Periodic solutions for a neutral delay predator-prey model with nonmonotonic functional response

Lian Duan

doi:10.14232/ejqtde.2012.1.48

Abstract

By using a continuation theorem based on coincidence degree theory, some new sufficient conditions are obtained for the existence of positive periodic solutions of the following neutral delay predator-prey model with nonmonotonic functional response: x 0 (t) = x(t)(r(t) −a(t)x(t −�(t)) −b(t)x 0 (t −�(t))) −g(x(t))y(t), y 0 (t) = y(t)(−d(t) +µ(t)g(x(t −�(t))). Moreover, an example is employed to illustrate the main results.

Highlights

In a classic study of population dynamics, the predator-prey models have been studied extensively
Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, the model (1.3) can be naturally extended to the following neutral delay predator-prey model with nonmonotonic functional response:
For the readers’ convenience, we introduce some concepts and results concerning the coincidence degree as follows

Summary

Introduction

In a classic study of population dynamics, the predator-prey models have been studied extensively. For a special case of this system, in view of time delay effect, Ruan [9] and Xiao [10] considered the bifurcation and stability of the following predator-prey model with nonmonotonic functional response x(t)[a bx(t)]. Fan and Wang [11] established verifiable criteria for the global existence of positive periodic solutions of a more general delayed predator-prey model with nonmonotonic functional response with periodic coefficients of the form x(t)[a(t) b(t)x(t)]. Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, the model (1.3) can be naturally extended to the following neutral delay predator-prey model with nonmonotonic functional response:.

The existence of a positive periodic solution

An Example