CAREER: Dynamics and harvesting of stochastic populations

环境波动已被证明可以驱使种群灭绝、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。研究环境波动(确定性和随机性)与相互作用物种的持久性之间的相互作用是重要的。开发一种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准确地确定收获和周期性或随机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，在下个世纪，温度、降水和风暴的时间变异性越来越大。该研究项目将为这一快速发展的领域提供亟需的理论支撑。与海洋动物捕捞有关的应用将是养护和管理脆弱或濒危物种的关键。在当今世界，围绕随机模型的最优控制的问题至关重要，因为在不断变化的环境中，存在着多个全球危机，以及物种的丧失。生态学家和进化生物学家将随机性作为从种群遗传学到物种灭绝风险的关键决定因素。但是，这些学科的科学家真正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交界处组织面向数学和生物专业的高级本科生和研究生的暑期班。为了有现实的物种共存模型，将周期性和随机的环境波动纳入其中是很重要的。结合动力系统和随机过程的思想，可以证明长期动力学是由生活在状态空间边界上的周期措施的入侵率(Lyapunov指数)决定的。然后，这些发展的想法将被用来研究非静态社区理论，其中系统的长期行为不能用均衡、吸引子或静态分布来描述。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，以便最大化产量。虽然有一些单一物种系统的结果，但在更现实的相互作用物种环境中却知之甚少。通过使用随机控制和马尔可夫链近似方法的新方法的组合，人们可以分析多物种捕捞问题，然后将结果应用于从渔业管理中获得重要的现实应用的洞察。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.