Biological Invasions, Range Pinning, and Adaptive Evolution of Quantitative Traits

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This research focuses on the development and analysis of mathematical models of adaptive evolution in populations distributed over a territory. The main focus is on quantitative traits (characteristics like body size affected by many genes), which have been studied less in this context than traits determined by a single gene. A combination of rigorous pencil-and-paper analysis and computer simulation is used to provide insight into how evolution affects the speed, success and ultimate extent of biological invasions. The results can inform thinking on environmental management issues, such as deciding how to allocate conservation measures among marginal or central populations. By elucidating general principles that can guide the construction and interpretation of more specific models for management of natural populations, this research helps to address the economically important environmental challenges of management of invasive species and response to climate change.Mathematically, this project will result in the development of new methods for proving existence and stability of localized solutions (representing populations whose territory is not expanding) and traveling waves (representing invasive species), for a class of differential equations and similar systems that do not admit analysis via commonly employed methods. Specifically, this analysis will result in the development of new methods for proving existence and stability of localized solutions and traveling waves, for a class of biologically meaningful systems that do not admit analysis via phase plane or comparison principle methods. Since those techniques cannot readily be extended to many systems or equations, progress in devising alternative approaches will help to fill a significant gap. Biologically, the resulting analysis of general range dynamics models involving quantitative traits will contribute to understanding of the interplay between gene flow and numerous other demographic, genetic and environmental factors that help shape a population's distribution in space and time. It will highlight factors, such as long-distance dispersal and habitat patchiness, that are not yet integrated into the relevant theory. This work will help show how incorporating adaptive genetic change into invasion models alters the predictions of such models.