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，逆转竞争排斥，改变遗传多样性，并改变传染病的传播。必须研究环境波动之间的相互作用，包括确定性和随机性，以及相互作用物种的持久性。开发一个严格的共存数学理论，结合数据驱动的应用程序，将有助于理论生态学家确定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。全球气候变化模型预测，在下一个世纪，气温、降水和风暴的时间变异性将增加。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。与海洋动物捕捞有关的应用将是保护和管理脆弱或濒危物种的关键。随机模型的最优控制问题在当今世界至关重要，在不断变化的环境中存在多个全球危机以及物种损失。生态学家和进化生物学家将随机性作为从种群遗传到灭绝风险的所有事情的关键决定因素。但是，来自这些学科的科学家实际上接触到支撑随机过程的数学概念是不完整的。教育目标的一个组成部分将是在生物学和随机学的界面上组织一个暑期学校，目标是数学和生物学的高年级本科生和研究生。为了有现实的物种共存模型，重要的是要纳入周期性和随机的环境波动。从动力系统和随机过程连接的想法，它将有可能表明，长期的动态是由入侵率（李雅普诺夫指数）的周期性措施生活在状态空间的边界。发展的想法，然后将被用来看看非平稳社区理论的长期行为的系统不能被描述的平衡，吸引子，或固定分布。保护生物学的一个重要问题是如何收获一个给定的种群，以最大限度地提高产量，同时不使种群灭绝。虽然有一些单物种系统的结果，很少有人知道在更现实的设置相互作用的物种。通过使用随机控制和马尔可夫链近似方法的新方法相结合，可以分析多物种捕捞问题，然后应用结果，以便深入了解渔业管理的重要现实应用。该奖项反映了NSF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.