Stable spikes for a reaction-diffusion system with two activators and one inhibitor

We consider a reaction-diffusion system with two activators and one inhibitor. We prove rigorous results on the existence and stability of spiky patterns. We show that for certain conditions on the parameters these solutions can be stable. The approach is based on analytical methods such as elliptic estimates, Liapunov-Schmidt reduction and nonlocal eigenvalue problems. This is joint work with Weiwei Ao and Juncheng Wei.