A stochastic individual velocity jump process modelling the collective motion of locusts
Seminar Room 1, Newton Institute
We consider a model describing an experimental setting, in which locusts run in a ring-shaped arena. With intermediate spatial density of the individuals, coherent motion is observed, interrupted by sudden changes of direction ("switching"). Contrary to the known model of Czirok and Vicsek, our model assumes runs of the individuals in either positive or negative direction of the 1D arena with the same speed, that are subject to random switches. As supported by experimental evidence, the individual switching frequency increases in response to a local or global loss of group alignment, which constitutes a mechanism to increase the coherence of the group. We show that our individual based model, although phenomenologically very simple, exhibits nontrivial dynamics with a "phase change" behaviour, and, in particular, recovers the observed group directional switching. Passing to the corresponding Fokker-Planck equation, we are able to give estimates of the expected switching times in terms of number of individuals and values of the model coefficients. Then we pass to the kinetic description, recovering a system of two kinetic equations with nonlocal and nonlinear right hand sides, which is valid when the number of individuals tends to infinity. We perform a mathematical analysis of the system, show some numerical results and point out several interesting open problems.