Modelling the Transition from Homogeneous to Columnar States in Locust Hopper Bands

Many biological systems form structured swarms, for instance in locusts, whose swarms are known as hopper bands. There is growing interest in applying mathematical models to understand the emergence and dynamics of these biological and social systems. We model the locusts of a hopper band as point particles interacting through repulsive and attractive social forces on a one dimensional periodic domain. The primary goal of this work is to modify this well studied modelling framework to be more biological by restricting repulsion to act locally between near neighbors, while attraction acts globally between all individuals. This is a biologically motivated assumption because repulsion in swarms is mainly a collision avoidance mechanism. We construct a discrete and continuum model in the limit of infinite individuals. Using an energy formulation of both discrete and continuum models, we find energy minimizing equilibrium configurations that are either constant density or clumped. Then, we perform a stability analysis of these swarm configurations to find transitions of stability and hysteresis. We show that with local repulsion and global attraction the constant density equilibrium has a bifurcation to instability for both the total mass of the swarm and the attraction strength of individuals