Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays

In this work we investigate a diffusive Gierer-Meinhardt system with gene expression time delays in the production of activators and inhibitors, and also a saturation in the activator production, which was proposed by Seirin Lee et al. (2010) [10]. We rigorously consider the basic kinetic dynamics of the Gierer-Meinhardt system with saturation. By using an upper and lower solution method, we show that when the saturation effect is strong, the unique constant steady state solution is globally attractive despite the time delays. This result limits the parameter space for which spatiotemporal pattern formation is possible.