Turing–Hopf bifurcation in a general Selkov–Schnakenberg reaction–diffusion system

Abstract

The dynamics of a general Selkov–Schnakenberg reaction–diffusion model is investigated. We first study the globally stability of the positive equilibrium and the existence of Hopf bifurcation for the corresponding ordinary differential equation (ODE). Based on that, Turing and Turing–Hopf bifurcations due to diffusion of reaction–diffusion model are demonstrated. Using the center manifold theory and the normal form method, the bifurcation diagram of Turing–Hopf bifurcation is obtained and the spatial–temporal patterns in the different regions are considered. Numerical simulations show that the spatially homogeneous periodic, the spatially inhomogeneous and the spatially inhomogeneous periodic solutions appear.