Ant mill

Abstract: We define here a \textit{directed edge reinforced random walk} on a connected locally finite graph. As the name suggests, this walk keeps track of its past, and gives a bias towards directed edges previously crossed proportional to the exponential of the number of crossings. The model is inspired by the so called \textit{Ant Mill phenomenon}, in which a group of army ants forms a continuously rotating circle until they die of exhaustion. For that reason we refer to the walk defined in this work as the \textit{Ant RW}. Our main result justifies this name. Namely, we will show that on any finite graph which is not a tree, and on $\mathbb Z^d$ with $d\geq 2$, the Ant RW almost surely gets eventually trapped into some directed circuit which will be followed forever. In the case of~$\mathbb Z$ we show that the Ant RW eventually escapes to infinity and satisfies a law of large number with a random limit which we explicitly identify.

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.