Mathematical modelling of pattern formation in activator–inhibitor reaction–diffusion systems with anomalous diffusion

Auto-wave solutions in nonlinear time-fractional reaction–diffusion systems are investigated. It is shown that stability of steady-state solutions and their subsequent evolution are mainly determined by the eigenvalue spectrum of a linearized system and level of anomalous diffusion (orders of fractional derivatives). The results of linear stability analysis are confirmed by computer simulations. To illustrate the influence of anomalous diffusion on stability properties and possible dynamics in fractional reaction–diffusion systems, we generalized two classical activator–inhibitor nonlinear models: FitzHugh–Nagumo and Brusselator. Based on them a common picture of typical nonlinear solutions in nonlinear incommensurate time-fractional activator–inhibitor systems is presented.