SIGEST

Abstract

The SIGEST paper in this issue, “Nonlocal Aggregation Models: A Primer of Swarm Equilibria” by Andrew J. Bernoff and Chad M. Topaz, is from the SIAM Journal on Applied Dynamical Systems. One of the many remarkable things in this paper is the authors' clear account of the very direct connections between first-kind Fredholm integral equations and apocalyptic plagues of locusts. Those of you who solve integral equations for a living will not be surprised that a first-kind Fredholm equation can be apocalyptic. While the results in the paper apply to many types of swarms, the authors focus on the particular behavior of Schistocerca gregaria, the swarming desert locust in Africa. These locusts self-organize into bubble-like swarms having a dense group of locusts on the ground, a flying group overhead, and an empty gap between them. The swarm moves with the wind with a rolling motion (see Figure 1 in the paper). This type of swarming is observed in nature and captured by the model in the paper. The paper looks at the relation between two models of swarming behavior. The discrete model is a system of ODEs for the time-dependent path of each individual bug. The forces on the locust are both exogenous (gravity, air resistance, food, predators) and endogenous (the preference to avoid collisions and stay with the group). If the swarm is at all large, the system of ODEs can become intractable, hence motivating a continuum approach. The continuum model, which the authors derive from the discrete model, is a variational principle for the density of locusts. The necessary conditions for stationarity of the energy functional can be expressed as a first-kind Fredholm equation. The analysis in the paper is for the continuous model. The paper looks deeply into the stability of the swarm configurations and the types of forces on the locusts, finds exact solutions for swarm equilibria in many parameter regimes, and carefully examines the regularity of the solutions of the continuous model. They find parameters for which the bubble-like swarms are stable solutions for both models. This is a satisfying paper with deep analysis, a great application, and a discrete formulation that is simple enough for a reader to play with individually or share with a class.