Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming.

Density-dependent social behaviors such as swarming and schooling determine spatial distribution and patterns of resource use in many species. Lagrangian (individual-based) models have been used to investigate social groups arising from hypothetical algorithms for behavioral interactions, but the Lagrangian approach is limited by computational and analytical constraints to relatively small numbers of individuals and relatively short times. The dynamics of "group properties", such as population density, are often more ecologically useful descriptions of aggregated spatial distributions than individual movements and positions. Eulerian (partial differential equation) models directly predict these group properties; however, such models have been inadequately tied to specific individual behaviors. In this paper, I present an Eulerian model of density-dependent swarming which is derived directly from a Lagrangian model in which individuals with limited sensing distances seek a target density of neighbors. The essential step in the derivation is the interpretation of the density distribution as governing the occurrence of animals as Poisson points; thus the number of individuals observed in any spatial interval is a Poisson-distributed random variable. This interpretation appears to be appropriate whenever a high degree of randomness in individual positions is present. The Eulerian model takes the form of a nonlinear partial integro-differential equation (PIDE); this equation accurately predicts statistically stationary swarm characteristics, such as expected expected density distribution. Stability analysis of the PIDE correctly predicts transients in the stochastic form of the aggregation model. The model is presented in one-dimensional form; however, it illustrates an approach that can be equally well applied in higher dimensions, and for more sophisticated behavioral algorithms.