A model of individual clustering with vanishing diffusion

TL;DR

This paper studies a one-dimensional population model with two reproduction rates and small diffusion, showing that solutions converge to a transport problem as diffusion vanishes.

Contribution

It proves the convergence of a coupled drift-diffusion and elliptic system to a limit transport problem in the zero-diffusion limit.

Findings

Solutions converge to the limit transport problem as diffusion tends to zero.

The model captures individual clustering behavior with two reproduction rates.

Mathematical proof of convergence in suitable topologies.

Abstract

We consider a model of individual clustering with two specific reproduction rates and small diffusion parameter in one space dimension. It consists of a drift-diffusion equation for the population density coupled to an elliptic equation for the velocity of individuals. We prove the convergence (in suitable topologies) of the solution of the problem to the unique solution of the limit transport problem, as the diffusion coefficient tends to zero.