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