Abstract:
I shall present models of mathematical biology describing the emergence of nontrivial patterns via collective actions of many individual entities: aggregating cockroaches, flocking birds and self-organizing phenomena in biological transportation networks. In the limits of large populations (mean field limits), they are modeled by systems of diffusive or kinetic partial differential equations. I shall demonstrate how our understanding of pattern emergence is supported by a variety of analytical methods: Lyapunov functional and stability estimates for (delay) differential equations, Ito calculus methods for systems with noise, energy dissipation and asymptotic stability methods for reaction-diffusion equations.