My current research consists of two main branches. One involves using integrodifference equations (IDEs) to model habitat movement due to climate change. The other uses mixed integer linear programs (MILPs) to address reserve selection under climate change. In the future, I hope to extend each branch in a number of ways and combine the two approaches.
Two-Dimensional Habitable Area Shift
Integrodifference equations are discrete-time, continuous-space analytical models that keep track of populations with a simple life-history including growth followed by dispersal. Previous investigators have used IDEs to great effect in invasion ecology and other fields. Recently, modelers used an IDE model to describe a population with a limited habitable area shifting poleward due to rising temperatures (Zhou and Kot 2011). The one-dimensional spatial model predicts whether a population will be able to persist in the face of climate change. It also unveils the relationship between population dynamics, dispersal ability, the rate of climate change, and population persistence.
My contribution to this approach has been to extend the model to two spatial dimensions, since most populations disperse over two-dimensional space. The extended model functions a gateway to exploring complex, realistic spatial scenarios such as migration corridors, topographical barriers, and highly fragmented landscapes. Even under simplistic assumptions, the two-dimensional model suggests a number of unintuitive hypotheses about the importance of habitat length (in the direction of climate change) and habitat width (in the perpendicular direction). Depending on the kurtosis of the dispersal kernel, a species may benefit more from a wide habitat or a long habitat. See our forthcoming manuscript in the Bulletin of Mathematical Biology for details.
Dynamic Reserve Selection
Reserve selection via mathematical programming has enjoyed much use among conservationists and land managers. Given a network of land areas for potential use as protected areas, the problem is to determine which combination of areas would maximize some measure of conservation or biodiversity if actually converted to protected areas. The power of mathematical programming comes from its ability to determine the optimal reserve design without testing each of the potentially enormous number of different land area combinations. 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 developing a MILP that explicitly includes the above biological and environmental factors. The model takes into account the feedback over time between conservation decisions and population dynamics, as well as the ability of the species to acclimate to new conditions. I will use the model to explore (i) the effects of acclimation ability on conservation goals; (ii) the advantage of constructing reserve selection models with built-in feedback between population processes and selection; and (iii) the costs and benefits of taking management action now vs. later.
Dispersal and acclimation are only two potential survival strategies for populations experiencing climate change. Adaptation (microevolution) and phenological shifts could also rescue populations from extirpation. I anticipate incorporating those two factors into both the range shift and reserve selection models. In particular, I plan to apply the IDE model to an age- or stage-structured population, which possesses distinct developmental phases. The IDE framework may also be a suitable avenue to model the evolution of dispersal during invasion or under climate change.
In addition to examining evolutionary and phenological effects, I plan to study the transient dynamics inherent in the above IDE model. While many models of population persistence during climate change are concerned with the long-term (asymptotic) survival, in many cases the population’s short-term (transient) dynamics are more important. Populations that may survive in the long term can exhibit dangerous dips in size in the short term, and population that cannot survive indefinitely may exhibit a boost in size in the short term. By developing the mathematical tools to understand the transient dynamics of species moving due to climate change, I hope to inform conservation efforts and assessments of species’ risk.