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

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