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VIGRE seminar, summer 2001: Ecological Modeling

Jay Walton, Paulo Lima-Filho
Students enrolled
Scott Pickens (undergraduate mathematics student); Dylan Copeland, Robert Main, John Ryan, Yanqiu Wang (graduate mathematics students); Meagan Clement — Washington and Lee, William Koppelman — University of Wyoming, Brandon Lindley — University of Central Arkansas REU students
The subject was ecological modeling; the mathematics utilized was very diverse including pointset topology, differential geometry, continuum mechanics, dynamical systems and partial differential equations. The overarching problem was predicting the impact upon an ecological environment of changes in the habitat size and structure, for example adding roads to a national park for the purpose of gaining access to natural resources. Ideas from topology and differential geometry were used to describe the topology and topography of habitats from the perspective of individual species. The interactions of species living within the habitat were modeled using ideas from continuum mechanics developed for that purpose by Walton and C. Turner for Turner's Ph.D. dissertation. The problems considered in this course are part of the activities supported by an NSF Biocomplexity Incubator grant (co-PI's: T. Lacher (Wildlife Ecology), Lima-Filho, Pilant, Stiller, and Walton). Previous models in this area frequently do not take into account the varied topology and topography and, in addition, make use of ODE's and spatially averaged competition parameters. These tend to predict one species completely wiping out the other. However, the PDE models that incorporate spatial effects give a much richer range of possible outcomes.
The modeling approaches that the students investigated in the VIGRE course are the subject of an upcoming science news report to be aired on the National Public Radio program Earth and Sky sometime in the future. In addition, one of the (then) first year graduate students who took the VIGRE course (supported as a VIGRE Fellow), Dylan Copeland, is now working on a problem coming from the class. The work is joint with B. Popov, J. Walton and M. Ziane, and Dylan is being supported on the NSF Biocomplexity grant mentioned above. This work will certainly lead to the first of many papers in this problem area.
One of the REU students, Bill Koppelman, has applied to the Texas A&M graduate math program (as well as others) to study mathematics applied to ecology. The lone Texas A&M undergraduate student in the course, Scott Pickens, was a sophomore at the time of the course. He now intends to pursue a mathematical career, and enthusiastically volunteered to be a member of our first ever COMAP Interdisciplinary Contest in Modeling team that competed in this year's competition in late January when he heard that the topic was wildlife and resource management. The contest problem turned out to involve species survival in a fragmented habitat to which Scott applied many of the ideas from last summer's VIGRE course.