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MBI Emphasis Year on Ecology and Evolution Tutorials
September 2005 - August 2006

Tutorial on Tree Reconstruction and Coalescence Theory
(September 7-9, 2005 and September 12-13, 2005)

Organizers: Dennis Pearl and Paul Fuerst

Phylogenetic trees are commonly used to describe the evolutionary history of a group of species, and may also be used to study rapidly evolving individual organisms such as certain viruses, bacteria or parasites. These trees are high-dimensional, non-real-valued data objects, with a specific pattern of built-in dependencies that violate the assumptions of many traditional methodologies and thus provide a rich source of statistical and mathematical challenges. This tutorial will provide an introduction to the area illustrated with some interesting and important biological problems that can be addressed using phylogenetic techniques.

Reference: Felsenstein, J (2003) Inferring Phylogenies. Sinauer Associates

Tutorial on Reaction - Diffusion Models (March 9-10, 2006)

Organizer: Chris Cosner

Reaction-Diffusion equations have been used extensively in mathematical ecology as models for the dynamics and interactions of spatially distributed populations. They provide a way of translating assumptions about local rates of movement, reproduction, and mortality into global conclusions about the persistence of populations and the structure of communities. They can be derived as continuum limits of spatially discrete stochastic processes. They can incorporate boundary conditions that describe edge-mediated effects. There are three major types of phenomena that can arise in reaction-diffusion models: traveling wavefronts, the formation of patterns in homogeneous space, and the presence of lower bounds on the sizes of domains that will support nonzero solutions or solutions with spatial patterns. Thus, they can be used to address issues related to biological invasions, spatial patterning, and critical patch size. The analysis of reaction-diffusion equations involves a mixture of ideas from dynamical systems and the theory of partial differential equations. Many reaction-diffusion equations have monotonicity properties arising from the maximum principle which allow comparisons between solutions. The stability of their equilibria is typically determined by the signs of principal eigenvalues of related elliptic partial differential operators. Information about the stability of equilibria often can be used to analyze the overall structure of the set of equilibria or the asymptotic behavior of solutions by means of bifurcation theory and persistence theory. The derivation, interpretation, and analysis of reaction-diffusion models will be discussed, along with the essential background ideas from partial differential equations and dynamical systems. Applications to biological invasions, spatial patterning, and spatial effects influencing the persistence or coexistence of populations will be described. The material will be drawn from various sources, a few of which are listed below.

1. R.S. Cantrell and C. Cosner (2003), Spatial Ecology via Reaction-Diffusion Equations. Wiley.
2. J. D. Murray (2004), Mathematical Biology I and II. Springer.
3. A. Okubo and S. Levin (2001), Diffusion and Ecological Problems: Modern Perspectives. Springer.