Pattern formation in individual-based systems with time-varying parameters

TL;DR

This paper investigates how patterns form in individual-based systems during finite-time symmetry-breaking bifurcations, showing that slow parameter changes lead to large patterns while fast changes produce many small domains, with noise playing a key role.

Contribution

It introduces a linear-noise approximation to predict pattern length scales in systems with time-varying parameters, validated across multiple models.

Findings

Large-scale patterns form during slow parameter sweeps.

Fast sweeps result in numerous small domains.

Theoretical predictions match simulation results.

Abstract

We study the patterns generated in finite-time sweeps across symmetry-breaking bifurcations in individual-based models. Similar to the well-known Kibble-Zurek scenario of defect formation, large-scale patterns are generated when model parameters are varied slowly, whereas fast sweeps produce a large number of small domains. The symmetry breaking is triggered by intrinsic noise, originating from the discrete dynamics at the micro-level. Based on a linear-noise approximation, we calculate the characteristic length scale of these patterns. We demonstrate the applicability of this approach in a simple model of opinion dynamics, a model in evolutionary game theory with a time-dependent fitness structure, and a model of cell differentiation. Our theoretical estimates are confirmed in simulations. In further numerical work, we observe a similar phenomenon when the symmetry-breaking bifurcation is triggered by population growth.