Dynamical behavior and bifurcation analysis of a homogeneous reaction–diffusion Atkinson system

In this paper, we are concerned with a homogeneous reaction–diffusion Atkinson oscillator system subject to homogeneous Neumann boundary conditions on a bounded spatial domain. Using the comparison principle and the techniques of invariant rectangle, we prove the existence of the attraction region of the solutions. We thus prove that under certain conditions, the solutions of the PDE system converge to the unique positive equilibrium solutions. We also derive precise conditions such that the system does not have nonconstant positive steady-state solutions. Finally, we use the bifurcation technique to show the existence of Turing patterns. The results provide a clearer understanding of the mechanism of formations of patterns.