Pattern formation and spatiotemporal chaos in a reaction-diffusion predator-prey system

In this paper, we consider a diffusive predator-prey system with modified Holling-Tanner functional response under homogeneous Neumann boundary condition. The qualitative analysis and Hopf bifurcation of the original ODE system are discussed, the conditions of the Turing instability for the reaction-diffusion system are derived, and the Turing space in the parameters space is achieved. We present the results of numerical simulations in order to validate the obtained analytical findings. We found some interesting spatiotemporal patterns when parameter values are taken in Turing-Hopf domain, in which the dynamics shows spatiotemporal behavior that is influenced by temporal oscillations as well as by Turing instabilities. With the help of numerical simulations, we identified the different types of spatial patterns in this diffusive predator-prey system, including stationary spatial pattern, periodic competing dynamics, and spatiotemporal chaos.