Math/EEB 466. Mathematical Ecology.
The fundamental question in ecology is a quantitative one: “What determines the distribution and abundance of organisms of different kinds?” Answers to this question take the form of mathematical models. Since ecological systems are heterogeneous and highly dynamic, such models are expected to account for the variability of organism abundance in both space and time. This course introduces the mathematics of dynamical models in ecology. We are interested in the relationships (a) between models and the principles of ecology, (b) among different model formulations, and (c) between models and ecological data.
Ecological systems are strongly and famously nonlinear, which means that ecological models cannot typically be “solved”. Rather, the techniques of nonlinear analysis, including dynamical systems theory, bifurcation analysis, and scientific computation, can be used to reveal model predictions. Deterministic nonlinear analysis alone is insufficient, however: careful consideration of ecological systems requires probabilistic thinking, both to account for the complexity of the real world and to make sense of data. In this course, we lay a solid foundation for the mathematical tools, both deterministic and stochastic, of the theoretical ecologist. We begin with deterministic nonlinear analysis of population models, proceed to the consideration of stochasticity in its various forms, and conclude by developing the tools needed to treat ecological models as scientific instruments. Throughout, emphasis is placed on the concepts that unify the subject and on a critical evaluation of the manner in which models express, or fail to express, aspects of biological complexity.
Students completing the course will have learned to create their own mathematical models, to analyze these models, and to test model predictions by confronting them with data.
Graduate and undergraduate students interested in the use of mathematical models and theoretical, statistical, and computational epidemiology and ecology are encouraged to enroll. The techniques introduced in the course will be useful to students from many disciplines, including ecology, evolutionary biology, complex systems, applied mathematics, physics, epidemiology, engineering, computer science, economics, natural resources, and others.
The course presumes mathematical maturity at the level of advanced calculus with prior exposure to ordinary differential equations, linear algebra, and probability. Students with concerns about their level of preparation are invited to consult the instructor before registering.
The course requirements consist of attendance at lecture, participation in discussion, and completion of periodically-assigned readings and homework exercises. Some homework problems will involve numerical computation in the R programming environment.