Spreading Properties of a City-Road Reaction-Diffusion Model on One-Dimensional Lattice

We propose and study a new model to describe biological invasions constrained on infinite homogeneous one dimensional metric graphs. Our model consists of an infinite PDE-ODE system where, at each vertex of the one-dimensional lattice $\mathbb{Z}$, we have a logistic equation, and connections between vertices are given by diffusion equations on the edges supplemented with Robin like boundary conditions at the vertices. We establish the main properties of the system and study the long time behavior of the solutions, especially by characterizing an asymptotic spreading speed for the system. In the fast diffusion regime, we derive a novel asymptotic model which exhibits similar propagation properties as the classical discrete Fisher-KPP on the one-dimensional lattice $\mathbb{Z}$.