Turing Instability of Spatially Homogeneous Periodic Solutions in a Reaction–Diffusion Bimolecular Model

ABSTRACT In recent years, Turing instability in coupled reaction–diffusion systems has garnered extensive attention from numerous researchers. Nevertheless, the majority of extant studies primarily concentrate on the Turing instability of constant steady states, with comparatively few delving into the Turing instability of spatially homogeneous periodic solutions. The main purpose of the present paper is to analyze Turing instability of spatially homogeneous periodic solutions to a bimolecular chemical reaction model with saturation law and linear spatial diffusion. By means of the perturbation theory of reaction–diffusion systems, a sufficient condition to Turing instability of spatially homogeneous periodic solutions in a general higher dimensional reaction–diffusion system with linear diffusion is provided. We generalize the associated result to the scenario of a general spatial domain featuring a smooth boundary, which is neither the entirety of space nor a star‐type spatial domain. In the case of two‐dimensional reaction–diffusion system, formulas for the first Lyapunov coefficient of Hopf bifurcating periodic solutions of the associated ODE system and the first‐order derivative of period of Hopf bifurcating periodic solutions at the critical parameter value of the needed perturbed ODE system are given. Then Turing instability of spatially homogeneous periodic solutions of the bimolecular reaction–diffusion model is analyzed. Numerical simulations are also carried out in order to verify the main theoretical predictions.