# Exploring the Ant Mill: Numerical and Analytical Investigations of Mixed Memory-Reinforcement Systems

## TL;DR

This paper models the complex motion of ant swarms, especially the death spiral phenomenon, using a novel continuous random walk framework that combines memory and reinforcement effects, supported by analytical and numerical stability analysis.

## Contribution

It introduces a new continuous PDE-based model integrating memory and reinforcement in random walks, capturing the death spiral phenomenon in ant swarms.

## Key findings

- Derived an axi-symmetric, steady-state solution resembling the death spiral.
- Proved the stability of the steady-state solution numerically.
- Demonstrated the model's ability to replicate complex biological motion patterns.

## Abstract

Under certain circumstances, a swarm of a species of trail-laying ants known as army ants can become caught in a doomed revolving motion known as the death spiral, in which each ant follows the one in front of it in a never-ending loop until they all drop dead from exhaustion. This phenomenon, as well as the ordinary motions of many ant species and certain slime molds, can be modeled using reinforced random walks and random walks with memory. In a reinforced random walk, the path taken by a moving particle is influenced by the previous paths taken by other particles. In a random walk with memory, a particle is more likely to continue along its line of motion than change its direction. Both memory and reinforcement have been studied independently in random walks with interesting results. However, real biological motion is a result of a combination of both memory and reinforcement. In this paper, we construct a continuous random walk model based on diffusion-advection partial differential equations that combine memory and reinforcement. We find an axi-symmetric, time-independent solution to the equations that resembles the death spiral. Finally, we prove numerically that the obtained steady-state solution is stable.