Spatiotemporal dynamics of a predation system with time delay and spatial diffusion

Abstract

This paper investigates the spatiotemporal dynamics of a Monod–Haldane type predator–prey interaction system that incorporates: (1) a time delay in the predator response term in the predator equation; and (2) diffusion in both prey and predator. We provide rigorous results of our system including the asymptotic stability of equilibrium solutions and the existence and properties of Hopf bifurcations with or without time delay and diffusion. The effect of diffusion on bifurcated periodic solutions is investigated. We show that analytically and numerically at a certain value the carrying capacity or time-delay-driven stability or instability occurs when the corresponding system has either a unique interior equilibrium or two interior equilibria. Moreover, we illustrate the spatial patterns out of an initially nearly homogeneous state via numerical simulations, which show that the system dynamics exhibits complex pattern replication: spiral wave patterns and chaotic spiral patterns by increasing the control parameter K and time delay τ respectively. In addition, we obtain further spiral patterns with different initial conditions. These results indicate that the carrying capacity and time delay play an important role in pattern selection. Our results may provide useful biological insights on population management for predator–prey interaction systems.