Math/EEB/Cmplxsys 466. Mathematical Ecology.
The study of ecology is the exploration of complex, intinsically dynamic systems. Ecologists formulate mathematical models to describe this complexity; the equations that result are interesting both for their biological predictions and their mathematical form. Full analytical solution of model equations is typically impossible, yet to the mathematically prepared mind, they can yield up their secrets.
This course is intended to provide students with the tools needed to formulate and analyze ecological models. It is an overview of the major categories of models and the mathematical techniques available for their analysis. Although the focus is on ecological dynamics, students in other disciplines—including evolutionary biology, natural resources, public health, chemical and bioengineering, economics—will find the methods readily applicable to their own fields. The course presumes mathematical maturity at the level of advanced calculus with prior exposure to ordinary differential equations, linear algebra, and probability.
An additional dimension to the course is its focus on the use of computer algebra systems in mathematical analysis. Students will gain practical skills in these techniques.
Topics covered depend on student interest and time available. The following is a list of topics typically covered.
- Reproduction, growth, and death: ab initio population models
- Dynamics of single-species population models:
- birth-death processes
- branching processes
- differential equations as limits of stochastic processes
- difference equation models
- delay equations
- Optimal control theory:
- Pontryagin's principle
- Optimal harvesting
- Models of interacting populations:
- Competition interactions
- Consumer-resource dynamics
- Epidemiological models
- Structured population models:
- Mathematical demography
- Matrix models and extensions
- Seasonality and its dynamical consequences
- Bifurcations
- Local bifurcations of equilibria
- Local bifurcations of periodic orbits
- Global bifurcations
- Numerical methods
- Delay differential equations
- Simulation of stochastic models
- Model-based inference
