Spatial patterns in non-locally interacting particle systems

Abstract

The influence of spatially non-local interactions on the aggregation, competition, and growth dynamics of interacting particle systems has been recently addressed. In this paper we survey recent results obtained for this kind of systems, focusing on two types of population dynamics models: a) density-dependent mobility particle systems, with conserved total number of individuals, and b) birth-death systems, where annihilation-creation events are allowed, so that the total number of particles is not conserved. Both models present a pattern forming instability leading to surprisingly similar spatial structures. The two levels of description, microscopic-particle and macroscopic-density, are analyzed. From the last one, a clear identification of the pattern forming instability is obtained.