Localized Swarms in Nonlocal Models: Dynamics and Equilibria

Monday, June 3, 2013 - 10:45am - 11:25am
Keller 3-180
Chad Topaz (Macalester College)
I discuss a partial integrodifferential equation model for the the collective motion of biological groups. The model describes a population density field u(x,t) advected by a velocity field v = q * u + f. The convolution q * u represents pairwise social interactions between swarm members and f(x) models exogenous forces such as food or light. Because social interactions are difficult to measure in experiment, one challenge in aggregation 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. For nonzero f, we use a variational formulation to find exact solutions for swarm equilibria in one spatial dimension. Typically, these are localized solutions with jump discontinuities or delta-concentrations at the group’s edges. We apply some results to locust swarms.
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