Mathematical Ecology - Syllabus 2006-2007
Math 581 Section#1 - EEB 581 Section#1
Dr. Louis Gross (gross@tiem.utk.edu)
Home Page: http://www.tiem.utk.edu/~gross/math581.html
Meeting time: 2:30-3:20
MWF Place: Ayres 111
Objectives: The goal of this course sequence is to provide an overview
of mathematical approaches in ecology. The emphasis is on developing
participants appreciation for the variety of approaches an applied
mathematician may take in addressing real-world problems. There is a
particular focus on the development of mathematical models to
elucidate general patterns arising in natural systems. Although the
emphasis is on ecological patterns, the approaches we will discuss are
readily applicable across the sciences. By the end of the sequence,
students should be capable of reading current research and be prepared
to pass a preliminary examination in the field. The course presumes
mathematical maturity at the level of advanced calculus with prior
exposure to basic differential equations, linear algebra, and
probability.
Textbook: Elements of Mathematical Ecology by Mark Kot. Cambridge
University Press, 2001. We will follow this text fairly closely. The
text will be supplemented by materials from several texts on the
accompanying reference list, as well as papers assigned in class.
Topics to be covered are given below, though these may be modified to a
certain extent by the interests of class participants.
Format: The course will be taught in lecture format, with occasional
in-class demonstrations. Class participants will be expected to attend
some special colloquia related to the topics of the course as they are
held during the semester. Students who audit must attend lectures, do
the assigned readings and participate in discussions. Students are
encouraged to attend Math 589 Section #1 to obtain additional
perspectives on the course topics.
Class Grading: I will regularly assign problems related to the course
material as homework. You may work on such problems with others from
the course, but you must independently write up your results, and make
it clear with whom you have collaborated on each homework set. I expect
to give one test at the end of the semester, to aid you in preparing
for the preliminary exam. This will likely be a take-home exam. There
will also be one computer-based project due at the end of the semester.
Course grading will be based upon: test (25% of grade), homework (50%),
project (25%).
Computer-based Project: There will be a computer-based project that
will be due at the end of the semester. The objective here is to ensure
that all participants are familiar with some standard methods to
numerically analyze a more complex problem in math ecology that
analytical methods are not able to address. It is also to encourage
participants to delve in some detail into a particular problem of
interest to them, and to provide an opportunity to practice technical
report writing. It is possible that this project could be used, with
further effort, as a basis for either a Masters thesis or a project for
the non-thesis Masters option in the Math Department.
Participants will be expected to choose a project by mid-semester, and
hand in to the instructor a one page description of what they intend to
pursue. The instructor will provide suggested project topics if a
participant so desires. The final project report should be produced as
a technical report, in standard scientific format, and should be in the
range of 10-20 typed pages. The report should include an abstract, an
introduction describing background material, a methods section
describing the tools applied, a results section, a conclusions section
that particularly includes future enhancements that are possible, and a
bibliography. Participants may make use of any of a number of tools
available on campus in carrying out this project, notably software
tools such as Maple, Mathematica, and Matlab, as well as specialized
ecological modeling tools such as RAMAS and Ecobeaker. Alternatively,
participants may write their own codes in any computer language of
their choosing. The report is expected to include include a disk
containing whatever codes or notebooks were used in producing the
results in the project.
Tentative Topic Coverage for the two semesters:
Single-species population models
Continuous-time ODE models
Continuous-time stochastic models - birth and death processes
Discrete-time deterministic models - difference equations
Discrete-time stochastic models - branching processes
Interacting population models
Predator-prey
Chemostat models
Competition models
Mutualism
Population harvesting and optimal control - introduction to bioeconomics
Spatially-structured population models
Patch and metapopulation models
Reaction-diffusion models
Linear models and spatial steady-states
Nonlinear models and spatial steady-states
Models of spread
Age-structured population models
Lotka integral equation and renewal equation
Leslie matrices and extensions
McKendrick- von Foerster equation
Simple non-linear models
Two sex models
What we will likely not cover: There are many topics within
mathematical ecology that are not included in this course sequence,
some of which are listed below. Any of these could serve as a basis for
the course projects. If there is particular interest on the part of
course participants in some of these, I can possibly rearrange the
schedule to briefly include them. Please inform me soon if you have a
particular interest in one of the below.
