On the role of the source terms in an activator-inhibitor system proposed by Gierer and Meinhardt

Abstract

Considered is a reaction-diffusion system consisting of an activator and an inhibitor which was proposed by Gierer and Meinhardt to model biological pattern formation. We prove that the initial-boundary value problem for the activator-inhibitor system has a unique solution for all $t \gt 0$ if the production rate of the activator is well-controlled by the inhibitor. Moreover, we prove that the solution stays in a bounded region if the source term for the activator becomes positive somewhere. We consider also how the source term for the activator affects the shape of stationary solutions in one spatial dimension.