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

Abstract

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 activatorinhibitor systems is presented.

Unconditionally convergent numerical method for the fractional activator–inhibitor system with anomalous diffusion

“…Auto‐wave solutions in nonlinear time‐fractional reaction–diffusion systems are investigated in Datsko et al. [24], authors generalized two classical activator–inhibitor nonlinear models to illustrate the influence of anomalous diffusion on stability properties and possible dynamics. Since diffusion rate manifests in both diffusion constant and diffusion exponent, Zhang et al.…”

“…Gafiychuk et al [23] analyzed the linear theory of instability in detail and finished the computer simulation of the fractional activator-inhibitor systems. Auto-wave solutions in nonlinear time-fractional reaction-diffusion systems are investigated in Datsko et al [24], authors generalized two classical activator-inhibitor nonlinear models to illustrate the influence of anomalous diffusion on stability properties and possible dynamics. Since diffusion rate manifests in both diffusion constant and diffusion exponent, Zhang et al explored their interactions on the emergence of Turing patterns [25].…”

Anomalous Solute Transport in a Cylindrical Two-Zone Medium with Fractal Structure

“…The complex trajectory of fluid particles and substances in the inter-aggregate media, fractures and porous blocks causes anomalous transport, so that conventional convective transport equations cannot adequately describe solute transport, transport equations must take this anomalous into account. Such media can be considered to be fractals [24,25].…”