Clustering, advection and patterns in a model of population dynamics with neighborhood-dependent rates

TL;DR

This paper presents a model of population dynamics with neighborhood-dependent birth and death rates, leading to pattern formation and cluster emergence, and explores the effects of fluid advection on these patterns.

Contribution

It introduces a new population model incorporating local neighborhood effects and analyzes pattern formation through stability analysis, including fluid advection influences.

Findings

Finite-wavelength instability causes pattern formation.

Clusters arrange periodically due to instability.

Advection influences global population properties.

Abstract

We introduce a simple model of population dynamics which considers birth and death rates for every individual that depend on the number of particles in its neighborhood. The model shows an inhomogeneous quasistationary pattern with many different clusters of particles. We derive the equation for the macroscopic density of particles, perform a linear stability analysis on it, and show that there is a finite-wavelength instability leading to pattern formation. This is the responsible for the approximate periodicity with which the clusters of particles arrange in the microscopic model. In addition, we consider the population when immersed in a fluid medium and analyze the influence of advection on global properties of the model.