On a Generalized Discrete Ratio-Dependent Predator-Prey System

Yong-Hong Fan,Lin-Lin Wang

doi:10.1155/2009/653289

Abstract

Verifiable criteria are established for the permanence and existence of positive periodic solutions of a delayed discrete predator-prey model with monotonic functional response. It is shown that the conditions that ensure the permanence of this system are similar to those of its corresponding continuous system. And the investigations generalize some well-known results. In particular, a more acceptant method is given to study the bounded discrete systems rather than the comparison theorem.

Highlights

Since the end of the 19th century, many biological models have been established to illustrate the evolutionary of species, among them, predator-prey models attracted more and more attention of biologists and mathematicians
The predator-prey interactions are described by φ x/y ; this function replaces the functional response function φ x in the traditional prey-dependent model
The authors mainly concentrated on the existence of periodic solutions and permanence for systems they considered

Summary

Introduction

Since the end of the 19th century, many biological models have been established to illustrate the evolutionary of species, among them, predator-prey models attracted more and more attention of biologists and mathematicians. Some special cases of system 1.8 have been studied, see 11, 12 and so forth In those papers, the authors mainly concentrated on the existence of periodic solutions and permanence for systems they considered. Fan and Wang see 17 considered the existence of positive periodic solutions for delayed periodic Michaelis-Menten type ratio-dependent predator-prey system. Later in , we proved that under conditions A1 and A2 , system 1.13 is permanent, so by the main result in , we can obtain Theorem 1.3, which gives another method to prove the existence of periodic solutions. It is not difficult to find: that for the continuous time model 1.3 and the discrete time model 1.13 , conditions that assure the existence of positive periodic solutions are exactly the same. In the third section, we give a proof to the main result

Preliminary

Findings

Proof of the Main Result