Lukeman Ryan, Li Yue-Xian, Edelstein-Keshet Leah
Department of Mathematics, and Institute of Applied Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2.
Bull Math Biol. 2009 Feb;71(2):352-82. doi: 10.1007/s11538-008-9365-7. Epub 2008 Oct 15.
Collective behavior of swarms and flocks has been studied from several perspectives, including continuous (Eulerian) and individual-based (Lagrangian) models. Here, we use the latter approach to examine a minimal model for the formation and maintenance of group structure, with specific emphasis on a simple milling pattern in which particles follow one another around a closed circular path.We explore how rules and interactions at the level of the individuals lead to this pattern at the level of the group. In contrast to many studies based on simulation results, our model is sufficiently simple that we can obtain analytical predictions. We consider a Newtonian framework with distance-dependent pairwise interaction-force. Unlike some other studies, our mill formations do not depend on domain boundaries, nor on centrally attracting force-fields or rotor chemotaxis.By focusing on a simple geometry and simple distant-dependent interactions, we characterize mill formations and derive existence conditions in terms of model parameters. An eigenvalue equation specifies stability regions based on properties of the interaction function. We explore this equation numerically, and validate the stability conclusions via simulation, showing distinct behavior inside, outside, and on the boundary of stability regions. Moving mill formations are then investigated, showing the effect of individual autonomous self-propulsion on group-level motion. The simplified framework allows us to clearly relate individual properties (via model parameters) to group-level structure. These relationships provide insight into the more complicated milling formations observed in nature, and suggest design properties of artificial schools where such rotational motion is desired.
群体和鱼群的集体行为已经从多个角度进行了研究，包括连续（欧拉）模型和基于个体（拉格朗日）的模型。在这里，我们使用后一种方法来研究群体结构形成和维持的最小模型，特别强调一种简单的 milling 模式，即粒子沿着封闭的圆形路径相互跟随。我们探讨个体层面的规则和相互作用如何在群体层面导致这种模式。与许多基于模拟结果的研究不同，我们的模型足够简单，以至于我们可以获得解析预测。我们考虑一个具有距离依赖的成对相互作用力的牛顿框架。与其他一些研究不同，我们的 milling 编队不依赖于域边界，也不依赖于中心吸引力场或转子趋化性。通过关注简单的几何形状和简单的距离依赖相互作用，我们对 milling 编队进行了表征，并根据模型参数推导了其存在条件。一个特征值方程根据相互作用函数的性质指定了稳定区域。我们对这个方程进行了数值探索，并通过模拟验证了稳定性结论，展示了稳定区域内部、外部和边界上的不同行为。然后研究了移动的 milling 编队，展示了个体自主自推进对群体层面运动的影响。简化的框架使我们能够清楚地将个体属性（通过模型参数）与群体层面的结构联系起来。这些关系为自然界中观察到的更复杂的 milling 编队提供了见解，并为期望这种旋转运动的人工群体提出了设计特性。
