Nonlocal interaction equations are continuum models for large systems of particles where every particle can interact not only with its immediate neighbors but also with particles far away. In biology, they arise as model for the movement of large swarms of individuals (insects, bacteria, birds, fish, etc...). The research will involve rigorous analysis of nonlinear partial differential equations, numerical simulations, and modeling. More specifically the following problems will be investigated: 1) Study of the formation of singularities and the formation of patterns in nonlocal interaction equations. 2) Study of nonlocal interaction equations in a domain with boundaries and use of optimal transport techniques in order to study phenomena of mass concentration on the boundaries (such phenomena are observed, for example, in continuum models for swarms of locusts). 3) Modeling of the formation of "foraging trails'' in an ant colony.
This project investigates equations describing the collective behavior of a large number of individuals, such as a swarm of insects, a flock of birds, a school of fish, or a colony of bacteria. Understanding basic rules of these swarming behaviors has potential for the design of flexible and robust algorithms for unmanned vehicles, with possible applications, for example, to the surveillance of hazardous areas. The equations investigated in this research also arise in various other contexts: evolution of vortex densities in superconductors, adhesion dynamics, simplified model for granular flow and orientational distribution of filaments in cells.
