  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.
