Periodic Pattern Formation Reaction-Diffusion System

Abstract

In this study, we dealt with the analysis and mathematical modelling of the phenomenon of periodic pattern formation in reaction and diffusion systems by studying the Procellator model as one of the simple chemical models that have been proposed, where we reviewed the theoretical aspects related to partial differential equations and in particular reaction and diffusion systems that how these reactions with diffusion lead to the emergence of complex spatial and temporal structures such as spots and waves. We focused on numerically studying the behaviour of the Brusselator model using the ADI and FDM method, how patterns, especially Cheetah spots, evolve over time and showed through simulation results how periodic travelling waves contribute to the formation of these patterns and their propagation from one side to the other, and how these interactions and propagation lead to the emergence of complex spatial and temporal structures such as spots and waves. This study enabled us to understand the mathematical mechanisms behind the phenomena of self-morphology and demonstrated the ability of mathematical models to explain and represent complex natural phenomena. In the future, this study can be extended to study the effect of non-periodic terms on pattern formation or extend the study to three dimensions to understand the evolution of more complex patterns. Other techniques can be used extensively to analyse the presence and behaviour of periodic waves on larger scales and explore other simpler or more complex models that mimic multi-component chemical or biological reactions.