Ecological dynamics in random environments

本计画探讨非线性动力学与随机非生物波动对相互作用族群分布与丰度的影响。为了研究这种相互作用，随机动力系统的理论相结合的动力系统技术与概率机械适用于一类随机生态地图足够一般的空间结构，阶段结构，和多物种的相互作用。新的数学方法验证随机耗散，持久性和灭绝将被开发。这些方法的应用，分散在随机环境中的人口，存储效应和Allee效应在阶段结构的人口，随机复制动态，和理想的自由人口在源-汇环境中的拮抗作用将被追求。对于结构化种群模型，将开发新的方法来理解非负矩阵随机乘积的参数化族的主Lyapunov指数的单调性和凸性。由于这些主要的李雅普诺夫指数决定了种群在低密度下的渐近增长率，它们对于理解持久性至关重要。环境波动之间的相互作用，例如由于人为干扰或天气，和生物过程可以显着影响物种相互作用的结果。因此，本项目所发展的数学技术的应用可能对生态群落的保护或恢复、生物入侵的预防、自然资源的管理和农业害虫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.