Stable Spikes for a Reaction-Diffusion System Modeling Color Pattern Formation

Abstract

We consider a reaction–diffusion system for color pattern formation with two activators and one inhibitor. Each of the activators models one of the colors being switched on, for example the first activator could represent the color blue and the second activator the color yellow. If both colors are present the pattern will have green color since the color green is achieved by a mixture of the colors blue and yellow. We prove rigorous results on the existence and stability of spikes for which one of the colors or both of them are switched on. To the best of our knowledge, this paper is the first study of spike solutions for a reaction–diffusion system with two activator and one inhibitor systems and arbitrary strength of the self-activation and cross-activation terms. We classify the different types of solutions which can exist depending on the choice of interaction parameters between the components and we show which of them are stable or unstable. In particular, solutions with spikes for both activators in the same position can be stable when cross-activation dominates over self-activation. On the other hand, solutions with a spike for only one activator and zero concentration for the other activator can be stable when self-activation dominates over cross-activation. The rigorous approach is based on analytical methods such as Green’s function, Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical results are confirmed by numerical simulations.