Diffusion instability region for systems of parabolic equations

We consider a system of two reaction-diffusion equations in a bounded region of $m$-dimensional space with Neumann boundary conditions on the boundary, for which the reaction terms $f(u,v)$ and $g(u,v)$ depend on two parameters $a$ and $b$. It is assumed that the system has a spatially homogeneous solution $(u_0, v_0)$, moreover, $f_u(u_0,v_0)>0$, and $-g_v(u_0,v_0)=F(\mathrm {Det (\mathrm {J })})$, where $\mathrm {J}$ is the Jacobi matrix of the corresponding linearized system in the diffusionless approximation, $F$ is a smooth, monotonically increasing function. A method is proposed for the analytical description of the region of necessary and sufficient conditions for Turing instability on the plane of the parameters of the system at a fixed diffusion coefficient $d$. It is shown that the region of necessary conditions for Turing instability on the plane $(\mathrm {Det (\mathrm {J})}, f_u)$ is bounded by the zero trace curve, the discriminant curve and the points $\mathrm {Det (\mathrm {J} )} = 0$. Explicit expressions for the curves of sufficient conditions are found and it is proved that the discriminant curve is the envelope of the family of these curves. It is shown that one of the boundaries of the Turing instability region, which consists of fragments of curves of sufficient conditions, is expressed in terms of the function $F$ and the eigenvalues of the Laplace operator in the considered region. The points of intersection of the curves of the sufficient conditions are found and it is shown that their abscissas do not depend on the form of the function $F$ and are expressed in terms of the diffusion coefficient and the eigenvalues of the Laplace operator. The particular case $F(\mathrm{Det(\mathrm{J})})=\mathrm{Det(\mathrm{J})}$ is considered. For this case, the range of wave numbers at which the Turing instability occurs is indicated. A partition of the semiaxis $d>1$ into half-intervals is obtained, each of which has its own minimum critical wave number. The intersection points of the curves of the sufficient conditions lie on straight lines independent of the diffusion coefficient $ d $. The Schnackenberg system and the Brusselator equations are considered as examples of applications of the proved statements.