Collective Motion Under Non-Reciprocal Pairwise Interactions

该项目旨在研究非互反粒子间力对长期集体运动的影响，其动机是与埃克塞特的实验行为生态学家合作。他们感兴趣的粒子是特立尼达孔雀鱼（Poecilia reticulata），这种非相互作用力是该物种雄性和雌性之间性冲突的结果。从长远来看，它们的行为也导致了关键的生态和进化过程，如人口分散和外来物种的入侵。初步的实验工作和一些初步的数学模型表明，这种由性冲突驱动的动态可以导致成对的孔雀鱼异常快速的扩散。要将特立尼达孔雀鱼的成对相互作用的生物学与整个系统的种群水平的后果结合起来，需要在不同的时间、空间和社会尺度上建立过程的数学模型，并以实验数据的经验证据为基础。该项目的目的是发展一个一般的数学框架，以便在一个社会、空间和时间尺度之间进行内插和推断，模拟非互惠（和互惠）社会力量对大规模人口过程、流动和结构的影响。用生物学的术语来说，这将是运动生态学和集体行为的交叉点，并且激发了解决“规模问题”的需要——理解个体层面的相互作用和决策如何随着时间的推移进入并驱动种群层面的模式和过程。由于孔雀鱼的活动在一定程度上是随机的，而且它们生活在“裂变-融合”社会中——松散的群体容易迅速凝聚和分裂，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：基于数据整合的代理模型，以开发和测试测量和建模的局部相互作用对更大规模结果的影响。位置和状态的变化将根据环境和其他附近个体的变化而发生，并添加噪音以说明不确定性；群体动力学的凝固-破碎模型，其中空间粗粒度将从个体动态映射到描述小群体内种群密度的随机PDE。群体结构和个体运动之间的反馈意味着这个方程将是McKean-Vlasov型的。该模型应得出群体形成和溶解的有效速率，从而将群体描述为可交换破碎凝聚过程（EFCP）；和人口水平模型，通过直接从McKean-Vlasov SPDE描述或从EFCP获取大尺度限制而获得（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，涌现的种群水平属性都是可以预测的。这包括具有不同浓度的互反和非互反动态的种群的全球扩散速度，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.