With Time Delay Refuge Predator - Prey System Of Hopf Bifurcation And Periodic Solutions

A Delayed Predator-prey System incorporating a prey refuge is considered in this paper. The part stability of the system is investigated, where τ is regarded as a parameter. It is found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability direction and other properties of bifurcating periodic solutions are derived. We have given a numerical simulation to verify some of the key results we have obtained. All the results indicated that refuge had a stabilizing effect on prey - predator interactions.This paper is organized as follows: In Chapter 1, some fundamental definitions and theories are introduced; In Chapter 2, the existence and stability of non-negative equilibriums are investigated, and the conditions of stability are obtained; In Chapter 3, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions on the center manifold are determined; In Chapter 4, some numerical simulations are performed to illustrate the analytical results found; the paper ends with a brief conclusion. In Chapter 5, we discussed whether refuge had a stabilizing effect on prey - predator interactions.