Dynamics of a diffusive predator–prey system with ratio-dependent functional response and time delay

Abstract

In this paper, we investigate the qualitative behaviors of a predator–prey system with ratio-dependent function. The system accommodates the diffusion effect to model the migration of individuals and the time delay induced by reproduction. We start with some basic properties of the system. Then the sufficient condition independent of time delay and diffusion effect for global asymptotical stability of the boundary equilibrium is obtained by using the comparison principle. Afterwards, based on the LaSalle’s invariance principle and Lyapunov functional, we investigate the global attractiveness of the positive equilibrium, arriving at its global asymptotical stability. Further, Hopf bifurcation induced by time delay around the positive equilibrium is explored. Finally, numerical examples are listed to verify the corresponding analytical results.