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

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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.