Stationary periodic patterns in the 1D Gray–Scott model

In this work, we study the existence and stability of a family of stationary periodic patterns in the ID Gray-Scott model. First, it is shown that these periodic solutions are born at a critical parameter value in a Turing/Ginzburg-Landau bifurcation, and an analysis of the appropriate Ginzburg-Landau normal form equation reveals that they exist below the critical parameter. Next, we analytically continue this family of periodic solutions in the ordinary differential equation for stationary solutions from the regime in which they are born and in which their spatial periods are O(l), to the regime where their spatial periods are asymptotically large. Depending on parameter values, the family terminates in global bifurcations via homoclinic orbits or in local bifurcations.