Pattern dynamics in a bimolecular reaction–diffusion model with saturation law and cross-diffusion

This paper is concerned with a bimolecular reaction–diffusion model with saturation law and cross-diffusion and subject to Neumann boundary conditions. Firstly, both the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of the positive constant steady state of model are established through the linearization analysis. Secondly, the amplitude equations of model in proximity to the positive constant steady state are obtained by means of the method of multiple-scale time perturbation analysis and successive approximations as the bifurcation parameters are confined to the interior of Turing instability region and near Turing bifurcation curve. Thirdly, the classification and stability of Turing patterns in the diffusion bimolecular model are analyzed based on the existence and stability of the stationary solutions to the amplitude equations. It is found that the appearance of spatial diffusion in the bimolecular chemical reaction model with saturation law can give rise to nonuniform spatial patterns and lead to more complex dynamical behaviors. When the bifurcation parameters are confined to the interior of Turing instability region and near Turing bifurcation curve, the spot patterns, the strap (maze) patterns as well as the mixture of spot and strap patterns can occur. Theoretical findings show that suitable reaction–diffusion systems can be used to explain the mechanism in formation of patterns in the natural world. Finally, in order to substantiate our theoretical findings, some suitable numerical simulations are also provided according to Matlab software package and difference methods solving the approximate solutions of partial differential equations of parabolic types.