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 behaviour of locusts. It exhibitsnontrivial dynamics with a "phase change" behaviour and recovers the observedgroup directional switching. Estimates of the expected switching times, interms of number of individuals and values of the model coefficients, areobtained using the corresponding Fokker-Planck equation. In the limit of largepopulations, a system of two kinetic equations with nonlocal and nonlinearright hand side is derived and analyzed. The existence of its solutions isproven and the system's long-time behaviour is investigated. Finally, a firststep towards the mean field limit of topological interactions is made bystudying the effect of shrinking the interaction radius in the individual-basedmodel when the number of individuals grows