. In the next section, we introduce some notations and the properties of the single species equation with impulsive effect which is used in this paper. In Section 3, we show the local stability of prey-free periodic solution and the existence of a positive periodic solution. Moreover, we give a sufficient condition for the permanence of system (1.2) by applying the Floquet theory. In Section 4, we illustrate bifurcation diagrams with respect to various parameters to show dynamical varieties of the model (1.2) by using numerical simulations. Finally, we give a conclusion.