Eftimie R, de Vries G, Lewis M A, Lutscher F
Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
Bull Math Biol. 2007 Jul;69(5):1537-65. doi: 10.1007/s11538-006-9175-8. Epub 2007 Feb 15.
We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.
我们构建并分析了一个用于群体形成的非局部连续统模型，并将其应用于均匀环境中动物的自组织群体。该模型由一个双曲守恒律系统组成，将个体运动描述为相关随机游走。转向率取决于三种社会力：对其他生物体的吸引力、对它们的排斥力以及与邻居对齐的倾向。线性分析用于研究社会相互作用力及其范围在群体形成中的作用。我们证明该模型可以产生多种模式，包括驻波脉冲、行波脉冲、行波列以及一种我们称为锯齿形脉冲的新型解。此外，数值模拟表明，所有三种社会力都是解释生物系统中观察到的复杂模式所必需的。然后，我们使用该模型研究可由这些不同模式描述的动物日常活动之间的转变。
