Analysis of local bifurcations in a two-component reaction-diffusion system of activator-substrate type with no-flux boundary conditions

In this paper, we investigate a reaction-diffusion system of the activator-substrate type, commonly referred to as the Gray-Scott system, under no-flux boundary conditions. First, we analyze the dynamics of the corresponding homogeneous system in the absence of diffusion. Subsequently, we examine the linear stability of a non-homogeneous steady state. By selecting an appropriate bifurcation parameter, we establish conditions under which a Hopf bifurcation occurs in the non-homogeneous system. We compute the normal form of this bifurcation up to the third order and specify the direction of the Hopf bifurcation using normal form theory. Finally, we present numerical simulations to illustrate our analytical results.