Determination of the limit sets of trajectories of the Gierer-Meinhardt system without diffusion

Abstract

We consider a reaction-diffusion system consisting of an activator and an inhibitor which models biological pattern formation. A complete description of the entire dynamics of the kinetic system, i.e., the system without diffusion terms, is given. In particular, the $\alpha$-limit sets and the $\omega$-limit sets of all trajectories are determined.