Résumé:
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.
