Slowly modulated two-pulse solutions in the Gray-Scott model II: Geometrictheory, bifurcations, and splitting dynamics

In this second paper, we develop a geometrical method to systematically stu dy the singular perturbed problem associated to slowly modulated two-pulse solutions. It enables one to see that the characteristics of these solution s are strongly determined by the ow on a slow manifold and, hence, also to identify the saddle-node bifurcations and bifurcations to classical traveli ng waves in which the solutions constructed in part I are created and annih ilated. Moreover, we determine the geometric origin of the critical maximum wave speeds discovered in part I. In this paper, we also study the central role of the slowly varying inhibitor component of the two-pulse solutions in the pulse-splitting bifurcations. Finally, the validity of the quasi-sta tionary approximation is established here, and we relate the results of bot h parts of this work to the literature on self-replication.