Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 218785, 8 pages

doi:10.1155/2012/218785

Research Article

A Lyapunov Function and Global Stability for a Class of Predator-Prey Models

Xiaoqin Wang¹ and Huihai Ma²

¹Faculty of Science, Shaanxi University of Science and Technology, Xi'an 710021, China

²College of Electrical and Information Engineering, Shaanxi University of Science and Technology, Xi'an 710021, China

Received 29 November 2012; Accepted 14 December 2012

Academic Editor: Junli Liu

Copyright © 2012 Xiaoqin Wang and Huihai Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We construct a new Lyapunov function for a class of predation models. Global stability of the positive equilibrium states of these systems can be established when the Lyapunov function is used.

1. Introduction

The dynamics of predator-prey systems are often described by differential equations, which represent time continuously. A common framework for such a model is [1–3]

where and are prey and predator densities, respectively, is the prey growth rate, is the functional response, for example, the prey consumption rate by an average single predator, and the per capita growth rate of predators (also known as the “predator numerical response"), which obviously increases with the prey consumption rate. The most widely accepted assumption for the numerical response with predator density restricting is as follows:

where is a per capita predator death rate, the conversion efficiency of food into offspring, the density dependent rate [2]. And prototype of the prey growth rate is the logistic growth

where is the carrying capacity of the prey.

When

where is the efficiency of predator capture of prey, model (1.1) is called Ivlev-type predation model, due originally to Ivlev [4]. And Ivlev-type functional response is classified to the prey-dependent; that is, is independent of predator [2].

Both ecologists and mathematicians are interested in the Ivlev-type predator-prey model and much progress has been seen in the study of the model [5–13]. Of them, Xiao [8] gave global analysis of the following model:

But, in paper [8], the author gave complex process to prove the global asymptotical stability of the positive equilibrium.

In this paper, we will establish a new Lyapunov function to prove the global stability of the positive equilibrium of model (1.5).

Our paper is organized as follows. In the next section, we discuss the existence, uniqueness of the positive equilibrium, and establish a new Lyapunov function to model (1.5). In Section 3, we will give some examples to show the robustness of our Lyapunov function.

2. Main Results

First of all, it is easy to verify that model (1.5) has two trivial equilibria (belonging to the boundary of , that is, at which one or more of populations has zero density or is extinct), namely, and . For the positive equilibrium, set

which yields

We have the following Lemma regarding the existence of the positive equilibrium.

Lemma 2.1 (see [8]). Suppose . Model (1.5) has a unique positive equilibrium if either of the following inequalities holds:

(i); (ii) and .

Lemma 2.2. Let , then is a region of attraction for all solutions of model (1.5) initiating in the interior of the positive quadrant .

Proof. Let be any solution of model (1.5) with positive initial conditions. Note that , by a standard comparison argument, we have

Then,

Similarly, since , we have

On the other hand, for all , we have and . Hence, is a region of attraction. As a consequence, we will focus on the stability of the positive equilibrium only in the region .

In the following, we devote to the global stability of the positive equilibrium for model (1.5) by constructing a new Lyapunov function which is motivated by the work of Hsu [3].

Theorem 2.3. If , the positive equilibrium of model (1.5) is globally asymptotically stable in the region .

Proof. For model (1.5), we construct a Lyapunov function of the form

Note that is non-negative, if and only if . Furthermore, the time derivative of along the solutions of (1.5) is

Substituting the expressions of and defined in (1.5) into (2.7), we can obtain

Define

then

So,

Then we can get

If

holds, which is equivalent to

Set , we obtain and

And set , we can get and

In view of , it follows that and in the region . Then is always true. It follows that , that is, . Consequently, the function satisfies the asymptotic stability theorem [14]. Hence, is globally asymptotically stable. This completes the proof.

3. Applications

In this paper, we construct a new Lyapunov function for proving the global asymptotical stability of model (1.5). The new Lyapunov function is useful not only to model (1.5), but also to other models.

In this section, we will give some examples to show the robustness of the Lyapunov function (2.6). The parameters of the following models are positive and have the same ecological meanings with those of in model (1.5).

Example 3.1. Considering the following Ivlev predator-prey model incorporating prey refuges (see [9]):

where is a refuge protecting of the prey. We can choose a Lyapunov functional as follows:

The proof is similar to that of the Section 2.

Example 3.2. Considering the following predator-prey model with Rosenzweig functional response (see [10]):

where is the victim’s competition constant. We can choose a Lyapunov functional as follows:

We omit the proof here.

Example 3.3. Considering the following model (1.1) with Holling-type functional response (see [11]):

where is known as a Holling type-II function, as a Holling type-III function and as a Holling type-IV function. We choose a Lyapunov function:

For more details, we refer to [12].

Example 3.4. Considering the following diffusive Ivlev-type predator-prey model (see [13]):

where the nonnegative constants and are the diffusion coefficients of and , respectively. , the usual Laplacian operator in two-dimensional space, is used to describe the Brownian random motion.

Model (3.7) is to be analyzed under the following non-zero initial conditions

and zero-flux boundary conditions:

In the above, is the outward unit normal vector of the boundary .

In order to give the proof of the global stability, we construct a Lyapunov function:

where

Then, differentiating with respect to time along the solutions of model (3.7), we can obtain

Using Green's first identity in the plane, and considering the zero-flux boundary conditions (3.9), one can show that

The remaining arguments are rather similar as Theorem 2.3.

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