Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions

In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates.