Mathematical Ecology
EEB/MATH/CMPLXSYS 466
Winter 2025
Syllabus
Mathematical models are the backbone of ecological theory; they form the basis for modern approaches to understanding, predicting, and managing the dynamics of ecological systems. This course provides an overview of the major classes of ecological models, with an emphasis on ecological dynamics. We will focus on principles guiding the formulation of models and on the mathematical techniques for their analysis. We will examine deterministic and stochastic models, structured and unstructured models, single- and multiple-species models. Because ecological systems are typically nonlinear, we cannot often “solve” model equations. Instead, we employ techniques of nonlinear, stochastic, and numerical analysis to obtain results. This course will introduce many of these techniques in the context of ecological theory.
An additional goal of the course is to develop skills in the use of mathematical software. We will make extensive use of R for numerical computations and Mathematica for symbolic computation.
Graduate and undergraduate students interested in the use of mathematical models and theoretical, statistical, and computational ecology are encouraged to enroll. The techniques introduced in the course will be useful to students from many disciplines, including evolutionary biology, applied mathematics, physics, complex systems, epidemiology, chemical engineering, economics, natural resources, and others.
Students should have a firm grasp of elementary linear algebra (e.g., Math 214, 217, 417, or 419) and ordinary differential equations (e.g., Math 214, 216, 256, 286, or 316), and have had some exposure to probability. Students without these prerequisites should consult the instructor before registering.
The lectures will cover material from the following set of topics. It is not anticipated that there will be sufficient time to cover all of these topics. The selection of topics covered will depend upon interest and time available.
The course will be taught using a mixture of lectures, readings, homework exercises, in-class computational demonstrations, and computer laboratory exercises. Attendance at all class meetings, participation in discussions and exercises, and completion of readings is expected.
Homework exercises will be assigned during the lectures. The course grade will be based upon these problem sets. Solutions should be self-contained, self-explanatory, and handed in via the Gradescope tool. When the solutions contain a computational element, carefully annotated code listings can be included with the write-up. When computer codes are provided, it must be possible for the instructor to run them on his machine without adjustment.
Students are encouraged to collaborate on the exercises. However, each student must hand in his or her own solutions, written in his or her own words, and must indicate the names of all collaborators on the finished document. Plagiarism and other lapses of academic integrity will not be tolerated (see below).
This course adopts a somewhat unusual approach to grading. The ability to reason correctly and insightfully is usually more important than the ability to reason quickly, although the latter has value as well. Therefore, on each homework assignment, students have two chances to complete the solution to the best of their ability. On the first draft, students will receive comments from the instructor, which will guide improvement of the final draft, due one week after it is returned. A penalty of 10% per day will be assessed for late homework.
Students who audit must attend lectures, do the assigned readings, and participate in discussions.
Because mathematical reasoning contains an essentially linguistic element, a subsidiary but very important aim of the course is to improve your mathematical writing ability. Feedback on your written work will be aimed at bringing your reasoning and writing to a high standard of excellence. The following guidelines are provided to assist you in achieving this goal:
You will not be expected to reach this standard immediately. Rather, expect constructive criticism about not only the content, but also the style, of your written work. Eventually, however, your grade will reflect both the substance and the style of your submissions.
Aaron A. King, Ph.D.
Nelson G. Hairston Collegiate Professor
Department of Ecology & Evolutionary Biology
Center for the Study of Complex Systems
Department of Mathematics
Office: 3038 Biological Sciences Building
WWW: https://kinglab.eeb.lsa.umich.edu/
Email: kingaa@umich.edu
Telephone: +1.734.936.7861
Tuesdays and Thursdays, 16:00–17:30 in 1508 NUB
The nature of the course is such that attendance at lectures is required. Lectures will be recorded to help those who are forced to miss the occasional lecture catch up. Viewing lecture recordings will not prove to be an adequate substitute for attendance.
Please direct-message (DM) the instructor on Slack if you are forced to miss a class meeting.
Mondays 16:00–17:00, Fridays 10:00–11:00, and by appointment. Office hours will be conducted via Zoom. The link will be posted in the course Slack channel. Contact Prof. King via direct message to set up an appointment.
The course Slack workspace is um-wn25-466-w2025.slack.com. You can access Slack via dedicated apps or via a web browser. To install Slack on your local device, follow the instructions given by the University. If you are registered for the course, you have been automatically included in this workspace. We will use this for all online communication about the course, including:
To contact the instructor privately, use DM over this Slack channel in preference to email.
Recommended text: Mark Kot, Elements of Mathematical Ecology, Cambridge University Press (2001). ISBN: 978-0-521-00150-2. This text should be available for purchase or rental via the Course Guide.
Additional readings will be made available via the course Canvas from time to time.
Much of the course work will be done using the software packages Mathematica and R. A site license for Mathematica is available on campus computers; because the reduced student rate is attractive, students are encouraged to consider purchasing a personal copy of Mathematica. Instructions for installing Mathematica using the University’s site license, are available online. R is free and open-source software: instructions for installing R can be found at R-Project.org.
The following are some reference materials that you may find useful.
Linear algebra: Strang, G. Linear algebra and its applications. Thomson, Brooks/Cole, 2006. An excellent introduction to applications.
Applied mathematics: Strang, G. Introduction to Applied Mathematics. Wellesley-Cambridge Press, 1986. A treasury of applied mathematics technique.
Differential equations: Strang, G. Differential equations and linear algebra. Wellesley-Cambridge Press, 2014. Haven’t read this, but if it’s as good as his other books, it will be excellent.
Prof. Strang also has an excellent series of lectures available for streaming through Open Courseware and YouTube.
Differential equations: Hirsch, M. W. & Smale, S. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, 1974. An excellent mathematical introduction to the subject.
Differential equations: Arnol’d, V. I. Ordinary Differential Equations. MIT Press, 1973. A high-level perspective from one of the great dynamical systems theorists.
Probability: Grimmett, G. & Stirzaker, D. Probability and random processes. Oxford University Press, 2001.
Probability: Feller, W. An introduction to probability theory and its applications. Wiley, 1968. The classic two-volume reference.
Stochastic processes: Gardiner, C. Stochastic Methods. Springer, 2009. A handbook, written from a physicist’s point of view.
Mathematical biology: Ellner, S. P. & Guckenheimer, J. Dynamic Models in Biology. Princeton University Press, 2006. An introduction to biological dynamics.
Mathematical ecology Kot, M. Elements of Mathematical Ecology. Cambridge University Press, 2001. A good introduction to models and methods, including demography and some eco-economics.
Mathematical epidemiology Keeling, M. & Rohani, P. Modeling infectious diseases in humans and animals. Princeton University Press, 2007.
Matrix models: Caswell, H. Matrix Population Models. Sinauer, 2001. The definitive text on these versatile models.
Population dynamics: Turchin, P. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Princeton University Press, 2003.
Calculus for biologists: Garfinkel, A., Shevtsov, J. & Guo, Y. Modeling life : the mathematics of biological systems. Springer, 2017. A very good lower-division introduction to calculus in the life sciences.
All submitted course work must be accompanied by the following signed pledge: “Upon my honor, I have neither given nor received any unauthorized aid on this problem set.” Work submitted without a signed pledge will not be graded.
The University of Michigan community functions best when its members treat one another with honesty, fairness, respect, and trust. The college promotes the assumption of personal responsibility and integrity, and prohibits all forms of academic dishonesty and misconduct. All cases of academic misconduct will be referred to the LSA Office of the Assistant Dean for Undergraduate Education. Being found responsible for academic misconduct will usually result in a grade sanction, in addition to any sanction from the college. For more information, including examples of behaviors that are considered academic misconduct and potential sanctions, please see lsa.umich.edu/lsa/academics/academic-integrity.html.
Students are prohibited from recording/distributing any class activity without prior, explicit, written permission from the instructor, as part of approved accommodations for students with disabilities. Approved recordings will be only for the student’s own private use.
Students in good standing (i.e., grade of B or better), forced by circumstances beyond their control to withdraw entirely from school after the withdrawal deadline may request a grade of Incomplete. The instructor may request proof of the necessitating circumstances and will, in consultation with the student, set the terms for the timely completion of course requirements.
The University of Michigan is committed to providing equal opportunity for participation in all classes, programs, services and activities. Requests for accommodations by persons with disabilities may be made by contacting the Services for Students with Disabilities (SSD) Office located at G664 Haven Hall. The SSD phone number is +1.734.763.3000. Once your eligibility for an accommodation has been determined you will be issued a verified individual services accommodation (VISA) form. Please present this form to me at the beginning of the term, or at least two weeks prior to the need for the accommodation (i.e., examination).
University Students may experience stressors that can impact both their academic experience and their personal well-being. These may include academic pressures and challenges associated with relationships, mental health, alcohol or other drugs, identities, finances, etc. If you are experiencing concerns, seeking help is a courageous thing to do for yourself and those who care about you. If the source of your stressors is academic, please contact me so that we can find solutions together. For personal concerns, U-M offers a variety of resources, many which are listed on the Student Well-being webpage.
Title IX prohibits discrimination on the basis of sex, which includes sexual misconduct—including harassment, domestic and dating violence, sexual assault, and stalking. I understand that sexual violence can undermine your academic success, so I encourage anyone dealing with sexual misconduct to talk to someone about their experience, so they can get the support they need. Confidential support and academic advocacy can be found with the Sexual Assault Prevention and Awareness Center (SAPAC) on their 24-hour crisis line, +1.734.936.3333 and at sapac.umich.edu.
Violations can be non-confidentially reported to the Office for Institutional Equity (OIE) at institutional.equity@umich.edu.