Effective hyperuniformity in time-integrated stochastic Turing patterns

Abstract

Demographic noise generates stochastic Turing patterns even when reaction-diffusion systems are deterministically stable. We show analytically and verify numerically in the Levin-Segel model that temporal integration of configurations reveals emergent large-scale organization. The intensive number variance in a window of size $R\gg 1$ approaches a finite reaction-kinetic floor as $1/R$, over a spatial range growing by orders of magnitude near the Turing instability. This yields an effectively hyperuniform, fine-tuning-free regime previously unidentified in non-conserved multispecies stochastic systems.