Biological aggregation driven by social and environmental factors: A nonlocal model and its degenerate Cahn-Hilliard approximation

TL;DR

This paper derives a local degenerate Cahn-Hilliard model from a nonlocal aggregation model to analyze biological group formations and explores how external food sources influence population density peaks.

Contribution

It introduces a simplified, well-posed local model that captures key features of nonlocal aggregation dynamics and examines control strategies using external potentials.

Findings

Energy minimizers of the local model resemble those of the nonlocal model.

Periodic food sources can reduce peak population density.

Random food distributions may increase peak density.

Abstract

Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. Beginning with a particular nonlocal aggregation model [Topaz et al., Bull. Math. Bio., 2006], we derive the minimal well-posed long-wave approximation, which is a degenerate Cahn-Hilliard equation. Energy minimizers of this reduced, local model retain many salient features of those of the nonlocal model, especially for large populations and away from an aggregation's boundaries. Using the Cahn-Hilliard model as a testbed, we investigate the degree to which an external potential modeling food sources can be used to suppress peak population density, which is essential for controlling locust outbreaks. A random distribution of food sources tends to increase peak density above its intrinsic value, while a periodic pattern of food sources can decrease it.