Turing instability and formation of temporal patterns in a diffusive bimolecular model with saturation law

In the present article, a bimolecular chemical reaction-diffusion model with autocatalysis and saturation law is considered. The local asymptotic stability and instability of the unique feasible equilibrium of the local system, and the existence of Hopf bifurcation of the local system at this unique equilibrium are analyzed in detail. In the stability domain of the equilibrium of the local system, the effect of the spatial diffusion including the variation of the size of the space domain and the diffusion coefficient on the stability is studied and Turing instability is demonstrated. In the instability domain of the local system, time-periodic patterns of the original reaction-diffusion system bifurcating from the constant positive steady state are found according to the Hopf bifurcation theorem for reaction-diffusion dynamical systems with homogeneous Neumann boundary conditions by considering various different bifurcation parameters. Finally, to verify the obtained theoretical prediction, some numerical simulations are also included.