Self-organized hyperuniformity in a minimal model of population dynamics

TL;DR

This model, generalizing a class of models recently introduced to account for protracted transients in biological systems, identifies a novel mechanism for hyperuniformity, and develops a hydrodynamic theory that conforms closely with the results of stochastic simulations.

Abstract

By generalizing a class of models recently introduced to account for protracted transients in biological systems, we identify a novel mechanism for hyperuniformity. In this model, competition of particles over a shared resource guides the population towards a critical steady state with prolonged individual life time. We show that, in its spatially extended form, this many-particle model exhibits hyperuniform density fluctuations. Through explicit coarse-graining, we develop a hydrodynamic theory that conforms closely with the results of stochastic simulations. Unlike previous models for non-equilibrium hyperuniform states, our model does not exhibit conservation laws, even when approaching criticality. Instead, hyperuniformity arises from the divergence of the interaction range as the system approaches the critical point. These findings may find applications in engineering, cellular population dynamics, and ecology.

Figures (6)

• Figure 1: (a) Snapshot of a two-dimensional realization of the population dynamics model (\ref{['nu']}-\ref{['rho']}), where particles are marked in black dots and the resource field $c$ in a color map displaying small variations around the critical value $c_{\text{crit}}=1$. (b) The instantaneous flow of the internal dynamics \ref{['nu']} for two representative particles that are indicated by arrows. All parameters are set to unity except for $k=7$, corresponding to $\mu^*\simeq0.05$\ref{['muss']}.
• Figure 2: Structure factor \ref{['sofq']} at increasing values of $k$ approaching criticality, with all other parameters set to unity. Solid lines are the theoretical prediction from the hydrodynamic theory (\ref{['yhydro']}-\ref{['rhohydro']}) SeeSupplementalMaterial. Symbols denote numerical simulations of (\ref{['nu']}-\ref{['rho']}) in $1d$ (circles) and $2d$ (squares). The dashed black line is the asymptotics \ref{['smallq']} at $k=60$.
• Figure 3: Scaled number variance \ref{['hyperuniformity']} color coded as in Fig.\ref{['structure']}, showing good agreement with the large $\ell$ asymptotics given by the first line in \ref{['largel2']}, here in solid black line. Dashed black curves account for the non-perturbative corrections in the second line of \ref{['largel2']}, which become dominant beyond a diverging crossover length scale $\sqrt{\ell_D\rho^*}\sim\mu^*{}^{-1/2}$.
• Figure S1: Data points are numerical simulations in $2d$ of the original model with the addition of diffusion of agents with diffusion coefficient $D_{\rho}=0.05$ and with $k=7$. All other parameters are set to unity. It displays only mild variations compared to the dynamics without diffusion (solid line). For any value of the diffusion coefficient, the hydrodynamics \ref{['yhydrodiff']} predict that $S(0)$ will coincide with the value of the model without diffusion.
• Figure S2: Data points correspond to numerical simulations of the model with the Gaussian noise \ref{['nun']} with the amplitude $D_\nu=0.01$. All other parameters, except $k=7,10$ are set to unity as in the main text. The numerical results display deviation from the theoretical prediction for the dynamics at zero noise amplitude $D_\nu=0$ in solid lines. Nevertheless, the plots suggest a trend towards hyperuniformity as $k$ is increased.