Point-condensation for a reaction-diffusion system

We consider stationary solutions of a reaction-diffusion system for an activator and an inhibitor. Let d₁ and d₂ be the respective diffusion coefficients of the activator and the inhibitor. Assuming that d₂ is sufficiently large, we construct stationary solutions which exhibit spiky patterns when d₁ is near zero. Moreover, we study the global (in d₁) structure of the solution set and show that if d₂ is sufficiently large then (a) whenever bifurcation from the constant solution occurs, there exists a continuum of nonconstant solutions which connects the point-condensation solutions with the bifurcating solutions; and (b) when no bifurcation from the constant solution occurs, the point-condensation solutions are connected to another family of point-condensation solutions.