Existence of spot and lane stationary solutions for an ant active matter PDE model

TL;DR

This paper proves the existence of multiple stationary solutions, including spot and lane patterns, in an ant collective behavior PDE model, and analyzes their stability as parameters vary.

Contribution

It establishes the existence of spot and lane stationary solutions along a bifurcation sequence and analyzes their stability for small anticipation parameters.

Findings

Spot solutions are locally stable for small anticipation.

Lane solutions are unstable for small anticipation.

Solutions emerge along a bifurcation sequence as interaction strength increases.

Abstract

This paper studies the existence of multiple non-trivial stationary solutions of a partial differential equation (PDE) model introduced in [3], motivated by collective ant behavior. Previous work suggested the presence of two types of non-trivial stationary solutions for this PDE system: spot and lane solutions. In this paper, we establish the existence of these families of solutions along a bifurcation sequence as the interaction strength grows, with progressively increasing numbers of clusters and parallel lanes, respectively. Finally, we show that, for small values of the anticipation parameter, the first bifurcating spot solutions are locally dynamically stable, while the lane solutions are unstable.