Linear threshold and directional shell selection for anisotropic caputo reaction–diffusion systems

We study the first loss of linear stability in a two-component reaction–diffusion system with constant anisotropic diffusion tensors and a common Caputo order between one and two. On the flat torus, identifying the critical spatial direction requires comparison of the angular stability thresholds over the entire Fourier lattice. Away from neutral eigenvalues and critical angular boundaries, uniform matrix Mittag–Leffler estimates justify this comparison for Sobolev initial displacement and velocity. The homogeneous mode is distinguished from the nonconstant modes. An explicit score orders the coordinate directions in the first Fourier shell for diagonal diffusion tensors, while a quantitative high-frequency bound makes the global search finite below any prescribed order less than two. A strictly selected pair of opposite modes persists under small coefficient perturbations. Examples distinguish homogeneous from nonconstant onset and exhibit an exact change of the preferred direction. Predictor–corrector computations are compared with the modal Mittag–Leffler solution. For a fixed finite family of modes, second-order convergence follows from the integrability of the initial second-derivative singularity, without assuming a bounded second derivative.