The dynamics of a kinetic activator–inhibitor system

In this paper we give a complete description of the entire dynamics of the kinetic system of a reaction–diffusion system proposed by A. Gierer and H. Meinhardt. In particular, the α-limit sets and ω-limit sets of all trajectories are determined, and it is shown that the dynamics of the system exhibits various interesting behaviors, including convergent solutions, periodic solutions, unbounded oscillating global solutions, and finite time blow-up solutions.