COLLECTIVE BEHAVIOR OF BIOLOGICAL AGGREGATIONS IN TWO DIMENSIONS: A NONLOCAL KINETIC MODEL

Abstract

We construct and investigate a new nonlocal kinetic model for the formation and movement of animal groups in two dimensions. The model generalizes to two dimensions, the one-dimensional hyperbolic model from (R. Eftimie, G. de Vries, M. A. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol.69 (2007) 1537–1566). The main modeling aspect in the present approach concerns the assumptions we make on the turning rates, to include, in a nonlocal fashion, the three types of social interactions that act among individuals of a group: attraction, repulsion and alignment. We show that solutions to the new mathematical model are bounded, along with their gradients. We also present numerical results to illustrate three types of group formations that we obtained with the new model, starting from random initial conditions: (i) swarms (aggregation into a group, with no preferred direction of motion), (ii) parallel/translational motion (uniform spatial density, movement in a certain preferred direction) and (iii) parallel groups (aggregation into a group, with movement in a preferred direction).