Stability and bifurcation diagram for a shadow Gierer–Meinhardt system in one spatial dimension

Abstract

We are concerned with a Neumann problem of a shadow system of the Gierer–Meinhardt model in an interval I=(0,1) . A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each n⩾1 , two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, n⩾2 , are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.