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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 218785, 8 pages
A Lyapunov Function and Global Stability for a Class of Predator-Prey Models
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.
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.
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 . 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 . And Ivlev-type functional response is classified to the prey-dependent; that is, is independent of predator .
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  gave global analysis of the following model:
But, in paper , 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 2.2. 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 .
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
Then we can get
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 . Hence, is globally asymptotically stable. This completes the proof.
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 ): 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 ): 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 ): 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 .
Example 3.4. Considering the following diffusive Ivlev-type predator-prey model (see ): 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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