## Summer 2002 Program in Modeling Complex Ecosystems.

Instructors: Jay Walton, Paulo Lima-Filho and with the help of Pat Wilber and Dave Swanson.

This VIGRE seminar was offered in conjunction with our NSF-funded Summer REU program .

General Description. In this interdisciplinary seminar we will discuss some mathematical aspects of ecosystems modeling. Its main emphasis will lie on the impact of changes in the landscape topology on populations, including its distribution and dynamics.

The seminar is intended for (graduate and advanced undergraduate) students from both Mathematics and Biological Sciences programs. Students in the Mathematics program will be introduced to a variety of situations and problems related to ecosystems where basic concepts of topology, geometry and differential equations will be applied. Students in Biological Sciences programs will learn new mathematicazl tools and concepts that will form a solid basis for modeling ecosystems. In either situation, the students will broaden their education and have a glimpse of some of the many contributions that Mathematics can provide to the study of biological systems.
Throughout the course, various modeling problems will be presented, and students will learn how to use the concepts presented in the seminar to solve these problems. Student participation will be a key ingredient in the seminar, as an attempt to approach the solutions to proposed problems via a broad discussion among the participants.

Outline of Topics.

• Topology of metric spaces:

The notion of metric space will be the starting point of the seminar. The abstract concept of distance can incorporate many features of a landscape and express how distinct species may perceive the topology of a landscape differently, despite living in the same geographic area.

• Metric spaces and their subspaces;
• Open sets, closed sets and their boundaries;
• Continuity, connectedness and compactness;
• Main examples: planar domains, and surfaces in three dimensional Euclidean spaces (as theoretical models for topographical descriptions of landscapes); planar grids and discrete sets (as models for digitalized landscapes).

• Basics of differential geometry of surfaces

This segment provides a presentation of particular examples of metric spaces, via a careful examination of surfaces in Euclidean space vs. planar domains. We show how to codify distances in the former via the notion of Riemannian metric on the latter. A particular manifestation of the concept arises when one analyzes topographical maps, where altitude is an additional data attached to a planar domain.

• Surfaces in Euclidean space; graphs;
• First fundamental form; lengths of curves;
• The underlying metric space; discrete analogues;
• Basic differential equations on surfaces.

• Topology of landscapes

Here we formalize the concepts of patches and habitats, as particular subspaces of a region (a metric space in this case). We generalize the usual notions of patches to incorporate scale sensitivity to our analysis. The scale sensitivity of the model will be demonstrated with the GIS program IDRISI.

• Compact subsets of a metric space;
• Various notions of distance between subsets;
• Connected components and patches; generalized patches and corridors;
• Variations in scale and computers demonstrations.

• Impact on dynamics of populations

Aspects of population dynamics will be presented, including proposed differential equations to model its behavior. This will be placed under the differential-geometric context described above.

• Issues and problems in population dynamics;
• Modeling the dynamics of populations;
• Basic differential equations governing the dynamics.