Abstract:
In this talk I will discuss nonlinear Fokker-Planck models describing diffusion processes with particle interactions. These models are motivated by the study of systems in biology and ecology composed of many interacting individuals, and arise as the population-level description of a stochastic particle-based model. In particular, we consider a system finite-sized hard-core interacting Brownian particles and use the method of matched asymptotic expansions to obtain a systematic model reduction. The result is a nonlinear Fokker-Planck equation, with the nonlinear term accounting for the size-exclusion interactions. We discuss the gradient-flow structure of the macroscopic system and compare its numerical solution with the stochastic particle system using Monte Carlo simulations. As an extension, we consider a system with two interacting species and study how the inter-species competition emerges at the population level. Now, the macroscopic model is a nonlinear cross-diffusion system that captures the enhancement of the effective diffusion rate due to excluded-volume interactions between particles of the same species, and the diminishment due to particles of the other species. |