Spherically symmetric internal layers for activator– inhibitor systems-I. Existence by a Lyapunov–Schmidt reduction

Reaction–diffusion systems of activator–inhibitor type are studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. Under the condition that the activator diffuses slowly, reacts rapidly and the inhibitor diffuses rapidly, reacts moderately, we show that the system admits a family of spherically symmetric internal transition layer equilibria. The method of proof consists of rigorous asymptotic expansions and a Lyapunov–Schmidt reduction.