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

与已知的人口规模波动相比，对人口的空间动态仍然知之甚少。例如，当种群在景观中扩散时，我们对多样性是如何维持的，为什么种群经常在空间中零星分布，以及物候学（生物事件的时间）如何影响这些过程和模式的理解非常有限。人口增长的Allee阈值或临界大小在许多人口的扩散中起着特殊的作用。该项目开发并分析了数学模型，以研究Allee效应和过度补偿生长的组合如何在参数空间区域内产生扩散速度和鲁棒非扩散解决方案的振荡，并从生物学角度确定何时应该预期这些结果。该项目旨在扩展这些结果，以解决诸如“入侵模式”如何在无界栖息地内的多个空间分离斑块中产生物种的生长和持久性，以及物候，阶段结构和屏障带如何影响空间扩展系统等重要问题。这项研究的结果有望对控制入侵物种的传播或促进本地物种重新引入灭绝地区产生直接的、关键的影响。研究生将通过参与数学和生物学交叉领域的研究来接受训练。该项目将通过在四个数学领域的严格努力，促进我们对空间人口动态的理解。这些是：(i)分析积分差分方程，以表征扩散速度的振荡和非扩散解的存在性；（ii）发展和分析由物候产生的狭道效应和过度补偿以及分散对种群持久性至关重要的空间模型；（iii）具有Allee效应的阶段结构模型的构建和检验；（iv）创建和探索具有小巷效应和静止或移动屏障区的空间模型。对于这些模型，将建立恒定速度和振荡速度行波解的存在性和稳定性，并探讨Allee效应与种群扩散之间的联系。将确定阻止、减缓或逆转人口入侵所需的最小屏障宽度。微分方程、积分方程和动力系统的方法将被用来确定解在开放空间中扩散或停止入侵的条件。这些模型在生物学中的应用将通过研究舞毒蛾等物种的传播来解决。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。

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.