Pattern formations and instability waves for a Reaction-Diffusion system

Abstract

The mechanism of pattern formations has been widely studied and for different types of Reaction-Diffusion equations. This phenomenon has a wide range of applications in the fields of biology, chemistry, engineering, etc. In this paper, we have studied the pattern formation for a Reaction-Diffusion model with nonlinear reaction terms; this equation is different from RDM which has been studied before, and which derived from the interaction between Turing stationery and wave instability. Next, we study the possible traveling wave solution for our RDM and their stability close to the steady states. We discretize the system of Reaction-diffusion equations in one dimension using Semi-Implicit second-order difference method and we investigate the different types of travelling wave solutions (TWS). A finite element package namely COMSOL Multiphysics is used to show some types of pattern formations and for two types of initial conditions. The initial conditions are chosen to investigate the type of spots that can be formulated from the interaction. In parallel, we have proved theoretically the regions where those pattern formations can be found depending on the value of the diffusion coefficients and wave number.