Spatial inhomogenity due to Turing bifurcation in a system of Gierer-Meinhardt type
Following methods used in \textit{M. Cavani} and \textit{M. Farkas} [Acta Math. Hung. 63, 375-393 (1994; Zbl 0809.92018)] and \textit{S. Kovács} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 42, 185-203 (1999; Zbl 0990.35063)], the paper investigates conditions of existence and (in)stability of stationary solutions of the ``predator-prey'' reaction-diffusion system \[ \begin{aligned} \partial S_1 / \partial t &= d_{11} \triangle S_1 - d_{12} \triangle S_2 + f_1(S_1,S_2),\\ \partial S_2 / \partial t &= \alpha (-d_{21} \triangle S_1 + d_{22} \triangle S_2) + f_2(S_1,S_2)\end{aligned} \] in \(\Omega \times \mathbb{R}_0^+\) with \(\Omega \subset \mathbb{R}^2\) chosen as 1) rectangle, 2) hexagon and 3) isosceles rectangular triangle with homogeneous Neumann conditions on the boundary, diffusion coefficients \(d_{11} d_{22} > d_{12} d_{21}\), \(\alpha > 0\), and reaction terms \(f_1 (S_1,S_2) = (S_1^a - S_1 S_2^b)/S_2^b\) and \(f_2 (S_1,S_2) = (S_1^c - S_2^{d+1})/S_2^d\) such that \(a > 1\), \(b,c > 0\), \(d\geq 0\), and \(bc > (a-1)(d+1)\). Among others, the results show that a diffusion-driven non-homogeneous pattern may arise, and its first order approximation is provided.
scientific article; zbMATH DE number 1907721
scientific article