Asymptotic dynamics of attractive-repulsive swarms

TL;DR

This paper classifies and predicts the long-term behavior of one-dimensional swarming models with attractive-repulsive interactions, revealing conditions for spreading, contracting, or steady states, and deriving related analytical bounds and wave properties.

Contribution

It provides a comprehensive classification of asymptotic dynamics based on the kernel's moments, linking population behavior to porous medium and Burgers' equations, and deriving explicit wave and blow-up bounds.

Findings

Population spreads or contracts depending on kernel moments.

Edges of spreading populations behave like traveling waves.

Finite-time blow-up bounds are analytically derived.

Abstract

We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel's first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady-state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly-supported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers' equation. We derive an analytical upper bound for the finite blow-up time after which the solution forms one or more $\delta$-functions.