Category: Nonlinear Dyn.

Summary: Estimating how much shell-to-bulk anisotropic diffusion is needed to erase a shell-localized unstable mode in a random Turing system.

Turing instabilities are a standard route to self-organized pattern formation, but disorder can make those unstable modes concentrate in specific regions instead of spreading smoothly through space. This experiment asks when an unstable mode that is initially concentrated near a shell-like boundary stops being localized once anisotropic diffusion mixes the shell and the bulk too strongly.

The script builds dense random reaction-diffusion Jacobians with separate shell and bulk structure, then increases an anisotropy control parameter while carrying a threshold bracket across larger system sizes. By tracking whether the unstable mode still places enough weight on the shell, it turns the question into a finite-size collapse threshold rather than a broad parameter sweep.

That matters because anisotropy and heterogeneity are both known to reshape pattern formation, but their combined effect on localization is much less mapped. The result is a direct estimate of when directional mixing destroys a localized Turing pattern precursor.

Method: Dense real eigensolves on 2N x 2N random Turing Jacobians with iterative deepening and bisection on shell-bulk anisotropy.

What is measured: Critical anisotropy threshold, shell mass of the unstable mode, localization score, persistence of instability, and bracket width.