Category: Nonlinear Dyn.

Summary: Finding when a fast-diffusing boundary shell pulls a heterogeneous reaction-diffusion system into a shell-localized Turing instability.

Turing instabilities explain how diffusion can destabilize an otherwise stable reaction system and create spatial patterning. This experiment asks a more specific question: in a random nonlocal activator-inhibitor operator, how strong must a fast-diffusing boundary shell be before the leading unstable mode relocates to that shell instead of remaining a bulk pattern.

The model constructs dense reaction-diffusion Jacobians with heterogeneous couplings and a tunable shell diffusion boost. It then uses eigensolves, iterative deepening, and bisection on the shell parameter to locate the onset of a shell-localized unstable mode across larger system sizes.

This matters because random Turing thresholds and localization are usually studied separately. The experiment combines them into a direct finite-size threshold problem for boundary-driven pattern formation in disordered media.

Method: Dense eigensolves on real 2N x 2N reaction-diffusion Jacobians with iterative deepening and bisection on shell diffusion strength.

What is measured: Critical shell-instability threshold, shell mass of the unstable mode, localization score, leading eigenvalue behavior, and bracket width.