Equilibria of biological aggregations with nonlocal repulsive-attractive interactions

TL;DR

This paper analyzes the equilibrium states of a biological aggregation model with nonlocal repulsive and attractive forces, proving symmetry and monotonicity, and exploring asymptotic behaviors through analytical and numerical methods.

Contribution

It establishes the existence, uniqueness, symmetry, and monotonicity of equilibrium solutions for a class of aggregation equations with combined Newtonian repulsion and power-law attraction.

Findings

Existence of unique, radially symmetric equilibria supported on a ball.

Equilibria are monotone decreasing functions of the radius.

Numerical simulations indicate equilibria act as global attractors.

Abstract

We consider the aggregation equation $\rho_{t}-\nabla\cdot(\rho\nabla K\ast\rho) =0$ in $\mathbb{R}^{n}$, where the interaction potential $K$ incorporates short-range Newtonian repulsion and long-range power-law attraction. We study the global well-posedness of solutions and investigate analytically and numerically the equilibrium solutions. We show that there exist unique equilibria supported on a ball of $\mathbb{R}^n$. By using the method of moving planes we prove that such equilibria are radially symmetric and monotone in the radial coordinate. We perform asymptotic studies for the limiting cases when the exponent of the power-law attraction approaches infinity and a Newtonian singularity, respectively. Numerical simulations suggest that equilibria studied here are global attractors for the dynamics of the aggregation model.