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

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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).

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Acknowledgments.

It is a great pleasure to acknowledge fruitful discussions with Akira Okubo who inspired this research. Thanks are also due to Simon Levin, Andrea Di Liddo and Silvia Boi for relevant discussions. During the revision process we have noticed the publication of a relevant paper [21], that might be of interest to the reader.

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Authors and Affiliations

  1. MIRIAM & Department of Mathematics, University of Milan, Italy

    Daniela Morale & Vincenzo Capasso

  2. SFB 359, Istitut für Angewandte Mathematik -Heidelberg, Germany

    Karl Oelschläger

Authors
  1. Daniela Morale
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  2. Vincenzo Capasso
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  3. Karl Oelschläger
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Morale, D., Capasso, V. & Oelschläger, K. An interacting particle system modelling aggregation behavior: from individuals to populations. J. Math. Biol. 50, 49–66 (2005). https://doi.org/10.1007/s00285-004-0279-1

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  • DOI: https://doi.org/10.1007/s00285-004-0279-1

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