# Electives – Mathematical Modeling of Ecological Systems

Ecology is a relatively young science that grew from the largely descriptive discipline of Natural History. As the science has matured, it has begun to develop a firm quantitative foundation. For the most part, this foundation has been statistical (Regression, Correlation, Analysis of Variance, Ordination). The purpose of this course is to introduce the students to the other component of this quantitative foundation, dynamic simulation modeling of ecological processes.

The students will first be exposed to the role of models in science and the relationship of models to scientific theories. Then the basics of calculus are reviewed in the context of the mass-balance concept. Next the students are introduced to numerical (as opposed to analytical) solutions of the mass-balance equation; that is, they are taught how to get a computer to do all the hard math. They then apply these techniques to a series of examples like the growth of an individual organism and of a population of organisms, the interactions within species communities (competition for resources, predator-prey systems), the cycling of elements within ecosystems, the hydrology of a watershed, and an analysis of the CO2 balance of the atmosphere.

The students will use what they learn over the course of the semester to develop their own simulation model of an ecosystem. They are provided with a model shell that includes a Windowsâ„˘ interface, integrator, and graphical-output package. The student then provides a set of equations describing the ecological processes they want to simulate. These equations are typically based on the simple concept of a mass balance and can be applied to ecosystem element cycles, population dynamics, or community interactions.

**Required Text**

Students will be assigned chapters from selected texts and papers from the primary literature.

**What to expect**

The students will complete about six programming problems that illustrate the topics covered in lecture. Students will be evaluated predominantly on a term project. In that project they will develop their own simulation model, address some ecologically significant question with the model, and write a manuscript describing the model and analysis. The manuscript is to be written as if it were to be submitted to a scientific journal (e.g., *Ecology*). Students are encouraged to relate their project to topics covered in other courses, and may use the model they develop as part of their independent research project undertaken during the last four to six weeks of the semester. Students are encouraged to discuss their projects with one another during the semester and seek one another’s advice. In addition, each student will make an oral presentation describing her or his project to the class.

**Syllabus (2 sessions per week)**

Session 1: Computer & software orientation, model examples and intro to the original Multiple Element Limitation Model (MEL)

Session 2. Why Model?

Session 3. Fountain experiment and MS Excel model

Session 4. Mass Balance and fountain model in Lazarus

Session 5. Numerical integration

Session 6. Class model

Session 7. Class model

Session 8. Forest Nitrogen budget

Session 9. The Michaelis-Menton equation

Session 10. Catchment hydrologic budget

Session 11. Population logistic and competing populations

Session 12. Predator-prey systems

Session 13. Parameter estimation and curve fitting

Session 14. In class help with projects

Session 15. The Multiple Element Limitation (MEL) Model

Session 16. Island model

Session 17. Forest model

Session 18. In class help with projects

Session 19. Student Presentations

Session 20. Student Presentations and Final Modeling Projects due