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.Comment: Some typos and some problems with the figures correcte