Permanence for a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and feedback controls

Qin Yue

doi:10.1186/s13662-015-0426-6

Abstract

A modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response and feedback controls is studied. By applying the differential inequality theory, sufficient conditions which guarantee the permanence of the system are obtained. Our results improve the main results of Zhang et al. (Abstr. Appl. Anal. 2014:252579, 2014). One example is presented to verify our main results.

Highlights

Let f (t) be a continuous bounded function on R, and we set f l = inf f (t), t∈R f u = sup f (t). t∈RLeslie [, ] introduced the following two species Leslie-Gower predator-prey model: x(t) = (r – b x(t))x(t) – p(x(t))y(t), y(t) = (r a y(t) x(t) )y(t), ( . )where x(t), y(t) stand for the population of the prey and the predator at time t, respectively
The predator consumes the prey according to the functional response p(x) and grows logistically with growth rate r and carrying capacity r x a proportional to the population size of the prey
Many papers have showed that feedback control variables have no influence on the permanent property of continuous system with feedback control

Summary

Introduction

In order to overcome this shortcoming, recently, AzizAlaoui and Daher Okiye [ ] suggested to add a positive constant d to the denominator and proposed the following predator-prey model with modified Leslie-Gower and Hollingtype II schemes: x (t) ). Zhu and Wang [ ] obtained sufficient conditions for the existence and global attractivity of positive periodic solutions of system Yu and Chen [ ] further considered the permanence and existence of a unique globally attractive positive almost periodic solution of system Sufficient conditions on the global asymptotic stability of a positive equilibrium were obtained by Yu [ ].

Results

Conclusion