Permanence in a food chain system with impulsive perturbations

Abstract

In this article, we investigate three species food chain system with Holling II functional responses and periodic constant impulsive perturbations of predator at different fixed time. The conditions for extinction of prey and top predator are given. By using the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we consider the local stability of prey and top predator eradication periodic solution. Further, we obtain the conditions of permanence of the system. The complex dynamics behavior of the system (1.3) is found by using numerical method.

Introduction

Based on experiments, Holling [1] suggested three different kinds of functional responses for different kinds of species to model the phenomena of predator, which made the standard Lotka-Volterra system more realistic. Biologically, it is quite natural the existence and asymptotical stability of equilibria and limit cycles for autonomous predator–prey systems with these functional responses. But, countless organisms live in seasonally or diurnally forced environments. Periodic forcing may also affect predators and prey. Periodically forced predator–prey models exhibit a wide variety periodic solutions [2], [3]. Considering the exploited predator–prey system (harvesting or stocking) is very valuable, for it involves the human activities. It can be referred to papers [4], in which the human activities always happen in a short time or instantaneously. The continuous action of human is then removed from the model, and replaced with an impulsive perturbation. These models are subject to short-term perturbations which are often assumed to be in the form of impulsive in the modelling process. Consequently, impulsive differential equations provide a natural description of such systems [5].

Recently, it is of great interests to investigate the models with impulsive perturbations in biological populations. Funasaki and Kot [6] have studied invasion and chaos in periodically pulsed mass-action chemostat. Shulgin et al. [7] have investigated pulse vaccination strategy in SIR epidemic model. Sanyi and Lansun [8] studied single-species population models with stage structure, density-dependent birth rate and birth pulses. Shuwen et al. [9] have given the study of predator–prey system with defensive ability of prey and impulsive perturbations on the predator.

We adopt it here as a basis to develop the following Lokta-Volterra three species food chains model [10]:

$$\left\{\begin{array}{c}{x}^{\prime }(t)=x(t)(1-x(t))-\frac{a_{1}x(t)y(t)}{1+b_{1}x(t)}\hfill \\ y^{\prime }(t)=\frac{a_{1}x(t)y(t)}{1+b_{1}x(t)}-\frac{a_{2}y(t)z(t)}{1+b_{2}y(t)}-d_{1}y(t)\hfill \\ z^{\prime }(t)=\frac{a_{2}y(t)z(t)}{1+b_{2}y(t)}-d_{2}z(t)\hfill \end{array}\right.$$

where $x(t)\text{,}y(t)\text{,}z(t)$ are the densities of the prey, predator and top predator at time t, respectively, $a_1\text{,}{a}_2\text{,}{b}_1\text{,}{b}_2\text{,}{d}_1\text{,}{d}_2$ are positive constants. Observe that the simple relation of these three species: z prey on y and only on y, and y prey on x and nutrient recycling is not accounted for.

The three species food chain model (1.1) has at most five non-negative equilibrium:

$$A(0\text{,}0\text{,}0)\text{,}B(1\text{,}0\text{,}0)\text{,}C\left(\frac{d_1}{a_1-b_1{d}_1}\text{,}\frac{a_1-b_1{d}_1-d_1}{(a_1-b_1{d}_1{)}^2}\text{,}0\right)\text{,}{D}_1(x_i\text{,}{y}_i\text{,}{z}_i)\text{,}i=1\text{,}2\text{.}$$

where

$$\begin{array}{cc}\hfill & x_i=\frac{b_1-1}{2b_1}+(-1{)}^{i-1}\frac{\sqrt{(b_1+1{)}^2-\frac{4a_1{b}_1{d}_2}{a_2-b_2{d}_2}}}{2b_1}\hfill \\ \hfill & y_i=\frac{d_2}{a_2-b_2{d}_2}\hfill \\ \hfill & z_i=\frac{(a_1-b_1{d}_1)x_i-d_1}{(a_2-b_2{d}_2)(1+b_1{x}_1)}\hfill \end{array}$$

i = 1, 2. They have studied complex dynamics of system(1.1) using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of equilibrium where the species in the food chain is absent.

Shuwen and Lansun [11] have developed the model (1.1) and investigated a Holling II functional responses food chain system with periodic constant impulsive perturbation of top predator:

$$\left\{\begin{array}{c}\left.\begin{array}{c}{x}^{\prime }(t)=x(t)(1-x(t))-\frac{a_{1}x(t)y(t)}{1+b_{1}x(t)}\hfill \\ y^{\prime }(t)=\frac{a_{1}x(t)y(t)}{1+b_{1}x(t)}-\frac{a_{2}y(t)z(t)}{1+b_{2}y(t)}-d_{1}y(t)\hfill \\ z^{\prime }(t)=\frac{a_{2}y(t)z(t)}{1+b_{2}y(t)}-d_{2}z(t)\hfill \end{array}\right\}t\ne \mathit{nT}\hfill \\ \left.\begin{array}{c}\mathrm{\Delta }x(t)=0\hfill \\ \mathrm{\Delta }y(t)=0\hfill \\ \mathrm{\Delta }z(t)=p\hfill \end{array}\right\}t=\mathit{nT}\hfill \end{array}\right.$$

where $\mathrm{\Delta }x(t)=x(t^{+})-x(t)\text{,}\mathrm{\Delta }y(t)=y(t^{+})-y(t)\text{,}\mathrm{\Delta }z(t)=z(t^{+})-z(t)\text{,}p>0$ is the release amount of top predator at t = nT, T is the period of the impulsive effect, n ∈ N, N is the set of all non-negative integers.

It is shown complex dynamics of system (1.2) using bifurcation diagrams for impulsive perturbation p as bifurcation parameter.

In this paper, we will consider the following a Holling II functional responses food chain system with periodic constant impulsive perturbation of predator:

$$\left\{\begin{array}{c}\left.\begin{array}{c}{x}^{\prime }(t)=x(t)(1-x(t))-\frac{a_{1}x(t)y(t)}{1+b_{1}x(t)}\hfill \\ y^{\prime }(t)=\frac{a_{1}x(t)y(t)}{1+b_{1}x(t)}-\frac{a_{2}y(t)z(t)}{1+b_{2}y(t)}-d_{1}y(t)\hfill \\ z^{\prime }(t)=\frac{a_{2}y(t)z(t)}{1+b_{2}y(t)}-d_{2}z(t)\hfill \end{array}\right\}t\ne \mathit{nT}\hfill \\ \left.\begin{array}{c}\mathrm{\Delta }x(t)=0\hfill \\ \mathrm{\Delta }y(t)=p\hfill \\ \mathrm{\Delta }z(t)=0\hfill \end{array}\right\}t=\mathit{nT}\hfill \end{array}\right.$$

The organization of the paper is as follows. In Section 1, we introduce the basic three species food chain model with periodic constant impulsive perturbations of predator. In Section 2, we will give some notations and lemmas. In Section 3, we analyze the dynamic behavior of such a system. By using Floquet theorem and amplitude perturbation method, we show that there exists an asymptotically stable prey and top predator eradication periodic solution when impulsive effect satisfies an inequality. When the stability of the prey and top predator eradication is lost, we prove the system is permanent by analytic method. Lastly, we give a brief discussion.