Ghennam, MaissaOuannas, Adel2024-10-162024-10-162024http://dspace.univ-oeb.dz:4000/handle/123456789/19996The aim of this work is to study the discrete Gierer-Meinhardt reaction-di_usion model, which is used to represent pattern formation. We investigate the local asymptotic behavior in the presence and in the absence of di_usion after proving that the model has a unique steady state. In addition to, we study the global asymptotic stability with the help of Lyapunov's direct method. We have also investigate the problem of synchronization control in the Gierer-Meinhardt model by using Lyapunov theory, then we have determined the conditions of synchronization. And by using the numerical simulations we validate the practical use of the techniques.enSynchronization; Diffusion systems; Discrete-timeOther