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

Tadahisa Funaki; Hirofumi Izuhara; Masayasu Mimura; Chiyori Urabe; Tadahisa Funaki; Hirofumi Izuhara; Masayasu Mimura; Chiyori Urabe

doi:10.3934/nhm.2012.7.705

Networks and Heterogeneous Media

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

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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: 01 March 2012 Revised: 01 July 2012
Primary: 35K55, 60K35; Secondary: 82C22, 92D25.

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.
- Cross-diffusion system,
- reaction-diffusion system,
- particle system,
- hydrodynamic limit,
- chemotaxis
Citation: Tadahisa Funaki, Hirofumi Izuhara, Masayasu Mimura, Chiyori Urabe. A link between microscopic and macroscopic models of self-organized aggregation[J]. Networks and Heterogeneous Media, 2012, 7(4): 705-740. doi: 10.3934/nhm.2012.7.705

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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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