COLLABORATIVE RESEARCH: Spatial Spread of Stage-structured Populations

生态学的两个重大挑战是理解和准确描述物种的空间分布。这种空间扩散在各种生态环境中都很重要，例如当非本地物种入侵新的栖息地时，以及当物种为响应全球变化过程而改变其空间分布时。为了应对这些挑战，需要能够捕捉物种空间传播动态的基本方面的种群模型，包括人口统计学和扩散。积分-差分方程将用于描述具有单独生长和扩散阶段的种群的扩散，其中生命速率和扩散能力由年龄、大小或发育阶段决定。涉及反应-扩散方程和积分-微分方程的半离散模型（混合动力系统）将被用于研究种群的传播，其中不同的过程或不同的速率发生在物种繁殖期内外。对于传粉受限的植物群体，以及生殖不同步和不完全交配的两性群体，将建立具有Allee效应的模型。来自两个经过充分研究的现场系统的数据与特定模型的结构相匹配，将用于参数化关键模型组件。研究人员将为模型检验传播速度和行波的存在性，提供传播速度和行波速度的公式，并计算速度对人口统计学和传播参数变化的敏感性和弹性。将使用微分方程、积分方程和动力系统的方法来研究模型的空间动力学。这项研究的结果也将对其他研究波传播的科学学科产生更广泛的影响。新的严谨数学将与广泛的现场和实验室数据相结合，以弥合抽象数学结果与生态观测之间的鸿沟。为了进一步扩大这项研究的影响，研究人员还将开发一个与生态入侵动力学相关的MathBench模块（.umd.edu）。该模块将被纳入由nsf资助的MathBench计划，该计划旨在提高生物学本科生的定量素养，并让他们更深入地了解数学在理解生物学问题中的作用。通过数学和生物学结合的新研究，该项目将有助于增加物种空间传播的信息。物种空间扩散的动态研究对于了解资源管理者何时何地可以采取行动限制非本地入侵物种的传播和影响至关重要。同样，更好地了解空间扩散的动态对于预测物种对全球变化过程的反应至关重要。在这个项目中，研究人员将开发和分析包含物种出生、生长、死亡和运动的数学模型，以确定物种的点。生命周期对人口空间扩散速度至关重要。受花粉供应限制的植物物种和具有不完美交配的种群的模型将被探索，重点是了解特定种群过程对物种空间传播速度和性质的影响。除了新颖的数学外，该项目还侧重于两个生态案例研究，将有助于指出自然资源管理的具体目标和机会。

Two significant challenges in ecology are to understand and accurately describe the spatial spread of species. Such spatial spread is important in a variety of ecological contexts, such as when non-native species invade new habitat and when species shift their spatial distributions in response to global change processes. Meeting these challenges requires population models that capture essential aspects of the dynamics of spatially spreading species, including demography and dispersal. Integro-difference equations will be used to describe the spread of populations with separate growth and dispersal stages wherein vital rates and dispersal abilities are determined by age, size, or developmental stage. Semi-discrete models (hybrid dynamical systems) involving reaction-diffusion equations and integro-differential equations will be employed to study the spread of populations in which different processes or different rates occur inside versus outside a species' reproductive period. Models with Allee effects will be developed for plant populations with pollination limitation, and for two-sex populations with reproductive asynchrony and imperfect mate-finding. Data from two well-studied field systems matching the structure of specific models will be used to parametrize key model components. The investigators will examine the existence of spreading speeds and traveling waves for the models, provide formulas for spreading speeds and traveling wave speeds, and calculate the sensitivity and elasticity of the speeds to changes in demographic and dispersal parameters. Methods from differential equations, integral equations, and dynamical systems will be used to investigate the spatial dynamics for the models. The outcomes of this research will also have broader impacts in other scientific disciplines where wave propagation is addressed. New rigorous mathematics will be integrated with extensive field and laboratory data to bridge the gulf between abstract mathematical results and ecological observations. To further broaden the impacts of this research, the investigators will also develop a MathBench module (.umd.edu) relating to ecological invasion dynamics. This module will feed into the larger, NSF-funded MathBench Initiative, which is designed to improve the quantitative literacy of undergraduate biology students and give them a deeper appreciation of the role of mathematics in understanding biological problems. Through new research at the interface of mathematics and biology, this project will contribute to the growing body of information on the spatial spread of species. Research on the dynamics of species spatial spread is essential to understanding when and where resource managers can act to limit the spread and impacts of non-native, invasive species. Likewise, better understanding of the dynamics of spatial spread is essential for forecasting species responses to global change processes. In this project the investigators will develop and analyze mathematical models incorporating species birth, growth, death, and movement to identify points in species? life-cycles that are critical to the rates of population spatial spread. Models for plant species limited by pollen supply and for populations featuring imperfect mate finding will be explored, with a focus on understanding the effects that particular population processes have on the rate and nature of species spatial spread. By focusing on two ecological case-studies in addition to novel mathematics, this project will help to point out specific targets and opportunities for natural resources management.