Hopf bifurcation and Turing instability in a predator–prey model with Michaelis–Menten functional response

Abstract

In this paper, the predator–prey model with Michaelis–Menten functional response subject to the Neumann boundary conditions is considered. The stability of the positive equilibrium and the direction of the Hopf bifurcations for the corresponding ordinary differential equation and partial differential equation are studied, respectively. To this end, the normal form method and the center manifold theorem are applied to determine the direction of the Hopf bifurcation and the stability of the bifurcated limit cycle when the diffusion terms are present. It is shown that the corresponding system can undergo either the supercritical or subcritical Hopf bifurcation at the equilibrium point within certain parameter range. Meanwhile, the Turing instability conditions for the equilibrium and the limit cycle bifurcated from the Hopf bifurcation are derived. As a result, some patterns occur in the model. Numerical simulations are carried out to verify the theoretical analysis.