From individual to collective behaviour of coupled velocity jump processes: A locust example

Radek Erban; Jan Haskovec

doi:10.3934/krm.2012.5.817

Article Contents

2012, Volume 5, Issue 4: 817-842. Doi: 10.3934/krm.2012.5.817

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From individual to collective behaviour of coupled velocity jump processes: A locust example

Radek Erban^1, and
Jan Haskovec^2,

1.
University of Oxford, Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB
2.
King Abdullah University of Science and Technology, Computer, Electrical and Mathematical Sciences and Engineering, Thuwal 23955-6900

Received: August 31, 2011

Revised: July 31, 2012

Published: November 2012

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Abstract

A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on the behaviour of locusts. It exhibits nontrivial dynamics with a pitchfork bifurcation and recovers the observed group directional switching. Estimates of the expected switching times, in terms of the number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations (with nonlocal and nonlinear right hand side) is derived and analyzed. The existence of its solutions is proven and the system's long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model.

Keywords:

Collective behaviour,
stochastic individual-based model,
density-dependent directional switching,
kinetic equation.

Mathematics Subject Classification: 60G35, 35L40, 92B05.

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