An interacting particle system modelling aggregation behavior: from individuals to populations

Abstract

In this paper we investigate the stochastic modelling of a spatially structured biological population subject to social interaction. The biological motivation comes from the analysis of field experiments on a species of ants which exhibits a clear tendency to aggregate, still avoiding overcrowding. The model we propose here provides an explanation of this experimental behavior in terms of "long-ranged" aggregation and "short-ranged" repulsion mechanisms among individuals, in addition to an individual random dispersal described by a Brownian motion. Further, based on a "law of large numbers", we discuss the convergence, for large N, of a system of stochastic differential equations describing the evolution of N individuals (Lagrangian approach) to a deterministic integro-differential equation describing the evolution of the mean-field spatial density of the population (Eulerian approach).

The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels

“…The purely gradient flow case has been studied for self-interacting individuals via pairwise potentials arising in the modelling of animal collective behavior: flocks, schools or swarms formed by insects, fishes and birds. The simplest models based on ODEs systems [15,24,29,43,44] led to continuum descriptions [19,18,14,37,42,47,48] for the evolution of densities of individuals. It is this class of models that we focus on here, although we will draw parallels to well-known problems and results from the incompressible flow literature.…”

Partial differential equation models in the socio-economic sciences

“…Another interesting example in this line is non-local aggregation [21][22][23], e.g. appearing in animal swarms looking for a density, but also in macroscopic models of consensus formation.…”

Уравнение Агрегации С Анизотропной Диффузией

“…These models are ubiquitous in mathematical biology where they have been used as macroscopic descriptions for collective behavior or swarming of animal species, see [53,13,54,55,67,16] for instance, or more classically in chemotaxis-type models, see [60,44,40,39,11,10,20] and the references therein.…”