My research is at the interface between applied mathematics and ecology and evolutionary biology. I model the dynamics of biological populations, use mathematics to shed light on the abundance and distribution of biological populations, and use biology to motivate interesting mathematical problems.
In recent years, my research has centered on integrodifference equations. These are discrete-time, continuous-space models for the growth and spread of biological populations. Integrodifference equations readily incorporate a range of dispersal mechanisms. They appear to be both more flexible and more realistic than simple reaction-diffusion models. Integrodifference equations are extremely useful for modeling biological invasions and for modelling the effects of climate change.
In addition to my long-standing interest in integrodifference equations, I am also interested in understanding how simple ecological models can give rise to complex or chaotic dynamics and in a broad range of other ecological and evolutionary models, systems, and analyses.
I am interested in plant species response to climatic change. In particular, I study 1) the processes driving range dynamics and 2) how patterns, in particular, deviations from the norm, can provide new insights. Rather than considering variability to be noise, I consider variability to reflect previously unidentified or overlooked factors. My goal is to use deviations in expected patterns in plant species distribution and traits to understand and, more realistically, predict response to climatic change. To address this goal I use observation data, field experiments, and statistical modeling.
Currently I am using mathematical and statistical models to assess global variability of plant species response to climate change. To understand and predict climate change impacts on species range margins we need more sophisticated mathematical (for creating generalizations) and statistical (for creating predictions) models that take into account species sensitivity (functional traits) and ability to respond to climate change (demographic and dispersal rates) along with possible interactions with the rate of climate change.
The overall goals of this project are to: 1) use theoretical mathematical models, based on stage-structured integrodifference equations, to identify potential threshold limits of tolerance and relationships not readily identified through observational studies and 2) use statistical models, built within a hierarchical Bayesian framework, to incorporate complexity apparent in natural systems, providing greater realism in analyses and accuracy in predictions.
My current research consists of two main branches.
One involves using integrodifference equations (IDEs) to model range shift under climate change. The other uses mixed integer programs (MIPs) to address reserve selection under climate change.
Integrodifference equations are discrete-time, continuous-space analytical models that keep track of populations with a simple life-history including growth followed by dispersal. Recently, investigators used an IDE model to describe a population with a limited habitable area shifting poleward due to rising temperatures. The one-dimensional spatial model predicts whether a population will be able to persist in the face of climate change. My contribution to this approach has been to extend the model to two spatial dimensions, since most populations disperse over 2-D space. The extended model will function as a gateway to exploring complex, realistic spatial scenarios such as migration corridors, topographical barriers, and highly fragmented landscapes.
Reserve selection via mathematical programming has enjoyed much use among conservationists and land managers. Given a network of land areas for potential use as nature reserves, the problem is to determine which combination of land areas would maximize some measure of conservation or biodiversity if actually converted to reserves. In the context of climate change, conservationists may need to hit a moving target as populations shift poleward and upslope. In addition to shifting their distributions, species may also respond to warming through acclimation. As a consequence, there is a need to design reserves that take into account population dynamics, dispersal, acclimation, and changing environmental conditions. I am currently developing an MIP that explicitly includes the above biological and environmental factors.
I am studying the effects of climate change on the population dynamics of species that are competing for the same resources. I'm using a coupled system of integrodifference equations to create a spatially explicit model that can be used to predict the long-term stability of competing populations, and their ability to keep pace with a changing climate. The model helps to illustrate how key life history information such as growth rate, dispersal ability, and carrying capacity can influence survival. This project is funded by a grant from the National Science Foundation.