We study the subsystem fluctuation of the time-integrated prey and predator population in the small wave vector limit (q→0) of a stochastic reaction-diffusion model in the Turing pattern formation regime, revealing a unique form of hyperuniformity. The fluctuation of the intensive subsystem population exhibits a dual behavior: increasing for smaller subsystem sizes while decreasing for larger ones, eventually saturating to the fluctuation of the maximum subsystem size of L/2 in a periodic system of L. This dependence indicates a short-range positive correlation followed by a long-range effective negative correlation, consistent with the feedback mechanisms of the known ecological models. The fluctuation of larger subsystems of size l converges to the fluctuation of the subsystem of L/2 as l
-1, demonstrating class-I hyperuniformity for an infinite system size limit, L→∞. Using the van Kampen system size expansion, we compute the dynamic power spectrum (S(q,ω)) of steady-state configurations, which saturates to a finite value in the limit of small frequency (ω→0) and q→0. This persistent fluctuation in the infinite spacetime volume limit arises from reaction dynamics creating a screen effect in the dynamic power spectrum. In Fourier mode, the dynamic power spectrum for ω=0 converges to its saturation value for the limit of q→0 as q², reflecting the homogenization effect of diffusion. This class-I type of hyperuniformity should be universal in the reaction diffusion models with finite diffusion-coefficients. Unlike traditional hyperuniform systems where the structure factor decays as q
α with α>0 in the q→0 limit, reaction-diffusion systems with multiple demographic fluctuation sources require separate analysis of power spectrum decay for dynamics (in our case, diffusion) contributing to q-dependent fluctuations, as the reaction dynamics can screen the effect of long-range negative correlation through persistent fluctuation. This hidden hyperuniformity in stochastic reaction-diffusion systems provides insights for analyzing ecosystem stability and resilience under external perturbation by characterizing large-scale structural changes.
References
- Fluctuation-driven Turing patterns, Thomas Butler and Nigel Goldenfeld, Phys. Rev. E 84, 011112 (2011)
- Hidden universal hyperuniform properties of stochastic Turing patterns, Anirban Mukherjee and Hong-Yan Shih, work in progress.
Keywords: Stochastic Turing patterns, Hyperuniformity, Self-organization, Ecological models, Universality