Agent-Based and Continuous Models of Swarms: Insights Gained Through Energy Methods and Dynamical Systems
A common class of models for biological systems is to have a set of identical agents which interact through attraction, repulsion, and alignment. When these social interactions are pairwise and additive they yield a natural energy formulation. Models with these characteristics include Kuramoto’s synchronizing oscillators and the aggregation equation description of attractive/repulsive swarms. I will review a set of discrete models and their continuous analogs and show how energy methods can help identify equilibria and determine their stability both numerically and analytically. I will discuss the effect of noise, external potentials, domain geometry and boundaries and show that these systems can manifest a menagerie of behaviors including clumping, collapse, pattern formation and hysteretic behavior.