Travelling Wave and Turing Patterns in Subdiffusive Autocatalytic Systems

In this work, we analyse autocatalytic reactions in complex and disordered media which are governed by subdiffusion. The mean square displacement of molecules here scale as tγ where 0<γ<1. These systems are governed by fractional partial differential equations. Two systems are analysed i) in the first a logistic growth expression is used to represent the growth kinetics of bacteria. Here the system dynamics is governed by a single variable. ii) the second system is a two variable cubic autocatalytic system in a porous media. Here each reactant is involved in the autocatalytic generation of the other. These systems have multiple steady states. They exhibit traveling waves moving from an unstable steady state to a stable steady state. The minimum wave velocity has been obtained from phase plane analysis analytically for the first system. In addition, the two variable system also shows Turing patterns in selected regions of parameter space. The stability boundary for Turing patterns for subdiffusive system is found to be the same as that for regular diffusive systems obtained by Seshai et al. [1]. System behaviour as predicted by the stability analysis is verified using a robust implicit numerical method based on L1 scheme.

Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System

This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.