(305c) Effect of Nonlinear Diffusion on Travelling Wave and Turing Patterns in Autocatalytic Systems

Many self-regulated systems arise because of autocatalytic interactions among species. In several complex systems, these species exhibit nonlinear diffusion where the diffusion coefficient has a power-law dependency on the concentration of the species. We investigate spatio-temporal behaviour of a two-variable autocatalytic system. The two variable system represents species concentrations that autocatalytically generate each other.

The two variable system can exhibit both Turing patterns and travelling wave solutions under different conditions. A stability analysis of the steady state and phase plane analysis have been performed to obtain the travelling waves solutions. Numerical simulations show the effect of exponents on travelling waves solutions. Regions in parameter space where Turing patterns can be found are obtained using Singularity theory. The variation in critical boundary of Turing space due to nonlinear diffusion has also been obtained numerically. We have found that incorporating nonlinear diffusion effects decrease the region where Turing patterns are observed. These predictions are verified using numerical simulations for one-dimensional and two-dimensional domains. We perform a weakly nonlinear stability analysis of cubic autocatalytic system, and derive the normal form equations governing the amplitude of the patterns. The amplitude equations allow us to predict the nature of Turing patterns.