From individual to collective behaviour of coupled velocity jump processes: A locust example
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AbstractA 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 coefi-cients, 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 efiect of shrinking the interaction radius in the individual-based model. © American Institute of Mathematical Sciences.
SponsorsThis publication was based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). JH acknowledges the financial support provided by the FWF project Y 432-N15. The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ ERC grant agreement n<SUP>o</SUP> 239870. RE would also like to thank Somerville College, University of Oxford, for a Fulford Junior Research Fellowship; Brasenose College, University of Oxford, for a Nicholas Kurti Junior Fellowship; the Royal Society for a University Research Fellowship; and the Leverhulme Trust for a Philip Leverhulme Prize. This prize money was used to support research visits of JH to Oxford. Both authors would like to thank to the Isaac Newton Institute for Mathematical Sciences in Cambridge (UK), where they worked together during the program "Partial Differential Equations in Kinetic Theories". The authors also acknowledge several interesting discussions and valuable hints provided by Jan Vybiral (Technical University Berlin) and Christian Schmeiser (University of Vienna).
JournalKinetic and Related Models