Snakes, Ladders, and Breathers: Organization of Localized Patterns in Reaction-Diffusion Systems

A numerical investigation is undertaken into the existence and stability of localized patterned states of a two-species activator-inhibitor reaction-diffusion system posed on the real line. The specific system studied, originally proposed as a glycolisys model, was recently shown to feature a twoparameter bifurcation structure common to many Schnakenberg-like systems. This structure arises from a codimension-two point at which the fundamental pattern forming Turing bifurcation changes criticality. In this paper, we compute the rung states of a ``snakes-and-ladders"" bifurcation diagram. Owing to the nonvariational structure of reaction-diffusion systems, they represent traveling-wave solutions. Moreover, considering a parameter regime where the localized patterned solutions exist for a wide range of parameters, numerical evidence suggests a more complex topology of the bifurcation diagram reminiscent of, but different from, what has been described as foliated snaking. Finally, we investigate the emergence of branches of breather solutions, as the ratio of diffusion parameters is increased from small values. It is shown how these branches also inherit the snake structure which leads to multiplicity of stable localized breathers of arbitrary wide spatial extent. The results are argued to have wider implications for nonvariational pattern formation systems. In particular, the nascence of traveling and breather states is a first step in the onset for complex spatio-temp oral localized structures with permanent dynamics.