Motivated by biological swarms, we study a partial integrodifferential equation describing a population density field u(x,t) advected by the velocity field v = q * u + f. The convolution q * u describes pairwise social interactions between swarm members and f(x) describes exogenous forces such as food or light. Because social interactions are difficult to measure in experiment, one challenge in swarm modeling is to choose a social interaction kernel q(x) that produces qualitatively correct macroscopic behavior. For f = 0, we determine conditions on q for u to asymptotically spread, contract, or reach steady state. We further analyze the spreading case, for which u obeys porous-medium-like dynamics, and the contracting case, for which the integral of u obeys Burgers-like dynamics. For nonzero f, we use a variational formulation to find exact solutions for swarm equilibria. Typically, these are compactly supported with jump discontinuities or delta-concentrations at the group’s edges. We apply our methods to locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. This is joint work with Andrew Bernoff.
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