Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain

Abstract

In this paper, we investigate theoretically and numerically a 2-D spatio-temporal dynamics of a predator-prey mathematical model which incorporates the Holling type II and a modified Leslie-Gower functional response and logistic growth of the prey. This system is modeled by a reaction diffusion equations defined on a disc domain { ( x , y ) ? R 2 / x 2 + y 2 < R 2 } with Dirichlet initial conditions and Neumann boundary conditions. We study the local and global stability of the positive equilibrium point. We show that the diffusion can induce instability of the uniform equilibrium point which is stable with respect to a constant perturbation as shown by Turing in 1950s and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation in a spatial domain. We perform the computations and generalize, on a circular domain, the results presented in Camara and Aziz-Alaoui 6].