Turing instability in the one-parameter Gierer–Meinhardt system

The purpose of this work is to find the region necessary and sufficient conditions for diffusion instability on parameter plane (τ, d) Gierer–Meinhardt system, where τ relaxation parameter, d dimensionless coefficient; derive analytically dependence critical wave number characteristic size spatial region; obtain explicit representations secondary spatially distributed structures, formed as a result bifurcation homogeneous equilibrium position, in form series degrees supercriticality. Methods. To Turing instability, methods linear stability analysis are applied. solutions (Turing structures), Lyapunov– Schmidt method used developed by V. I. Yudovich. Results. Expressions coefficient terms eigenvalues Laplace operator an arbitrary bounded obtained. found explicitly two cases: when interval rectangle. Explicit expressions first expansions stationary with respect supercriticality constructed one-dimensional case, well rectangle, one numbers equal zero. In these cases, soft loss found, examples given. Conclusion. A general approach proposed finding constructing structures. This can be applied wide class mathematical models described system reaction–diffusion equations.