Ecological Modeling: From Individual Utilization of Space to Community Structure

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Cosner9973017 The investigators study mathematical models for the dynamicsof spatially distributed biological populations. The goal of theproject is to provide a better theoretical understanding of howthe spatial distribution and movement of individuals affect thepersistence or extinction of populations. The motivation is togain insight into the ways that spatial effects contribute to themaintenance of biodiversity and the ways that changes in thespatial aspects of the environment can threaten biodiversity.The project is primarily concerned with the construction andanalysis of models for edge-mediated effects and the developmentof methods for translating assumptions about the behavior ofindividuals on a small spatial scale into conclusions about thedynamics of populations on a large spatial scale. Edge-mediatedeffects occur when the presence of an interface between habitattypes has an effect on population dynamics. An example of anedge-mediated effect might occur if a fragment of habitat werecut off from a larger region by a road that some species wouldcross but others would not. If the presence of the edge (theroad) thus resulted in a change in the species composition of theecological community in the fragment, that would be anedge-mediated effect. Issues of scaling from local behavior tolarge-scale conclusions arise in ecology because it is oftenpossible to determine small-scale behavior by observing a fewindividuals, while directly observing the overall dynamics of alarge population is much more difficult. The methodology of theproject is to construct mathematical models for spatiallydistributed populations and then to analyze those models'predictions of persistence or extinction for the species theydescribe. The mathematical models are typically based onordinary or partial differential equations. The analysistypically involves ideas from dynamical systems, nonlinearfunctional analysis (especially bifurcation theory), and partialdifferential equations (especially comparison methods andelliptic spectral theory). A secondary goal is to obtain newmathematical results that are useful in the context ofapplications. The investigators study ways of using mathematics tounderstand ecological problems. The problems mostly involvespatial aspects of the environment; for example, how to designthe most effective nature reserves, or how to predict the likelyeffects of the fragmentation of natural habitats on thepopulations that inhabit them. The problems also involveinteractions between different biological species, for examplepredators and prey. The methodology is to express the essentialfeatures of a biological system in terms of equations and then tostudy the equations to determine what they predict. The analysisoften requires the development of new mathematical results, whichmay be of use to other investigators. A typical goal of theanalysis is to determine how various environmental and biologicalfactors affect the persistence or extinction of populations. Areason for using mathematical models is that in situationsinvolving endangered species, genetically engineered organisms,or large scale environmental change, it is often impossible to doexperiments but crucial to have some idea of what to expect.Some of the models are directly motivated by problems in refugedesign or pest control. Others are motivated by basic scientificquestions about the effects of the spatial aspects ofenvironments on populations. But even those may prove to berelevant to issues raised in the contexts of environmentalprotection and the biological effects of global change. Thepotential benefits of the project include improved decisionmaking in conservation, refuge design, land management, and urbanplanning.