Collaborative Research: Population Dynamics in Random Environments: Theory and Approximation

该项目将制定和分析一个通用的数学框架，以促进理解受随机环境波动影响的物种的持续和灭绝。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。随机的环境波动已被证明会导致种群灭绝，促进持久性，改变遗传多样性，并改变传染病的传播。因此，迫切需要开发工具来了解环境条件的随机时间波动对物种的影响。pi将发展数学理论，结合分析和数值近似方法，帮助理论生态学家确定环境波动如何影响生态群落的长期动态。在与麻省理工学院戈尔实验室的合作下，pi将通过与微生物生态学实验的比较来测试理论结果。研究人员计划让高中生和大学生参与项目，让他们发展编程技能，丰富他们的数学和生态知识。在外联方面，调查人员将组织研讨会和会议，并促进妇女和传统上代表性不足的少数民族成员参与科学。pi将研究经历随机时间环境变化的相互作用种群的连续和离散时间模型。在连续时间设置下，研究将集中在分段确定性马尔可夫过程-在不同的常微分方程系统之间随机切换的过程。在离散时间设置下，将分析随机差分方程。将开发新的方法来检查物种何时持续存在并收敛于它们的不变概率度量（描述物种亚群落的“随机平衡”），并以指数速度确定物种灭绝的条件。由于所有的理论模型都只是自然系统的近似，pi将研究在模型参数的小的、依赖于密度的扰动下，持续/灭绝结果是如何变化的。消失/持续标准将涉及李雅普诺夫指数，通常不能明确计算。为了解决这一问题，将发展估计李雅普诺夫指数的解析和数值近似方法。最后，pi将与麻省理工学院的戈尔实验室一起进行实验，以了解在环境波动下，分析结果如何与多物种微生物系统进行定性比较。该项目由数学科学部数学生物学计划和促进竞争研究的既定计划（EPSCoR）共同资助。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。

This project will formulate and analyze a general mathematical framework to facilitate understanding the persistence and extinction of species affected by random environmental fluctuations. Global climate change models predict increasing temporal variability in temperature, precipitation and storms in the next century. Random environmental fluctuations have been shown to drive populations extinct, promote persistence, change genetic diversity, and modify the spread of infectious diseases. It is therefore urgent to develop tools for understanding the effects of random temporal fluctuations in environmental conditions on species. The PIs will develop mathematical theory, in conjunction with analytical and numerical approximation methods, to help theoretical ecologists pinpoint how environmental fluctuations affect the long-term dynamics of ecological communities. In collaboration with the Gore lab at Massachusetts Institute of Technology the PIs will test theoretical results by comparison with microbial ecology experiments. The investigators plan to involve high school and undergraduate students in projects allowing them to develop programming skills and diversify their mathematical and ecological knowledge. For outreach, the investigators will organize seminars and conferences and promote the participation of women and members of traditionally underrepresented minorities within the sciences.The PIs will investigate continuous and discrete time models of interacting populations that experience random temporal environmental variations. In the continuous time setting the research will focus on Piecewise Deterministic Markov Processes - processes that switch between different systems of ordinary differential equations at random times. In the discrete time setting stochastic difference equations will be analyzed. New methods for checking when species persist and converge to their invariant probability measures (which describe the 'random equilibria' of subcommunities of species) will be developed, and conditions under which species go extinct exponentially fast determined. Since all theoretical models are merely approximations of natural systems, the PIs will study how the persistence/extinction results change under small, density-dependent, perturbations of the model parameters. The extinction/persistence criteria will involve Lyapunov exponents, which usually cannot be computed explicitly. In order to resolve this issue analytical and numerical approximation methods for estimating the Lyapunov exponents will be developed. Finally, together with the Gore lab at Massachusetts Institute of Technology, the PIs will run experiments in order to see how analytical results qualitatively compare with multi-species microbial systems under environmental fluctuations. This project is jointly funded by the Division of Mathematical Sciences Mathematical Biology Program the Established Program to Stimulate Competitive Research (EPSCoR).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.