FROM INDIVIDUAL TO COLLECTIVE BEHAVIOUR OF COUPLED VELOCITY JUMP PROCESSES: A LOCUST EXAMPLE

Authored by Radek Erban, Jan Haskovec

Date Published: 2012

DOI: 10.3934/krm.2012.5.817

Sponsors: European Union European Research Council (ERC)

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

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.

Tags

models flocking Convergence Equation Motion Stochastic particle approximation Keller-segel system