CAREER: Dynamics and harvesting of stochastic populations

环境波动已被证明会导致种群灭绝，促进持久性，逆转竞争性排斥，改变遗传多样性，并改变传染病的传播。研究确定性和随机性环境波动之间的相互作用以及相互作用物种的持久性是很重要的。发展一个严谨的共存数学理论，结合数据驱动的应用程序，将有助于理论生态学家精确地指出收获和周期性或随机的环境波动如何影响生态群落的长期动态。全球气候变化模式预测下个世纪温度、降水和风暴的时间变异性会增加。该研究项目将为这一快速发展的领域提供急需的理论基础。与捕捞海洋动物有关的应用将是保护和管理脆弱或濒危物种的关键。围绕随机模型最优控制的问题在当今世界至关重要，因为在不断变化的环境和物种丧失中存在多种全球危机。生态学家和进化生物学家认为，从种群遗传学到灭绝风险，随机性是决定一切的关键因素。但是，来自这些学科的科学家实际上对支撑随机过程的数学概念的了解是不完整的。教育目标的一个组成部分将是组织一所暑期学校，在生物学和随机学的界面上，针对数学和生物学的高级本科生和研究生。为了建立物种共存的现实模型，将周期性和随机的环境波动结合起来是很重要的。将动力系统和随机过程的思想联系起来，将有可能表明，长期动力学是由生活在状态空间边界上的周期性测度的入侵率（李雅普诺夫指数）决定的。这些发展出来的思想将被用于研究非平稳群落理论，在非平稳群落理论中，系统的长期行为不能用平衡、吸引子或平稳分布来描述。保护生物学的一个重要问题是，如何在不导致种群灭绝的情况下，收获一个给定的种群，以最大限度地提高产量。虽然在单物种系统中有一些结果，但在更现实的物种相互作用环境中却知之甚少。通过结合使用随机控制和马尔可夫链近似方法的新方法，可以分析多物种捕捞问题，然后应用结果，以便从渔业管理中获得重要的现实应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Environmental fluctuations have been shown to drive populations extinct, facilitate persistence, reverse competitive exclusion, change genetic diversity, and modify the spread of infectious diseases. It is important to study the interplay between environmental fluctuations, both deterministic and random, and the persistence of interacting species. Developing a rigorous mathematical theory for coexistence, in conjunction with data-driven applications, will help theoretical ecologists pinpoint how harvesting and periodic or random environmental fluctuations affect the long term dynamics of ecological communities. Global climate change models predict increasing temporal variability in temperature, precipitation and storms in the next century. The research project will provide much-needed theoretical underpinning for this fast-moving area. The application related to the harvesting of marine animals will be key for conservation and management of vulnerable or endangered species. Questions around optimal control of stochastic models are vital in today's world where there are multiple global crises in a changing environment as well as species loss. Ecologists and evolutionary biologists invoke stochasticity as a key determinant of everything from population genetics to extinction risk. But the exposure that scientists from such disciplines actually get to the mathematical concepts underpinning stochastic processes is incomplete. An integral component of the educational objectives will be the organization of a summer school at the interface of biology and stochastics targeted to advanced undergraduate and graduate students from mathematics and biology. In order to have realistic models for the coexistence of species it is important to incorporate both periodic and random environmental fluctuations. Connecting ideas from dynamical systems and stochastic processes, it will be possible to show that the long-term dynamics is determined by the invasion rates (Lyapunov exponents) of the periodic measures living on the boundary of the state space. The developed ideas will then be used to look at non-stationary community theory where the long term behavior of the system can not be described by an equilibrium, an attractor, or a stationary distribution. An important question from conservation biology is how to harvest a given population in order to maximize the yield while not driving the population extinct. While there are a few results for single-species systems, little is known in the significantly more realistic setting of interacting species. By using a combination of novel approaches from stochastic control and Markov chain approximation methods one can analyze multi-species harvesting problems and then apply the results in order to gain insight for important real-life applications from fishery management.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.