Range expansion by growth and congestion

TL;DR

This paper introduces a nonlinear, nonlocal model for population range expansion driven by growth and competition, deriving a singular limit that simplifies analysis of the spreading dynamics.

Contribution

It presents a novel mathematical model and rigorous analysis of the free boundary problem describing population expansion due to growth and congestion.

Findings

Characterization of the free boundary evolution.

Identification of traveling wave solutions.

Determination of asymptotic spreading speed.

Abstract

We introduce a nonlinear and nonlocal model that describes the range expansion of a population resulting from growth and competition for space. This type of phenomenon underlies the expansion of colonies of immotile cells which motivated this work; Similar mechanisms are at play in urban sprawling which we briefly discuss as well. We rigorously derive a singular limit of this model corresponding to a regime where dispersal occurs only from saturated areas. The limiting model, which has the structure of an obstacle free boundary problem in time, provides an effective approach to the description of the range expansion of a population as a result of growth, saturation and dispersion. We then establish the main mathematical properties of this singular problem. In particular, we characterize the evolution of a free boundary that delimits the saturated area. We identify traveling wave solutions and characterize the asymptotic speed of spreading of compactly supported solutions.