| Contents |
| We are concerned with the spatiotemporal patterns in a kind of reaction diffusion Lengyel-Epstein system, which accounts for the qualitative feature of the well-known Cholrite-Iodide-Malonic Acid and Starch reaction, through which the first experimental evidence of Turing patterns was observed by De Kepper et al. Firstly, we derived the precise conditions on the system parameters so that the spatially homogenous equilibrium solution becomes Turing unstable. Secondly, we constructed a Lyapunov function to show that the spatially homogenous equilibrium solution is globally asymptotically stable when the feeding rate of iodide is small. We also showed that for small spatial domains, all the solutions eventually converge to a spatially homogeneous and time periodic solution. Finally, we proved the existence of bifurcations of spatially non-homogeneous periodic solutions and steady state solutions. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics including Turing instability and oscillatory behavior. Examples of numerical simulation are also shown to support and strengthen the analytical approach. |
|