Non-local kinetic and macroscopic models for self-organised animal aggregations

TL;DR

This paper explores how different scaling methods transform kinetic models of animal aggregations into simpler macroscopic models, analyzing the preservation or loss of spatial patterns through asymptotic analysis and numerical simulations.

Contribution

It introduces and compares three scaling approaches for reducing complex biological aggregation models and examines pattern preservation using asymptotic preserving numerical methods.

Findings

Stationary aggregation patterns are preserved under parabolic scaling.

Moving aggregation patterns are lost as the scaling coefficient approaches zero.

Bifurcation diagrams explain the pattern transitions during scaling.

Abstract

The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. Here, we consider three scaling approaches (parabolic, hydrodynamic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Next, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient $\epsilon$ is varied from $\epsilon=1$ (for 1D transport models) to $\epsilon=0$ (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit $\epsilon\to 0$, while other patterns (describing moving aggregations) are lost in this limit. To understand the loss of these patterns, we construct bifurcation diagrams.