Dynamics of Activator Inhibitor Reaction-Diffusion Systems

Abstract

This thesis seeks primarily to contribute to an attempt to understand the patterns we see in nature, such as pigmentation in animals, branching in trees and skeletal structures, as well as how the vast range of patterns and structures emerge from an almost uniformly homogeneous fertilized egg, through survey and study as well as improve a number of mathematical models of reaction-di usion of the type activator-inhibitor and related systems. The main objective is to conduct mathematical analysis and numerical simulations of the such models under two di erent e ects, the rst one, is time-fractional derivative instead of classical derivative; Wherewe have established algebraic conditions for the asymptotic stability of time-fractional reaction-di usion systems in these cases, commensurate/incommensurate and linear/nonlinear. We have also presented in this context, new predictor-corrector numerical schemes suitable for fractional di erential equations with/without delay and time-fractional reaction-di usion systems. Numerical formulas are presented that approximate the Caputo as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. In addition, a case study is considered where the proposed schemes is used to obtain numerical solutions of the Gierer-Meinhardt activator-inhibitor model with the aim of assessing the system’s dynamics. The second one, is the e ects of growth of spatial domain, evolving of domains was incorporated into activator-inhibitor reaction-di usion systems and others then the existence as well as the asymptotic behavior of solutions is proved under certain conditions using Lyapunov functionals combined with the regularization e ect of the parabolic equation. To con rm and validate the analytical results, numerical simulations are employed.