Transport Models for Collective Dynamics in Biological Systems

Nonlocal transport vs nonlinear diffusion: from particle description to large time asymptotics

Marco Di Francesco

University of Bath


Aggregation phenomena in microbiology and animal biology can be often described by PDEs of "transport" type, with a "nonlocal" velocity field. I shall quickly provide a formal derivation of those PDEs from particle-based ODEs. I shall then highlight their variational structure, which often leads to well-posedness in a probability-measure sense. A major issue is providing a mathematical description of the emergence (or not) of collective behaviour, or "multiple" behaviour in the large-time asymptotics, depending on the choice of the initial conditions or other parameters. This issue has been partly investigated in the recent literature (cf. chemotaxis with two species). I will briefly describe recent results on the existence and uniqueness of non trivial steady states for a model with quadratic diffusion (in collaboration with M. Burger), and a recent work in preparation on the finite time blow up and "multiple collapse" for a "purely nonlocal" model with two species of agents (with S. Fagioli, PhD student from L'Aquila). Finally, I shall focus on the derivation of a "mildly" singular repulsive model as "large particle limit" of discrete ODE systems in one space dimension (in collaboration with G. A. Bonaschi, from Eindhoven, and J. A. Carrillo), and its interplay with the theory of entropy solutions for scalar conservation laws.