Fluctuations impact on a pattern-forming model of population dynamics with non-local interactions

TL;DR

This paper investigates how fluctuations influence pattern formation in a population dynamics model with non-local interactions, comparing stochastic particle simulations with deterministic mean-field approximations to understand their effects.

Contribution

It introduces a particle-based model with non-local interactions and analyzes the impact of stochastic fluctuations on pattern formation, contrasting it with mean-field predictions.

Findings

Deterministic mean-field captures some pattern features like finite-wavelength instability.

Stochastic fluctuations significantly alter the parameter thresholds for pattern emergence.

Fluctuations affect the nature of the homogeneous phase and pattern stability.

Abstract

A model of interacting random walkers is presented and shown to give rise to patterns consisting in periodic arrangements of fluctuating particle clusters. The model represents biological individuals that die or reproduce at rates depending on the number of neighbors within a given distance. We evaluate the importance of the discrete and fluctuating character of this particle model on the pattern forming process. To this end, a deterministic mean-field description, including a linear stability and a weakly nonlinear analysis, is given and compared with the particle model. The deterministic approach is shown to reproduce some of the features of the discrete description, in particular, the existence of a finite-wavelength instability. Stochasticity in the particle dynamics, however, has strong effects in other important aspects such as the parameter values at which pattern formation occurs, or the nature of the homogeneous phase.