In this paper, we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well as the available space. All individuals have the tendency to stay within their own group and avoid the others. These interactions lead to the formation of aggregates in case of a single species and to segregation in the case of multiple species. We derive two different mean-field models, which are based on these interactions and weigh local and non-local effects differently. We discuss existence and stability properties of solutions for both models and illustrate the rich dynamics with numerical simulations.

中文翻译：

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分离动力学的平均场模型

在本文中，我们推导并分析了随机游走个体群体动力学的平均场模型。个体的随机运动仅受存在的不同群体的感知密度以及可用空间的影响。所有的人都倾向于留在自己的群体中并避开其他人。这些相互作用在单一物种的情况下导致聚集体的形成，在多种物种的情况下导致分离。我们推导出两种不同的平均场模型，它们基于这些相互作用，并以不同的方式权衡局部和非局部效应。我们讨论了两种模型解的存在性和稳定性属性，并通过数值模拟说明了丰富的动力学。