Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

Abstract

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.