Collective Motion Under Non-Reciprocal Pairwise Interactions

这个项目是为了研究非互易粒子间力对长期集体运动的影响，该项目是由埃克塞特的实验行为生态学家合作推动的。它们感兴趣的颗粒是Trinidadian Guppies(Poecilia Reticata)，这种非互惠的力量是该物种雄性和雌性之间性别冲突的结果。从长远来看，它们的行为还会导致关键的生态和进化过程，如种群扩散和外来物种入侵。初步的实验工作和一些初步的数学模型表明，在性冲突的驱动下，这种动态可能会导致孔雀鱼成对的异常快速扩散。要将特里尼迪亚古比鱼成对相互作用的生物学和整个系统的种群水平后果结合起来，将需要不同时间、空间和社会尺度上的过程的数学模型，以及建立在实验数据的经验证据基础上的假设。该项目的目的是建立一个通用的数学框架，从一个社会、空间和时间尺度到另一个社会、空间和时间尺度进行内插和外推，模拟非相互(和相互)的社会力量对大规模人口过程、流动和结构的影响。在生物学方面，这将处于运动生态学和集体行为的交叉点，并激发了解决“规模问题”的需要--了解随着时间的推移，个体水平的相互作用和决定是如何进入和推动种群水平的模式和过程的。由于孔雀鱼的运动在一定程度上是随机的，而且它们生活在‘裂变-聚变’社会中--松散的群体容易迅速凝结和碎裂，因此需要广泛的数学工具和技术来生成一套模型。可能的方法包括但肯定不限于：一个基于数据综合因素的模型，以开发和测试测量和模拟的局部相互作用对更大规模成果的影响。位置和状态会随着环境和附近其他个体的变化而发生变化，并增加噪声来解释不确定性；群体动力学的凝聚-碎片化模型，其中空间粗粒化将从个体动态映射到描述小群体内种群密度的随机偏微分方程。群体构型和内部个体运动之间的反馈意味着这个方程将是McKean-Vlasov类型的。这个模型应该产生有效的基团形成和溶解速率，以提供将种群描述为可交换碎片凝聚过程(EFCP)的信息；以及种群水平模型，通过直接从McKean-Vlasov SPDE描述或从EFCP获得较大的比例限制获得(这些结果之间的一致性将是我们建模的重要一致性检查)。在任何一种情况下，都可以预测出现的种群水平特性。这包括在具有不同浓度的相互和非相互动态的种群中的全球扩散速度，或在不断扩大的种群中可能出现的表型梯度等数量。

This project, to investigate the effect on non-reciprocal inter-particle forces on long-term collective motion, is motivated by collaboration with experimental behavioural ecologists in Exeter. Their particles of interest are Trinidadian Guppies (Poecilia reticulata), and the non-reciprocal forces arise as a result of sexual conflict between males and females of the species. In the long-term, their behaviour also results in key ecological and evolutionary processes such as population dispersal and the invasion of alien species. Pilot experimental work and some preliminary mathematical modelling suggest that this dynamic, driven by sexual conflict, can give rise to anomalously fast diffusion of pairs of guppies. To combine the biology of pairwise interactions of Trinidadian Guppy fish through to population level consequences throughout the system will require mathematical models of processes at different time, spatial and social scales, with assumptions founded in empirical evidence from the experimental data. The aim of this project is to develop a general mathematical framework to interpolate between and extrapolate from one social, spatial and temporal scale to the next, modelling the effect of non-reciprocal (and reciprocal) social forces on large-scale population processes, movement and structure. In biological terms this will sit in the intersection of movement ecology and collective behaviour, and motivates the need to address the 'problem of scale' - understanding how individual level interactions and decisions, feed into and drive patterns and processes at the population level over time. As the movement of guppies is in part stochastic, and they live in 'fission-fusion' societies - with loose groups subject to rapid coagulation and fragmentation, a wide range of mathematical tools and techniques will be needed to generate a suite of models. Likely approaches include but are certainly not limited to: a data-integrative agent-based model to develop and test the effects of measured and modelled local interactions on larger-scale outcomes. Changes in position and state will occur in response to the environment and other nearby individuals, with noise added to account for uncertainty; coagulation-fragmentation model of group dynamics, in which spatial coarse graining would map from individual dynamics to a stochastic PDE describing population density within small groups. Feedback between the group configuration and the motion of individuals within it means this equation will be of the McKean-Vlasov type. This model should yield effective rates of group formation and dissolution to inform a description of the population as an exchangeable fragmentation coagulation process (EFCP); and population level models, obtained by taking large scaling limits either directly from the McKean-Vlasov SPDE description, or from the EFCP (agreement between these results will be an important consistency check for our modelling). In either case, emergent population level properties can be predicted. This includes quantities such as the global speed of dispersal in populations with different concentrations of reciprocal and non-reciprocal dynamics, or possible emergent phenotypic gradients in expanding populations.