This presentation explores the field-road diffusion model developed in 2012 by Berestycki, Roquejoffre, and Rossi. This parabolic system aims to capture the significant dispersal effects induced by lines of fast diffusion, with wide-ranging applications in population dynamics, ecology, and epidemiology.
Initially, we will introduce the model, emphasizing its ability to simulate accelerated spread phenomena. We will then concentrate on the explicit determination of the fundamental solution to the macroscopic system, achieved through the application of a double integral transform, namely Fourier and Laplace. This analytical framework offers clear insights into the model's dynamics and sets the stage for exploring non-linear issues such as "persistence vs. extinction" phenomena in the presence of reaction terms with the so-called Allee effect.
The second part of the talk will be dedicated to provide a stochastic foundation for the deterministic framework by deriving the governing equations of the diffusive field-road model from an interacting particle system. To introduce this approach, we will go back to the origins of the Symmetric Simple Exclusion Process (SSEP) which enables the rigorous derivation of solutions to the Heat equation on the torus. After outlining the principles of this type of particle system, we will see how it can be used to generate certain boundary conditions. This will allow us to introduce a microscopic dynamics for the field-road diffusion model and present our recent result on its hydrodynamic limit.
