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.
Highlights
Individual-based behaviour in biology can be often modelled as a velocity jump process [20]
We introduced an individual based model with velocity jumps aimed at explaining the experimentally observed collective motion of locusts marching in a ring shaped arena [2]
We showed that our model has the same predictive power as the model of Czirok and Vicsek, in particular, it exhibits the rapid transition to highly aligned collective motion as the size of the group grows and the switching of the group direction, with frequency rapidly decreasing with increasing group size
Summary
Individual-based behaviour in biology can be often modelled as a velocity jump process [20]. A classical example is bacterial chemotaxis [8] Individual bacteria change their frequency of velocity changes according to their environment. Our model is motivated by the recent experiments of Buhl et al [2] They studied an experimental setting, in which locust nymphs marched in a ring-shaped arena. Yates et al [24] analysed experimental data of Buhl et al [2] and proposed that the frequency of random changes in the direction of an individual increases when the individual looses the alignment with the rest of the group. We incorporate this observation into a stochastic individual-based model formulated as a velocity jump process.
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