Ecological dynamics in random environments

本项目研究了非线性动力学和随机非生物涨落对相互作用种群的分布和丰度的影响。为了研究这种相互作用，将动力系统技术和概率机制相结合的随机动力系统理论被应用于一类足够普遍的随机生态地图，以解释空间结构、阶段结构和多物种相互作用。将开发新的数学方法来验证随机耗散、持续和灭绝。这些方法将应用于随机环境中的种群分散、阶段结构种群中的存储效应和Allee效应、随机复制子动力学以及源-汇环境中理想自由种群的拮抗相互作用。对于结构总体模型，将发展新的方法来理解非负矩阵的随机积参数化族的主导Lyapunov指数的单调性和凸性。由于这些主导的Lyapunov指数决定了低密度下种群的渐近增长率，因此它们对于理解种群的持续性是至关重要的。环境波动，例如由于人为干扰或天气引起的环境波动与生物过程之间的相互作用可以显著影响物种相互作用的结果。因此，本项目开发的数学技术的应用可能对生态群落的保护或恢复、防止生物入侵、自然资源管理和农业病虫害的生物防治具有实用价值。

This project investigates the interplay of nonlinear dynamics and random abiotic fluctuations on distribution and abundance of interacting populations. To investigate this interplay, the theory of random dynamical systems which combines dynamical systems techniques with probabilistic machinery are applied to a class of random ecological maps sufficiently general to account for spatial structure, stage structure, and multispecies interactions. New mathematical methods for verifying stochastic dissipativeness, persistence, and extinction will be developed. The application of these methods to populations dispersing in random environments, the storage effect and the Allee effect in stage-structured populations, random replicator dynamics, and antagonistic interactions of ideal free populations in source-sink environments will be pursued. For models of structured populations, new methods will be developed for understanding monotonicity and convexity properties of the dominant Lyapunov exponent for parameterized families of random products of non-negative matrices. Since these dominant Lyapunov exponents determine the asymptotic growth rate of populations at low densities, they are critical for understanding persistence.The interaction between environmental fluctuations, for example those due to anthropogenic disturbances or weather, and biological processes can dramatically affect the outcomes of species interactions. Consequently, the applications of the mathematical techniques developed in this project may be of practical value to the conservation or the restoration of ecological communities, the prevention of biological invasions, the management of natural resources, and biological control of agricultural pests.