从个体行为到耦合速度跃变过程的集体行为：蝗虫的例子
From individual to collective behaviour of coupled velocity jump processes: a locust example
摘要：
一类随机个人为基础的模型，写在耦合速度跳跃过程，提出和分析。这种建模方法结合了最近关于蝗虫行为的实验结果。它表现出非平凡的动态与“相变”的行为，并恢复所观察到的组定向开关。使用相应的Fokker-Planck方程获得的预期切换时间的估计，在个人的数量和模型系数的值。在大种群的限制下，导出并分析了一个具有非局部和非线性右端的两个动力学方程组。证明了其解的存在性，并研究了系统的长期行为。最后，通过研究个体模型中相互作用半径随个体数增长而减小的效应，对拓扑相互作用的平均场极限进行了初步的探讨。
英文摘要：
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 behaviour of locusts. It exhibits nontrivial dynamics with a "phase change" behaviour and recovers the observed group directional switching. Estimates of the expected switching times, in terms of 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 when the number of individuals grows.
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