Localized pattern formation for mussel-algae model and other reaction-diffusion systems with saturation

We examine a class of reaction-diffusion systems where saturation drives pattern formation.A prime example is the classical model of mussel-algae interactions.We asymptotically construct a pattern consisting of N spikes.Unlike many other models where the inner spike profile is described by a nonlinear ODE w -w +w p = 0, here it is described by a linear ODE w + w -1 = 0. We then study the stability of resulting patterns.A new non-local eigenvalue problem (NLEP) is derived which controls the structural stability of N such spikes.We rigorously prove its stability in the relevant regime, taking advantage of the explicit formula for the shape of the spike.Self-replication is also observed, and we derive a new "core problem" which captures this phenomenon.Finally, the asymptotic stability band for existence of N spikes is given.Our methods are applicable to a wide range of systems that exhibit saturation; we provide a variant of Schnakenberg model as another example.Numerical simulations confirm our asymptotic predictions.