Spatiotemporal Dynamics and Pattern Formations of an Activator-Substrate Model with Double Saturation Terms

By using center manifold theory, Poincaré-Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.