Biophysical ecology and physiological ecology models
Stochastic community models
Food web models
Spatial community models
Cellular automata approaches
Individual-based models (these are in 681-2 typically)
Integro-differential equation models (general delay models)
Integro-difference equation models
Fluctuating environment models
Spatial branching and L-systems
Epidemic models
Neural nets, genetic algorithms, A-life models
Basic Reference List: The below texts are general ones that you may
find of most interest relative to the content of this course.
Additional references will be given in each section of the course.
Allen, L. J. S. 2003. An Introduction to Stochastic Processes with
Applications to Biology. Pearson. Upper Saddle River, NJ.
Allen, L. J. S. 2007. An Introduction to Mathematical Biology.
Pearson. Upper Saddle River, NJ.
Allman, E. S. and J. Rhodes. 2004. Mathematical Models in Biology:
An Introduction. Cambridge Univ. Press. Cambridge.
Brauer, F. and C. Castillo-Chavez. 2001. Mathematical Models in
Population Biology and Epidemiology. Springer. New York.
Caswell, H. 2001. Matrix Population Models. 2nd Edition. Sinauer.
Sunderland, MA.
Clark, Colin W. 1976. Mathematical Bioeconomics: The Optimal Management
of Renewable Resources. Wiley. New York.
Cushing, J. M. 1998. An Introduction to Structured Population Dynamics.
SIAM, Philadelphia, PA.
Denny, M and S. Gaines. 2000. Chance in Biology: Using Probability to
Explore Nature. Princeton Univ. Press. Princeton, NJ.
Edelstein-Keshet, L. 1988. Mathematical Models in Biology. Random
House, New York. (Reissued by SIAM 2005)
Ellner, S. P. and J. Guckenheimer. 2006. Dynamic Models in Biology.
Princeton Univ. Press. Princeton.
de Vries, G., T. Hillen, M. Lewis, J. Muller, and B. Schonfisch. 2006.
A Course in Mathematical Biology: Quantitative Modeling with Mathematical
and Computational Methods. SIAM. Philadelphia, PA
Gotelli, Nicholas J. 1995. A primer of ecology. Sinauer Associates,
Sunderland, MA. Second Edition 1998.
Haefner, J. W. 1996. Modeling Biological Systems: Principles and
Applications. Chapman and Hall, NY. (Reissued by Springer 2005)
Hallam, T. G. and S. A. Levin (eds.). 1986. Mathematical Ecology: an
Introduction. Springer-Verlag. Berlin.
Hastings, A. 1997. Population Biology: Concepts and Models.
Springer-Verlag, NY.
Hofbauer, J. and K. Sigmund. 1988. The Theory of Evolution and
Dynamical Systems. Cambridge University Press, Cambridge.
Jones, D. S. and B. D. Sleeman. 2003. Differential Equations and
Mathematical Biology. Chapman and Hall. Boca Raton, FL.
Levin, S. A., Hallam, T. G. and L. J. Gross (eds.). 1989. Applied
Mathematical Ecology. Springer-Verlag. Berlin.
Mangel, M. 2006. The Theoretical Biologist's Toolbox: Quantitative
Methods for Ecology and Evolutionary Biology. Cambridge Univ. Press.
Cambridge.
Maynard Smith, J. 1968. Mathematical Ideas in Biology. Cambridge Univ.
Press, Cambridge.
Maynard Smith, J. 1974. Models in Ecology. Cambridge University Press,
Cambridge.
Murray, J. D. 1989. Mathematical Biology. Springer-Verlag. New York.
Okubo, Akira (1980) Diffusion and ecological problems: mathematical
models. Springer-Verlag. Berlin. (Reissued with additions 2004)
Pielou, E. C. 1977. Mathematical Ecology. Wiley. New York.
Renshaw, E. 1991. Modelling Biological Populations in Space and Time.
Cambridge University Press.
Taubes, C. H. 2001. Modeling Differential Equations in Biology.
Prentice Hall. Upper Saddle River, NJ.
Key Journals in the Field:
American Naturalist
Bulletin of Mathematical Biology
Journal of Mathematical Biology
Journal of Theoretical Biology
Mathematical Biosciences
Mathematical Biosciences and Engineering
Theoretical Population Biology