Nair Gokul G, Senthilnathan Athmanathan, Iyer Srikanth K, Guttal Vishwesha
Department of Physics, Indian Institute of Science, Bengaluru, 560 012, India.
Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
Phys Rev E. 2019 Mar;99(3-1):032412. doi: 10.1103/PhysRevE.99.032412.
Many animal groups are heterogeneous and may even consist of individuals of different species, called mixed-species flocks. Mathematical and computational models of collective animal movement behavior, however, typically assume that groups and populations consist of identical individuals. In this paper, using the mathematical framework of the coagulation-fragmentation process, we develop and analyze a model of merge and split group dynamics, also called fission-fusion dynamics, for heterogeneous populations that contain two types (or species) of individuals. We assume that more heterogeneous groups experience higher split rates than homogeneous groups, forming two daughter groups whose compositions are drawn uniformly from all possible partitions. We analytically derive a master equation for group size and compositions and find mean-field steady-state solutions. We predict that there is a critical group size below which groups are more likely to be homogeneous and contain the abundant type or species. Despite the propensity of heterogeneous groups to split at higher rates, we find that groups are more likely to be heterogeneous but only above the critical group size. Monte Carlo simulation of the model show excellent agreement with these analytical model results. Thus, our model makes a testable prediction that composition of flocks are group-size-dependent and do not merely reflect the population level heterogeneity. We discuss the implications of our results to empirical studies on flocking systems.
许多动物群体是异质的，甚至可能由不同物种的个体组成，称为混合物种群。然而，集体动物运动行为的数学和计算模型通常假设群体和种群由相同的个体组成。在本文中，我们使用凝聚-破碎过程的数学框架，为包含两种类型（或物种）个体的异质种群开发并分析了一种合并和分裂群体动力学模型，也称为裂变-融合动力学。我们假设，与同质群体相比，异质程度更高的群体经历更高的分裂率，形成两个子群体，其子群体的组成从所有可能的划分中均匀抽取。我们解析推导了群体大小和组成的主方程，并找到了平均场稳态解。我们预测，存在一个临界群体大小，低于这个大小，群体更有可能是同质的，并且包含数量较多的类型或物种。尽管异质群体有更高的分裂倾向，但我们发现，群体更有可能是异质的，但这仅发生在临界群体大小之上。该模型的蒙特卡罗模拟结果与这些分析模型结果非常吻合。因此，我们的模型做出了一个可检验的预测，即鸟群的组成取决于群体大小，而不仅仅反映种群水平的异质性。我们讨论了我们的结果对鸟类群体系统实证研究的意义。
