A non-local model for a swarm

Alexander Mogilner, Leah Edelstein-Keshet

Research output: Contribution to journalArticle

Abstract

This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles.

Original languageEnglish (US)
Pages (from-to)534-570
Number of pages37
JournalJournal of Mathematical Biology
Volume38
Issue number6
StatePublished - Jun 1999

Fingerprint

swarms
Swarm
Density Dependence
Term
Singular Perturbation Theory
Nonlocal Interactions
Linear stability analysis
Advection-diffusion Equation
Disintegration
Linear Stability Analysis
Density Profile
Advection
Continuum Model
Unequal
Convolution
Integro-differential Equation
Model
Range of data
Time Scales
Interior

Keywords

  • Aggregation
  • Integro-differential equations
  • Non-local interactions
  • Swarming behavior
  • Traveling band solutions

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Mathematics (miscellaneous)

Cite this

Mogilner, A., & Edelstein-Keshet, L. (1999). A non-local model for a swarm. Journal of Mathematical Biology, 38(6), 534-570.

A non-local model for a swarm. / Mogilner, Alexander; Edelstein-Keshet, Leah.

In: Journal of Mathematical Biology, Vol. 38, No. 6, 06.1999, p. 534-570.

Research output: Contribution to journalArticle

Mogilner, A & Edelstein-Keshet, L 1999, 'A non-local model for a swarm', Journal of Mathematical Biology, vol. 38, no. 6, pp. 534-570.
Mogilner A, Edelstein-Keshet L. A non-local model for a swarm. Journal of Mathematical Biology. 1999 Jun;38(6):534-570.
Mogilner, Alexander ; Edelstein-Keshet, Leah. / A non-local model for a swarm. In: Journal of Mathematical Biology. 1999 ; Vol. 38, No. 6. pp. 534-570.
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