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

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