About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 218785, 8 pages
http://dx.doi.org/10.1155/2012/218785
Research Article

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.

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 [13] 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 [513]. 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.

References

  1. D. Alonso, F. Bartumeus, and J. Catalan, “Mutual interference between predators can give rise to Turing spatial patterns,” Ecology, vol. 83, no. 1, pp. 28–34, 2002. View at Publisher · View at Google Scholar
  2. P. Abrams and L. Ginzburg, “The nature of predation: prey dependent, ratio dependent or neither?” Trends in Ecology and Evolution, vol. 15, no. 8, pp. 337–341, 2000. View at Publisher · View at Google Scholar
  3. S. B. Hsu, “On global stability of a predator-prey system,” Mathematical Biosciences, vol. 39, no. 1-2, pp. 1–10, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, 1961.
  5. R. E. Kooij and A. Zegeling, “A predator-prey model with Ivlev's functional response,” Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 473–489, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Sugie, “Two-parameter bifurcation in a predator-prey system of Ivlev type,” Journal of Mathematical Analysis and Applications, vol. 217, no. 2, pp. 349–371, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Feng and S. Chen, “Global asymptotic behavior for the competing predators of the Ivlev types,” Mathematica Applicata, vol. 13, no. 4, pp. 85–88, 2000. View at Zentralblatt MATH · View at MathSciNet
  8. H. B. Xiao, “Global analysis of Ivlev's type predator-prey dynamic systems,” Applied Mathematics and Mechanics, vol. 28, no. 4, pp. 461–470, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. B. Collings, “Bifurcation and stability analysis of a temperature-dependent mitepredator-prey interaction model incorporating a prey refuge,” Bulletin of Mathematical Biology, vol. 57, no. 1, pp. 63–76, 1995.
  10. M. L. Rosenzweig, “Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,” Science, vol. 171, no. 3969, pp. 385–387, 1971. View at Publisher · View at Google Scholar
  11. C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 97, S45, pp. 5–60, 1965. View at Publisher · View at Google Scholar
  12. S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2000/01. View at Publisher · View at Google Scholar · View at MathSciNet
  13. W. Wang, L. Zhang, H. Wang, and Z. Li, “Pattern formation of a predator-prey system with Ivlev-type functional response,” Ecological Modelling, vol. 221, no. 2, pp. 131–140, 2010. View at Publisher · View at Google Scholar
  14. D. R. Merkin, F. F. Afagh, and A. L. Smirnov, Introduction to the Theory of Stability, vol. 24 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1997. View at MathSciNet