A link between microscopic and macroscopic models of self-organized aggregation

Tadahisa Funaki; Hirofumi Izuhara; Masayasu Mimura; Chiyori Urabe

doi:10.3934/nhm.2012.7.705

Article Contents

2012, Volume 7, Issue 4: 705-740. Doi: 10.3934/nhm.2012.7.705

This issue Previous Article PDE problems arising in mathematical biology Next Article Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary

A link between microscopic and macroscopic models of self-organized aggregation

1.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914
2.
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki, Kanagawa 214-8571
3.
FIRST, Aihara Innovative Mathematical Modelling Project, Japan Science and Technology Agency, Collaborative Research Center for Innovative Mathematical Modelling, Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505

Received: March 2012

Revised: July 2012

Published: December 2012

Abstract / Introduction Related Papers Cited by

Abstract

In some species, one of the roles of pheromones is to influence aggregation behavior. We first propose a macroscopic cross-diffusion model for the self-organized aggregation of German cockroaches that includes directed movement due to an aggregation pheromone. We then propose a microscopic particle model which is set into context with the macroscopic model. Our goal is to link the macroscopic and microscopic descriptions by using the singular and the hydrodynamic limit procedures. A hybrid model related to the macroscopic and microscopic models is also proposed as a cockroach aggregation model. This hybrid model assumes that each individual responds to pheromone concentration and moves by two-mode simple symmetric random walks. It shows that even though the movement of individuals is not directed, two-mode simple symmetric random walks and effect of the pheromone result in self-organized aggregation.

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Mathematics Subject Classification: Primary: 35K55, 60K35; Secondary: 82C22, 92D25.

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