Spreading Speeds and Non-Spreading Solutions for Spatial Population Models with Allee Effects

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Spatial dynamics of populations remain poorly understood when compared to what is known about fluctuations in population size. For example, we have very limited understanding of how variability is maintained when populations spread across landscapes, why populations are often patchily distributed in space, and how phenology (the timing of biological events) influences these processes and patterns. The Allee threshold or critical size for population growth plays a particular role in spread of many populations. This project develops and analyzes mathematical models to investigate how a combination of an Allee effect and over-compensatory growth can produce oscillations in spreading speeds and robust non-spreading solutions across regions of parameter space, and to determine biologically when these outcomes should be expected. The project aims to extend these results to address important questions such as how "invasion models" can yield growth and persistence of a species in multiple, spatially separated patches within an unbounded habitat, and how phenology, stage-structure, and barrier zones affect spatially spreading systems. The findings of this research are expected to have direct, critical implications for controlling the spread of invasive species or promoting the reintroduction of native species into areas of extirpation. Graduate students will be trained through involvement in research at the interface of mathematics and biology. This project will advance our understanding of spatial population dynamics through rigorous efforts in four mathematical areas. These are: (i) analysis of integro-difference equations to characterize oscillations in spreading speeds and existence of non-spreading solutions; (ii) development and analysis of spatial models where Allee effects and over-compensation are generated by phenology and where dispersal is critical to population persistence; (iii) construction and examination of stage-structured models with an Allee effect; and (iv) creation and exploration of spatial models with an Allee effect and a stationary or moving barrier zone. For the models, the existence and stability of traveling wave solutions with constant and oscillating speeds will be established, and the links between the Allee effect and population dispersal will be explored. The minimal width of a barrier zone needed to stop, slow, or reverse a population invasion will be determined. Methods from differential equations, integral equations, and dynamical systems will be employed to identify conditions under which solutions spread into open space or stop invading. Applications of the models to biology will be addressed through studies of spread of species such as the gypsy moth.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.