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.
