Introduction to Adaptive Systems
CMPLXYS 510/MATH 550, Fall 2025
Syllabus
This course is an introduction to evolutionary game theory and population dynamics, biologically motivated by questions in evolutionary biology and behavioral ecology and mathematically based in the theories of nonlinear dynamical systems and differential geometry. Topics include general Lotka-Volterra systems, normal-form games, replicator dynamics, evolutionarily stable strategies, adaptive dynamics, and the evolution of cooperation.
The course is suitable for theoretically and mathematically-minded science and engineering students. Applied mathematicians, statisticians, computer scientists, and engineers are especially welcome, as are students of biology, ecology, evolution, epidemiology, environmental sustainability, economics, social sciences, and bioinformatics with suitable mathematical backgrounds.
Students should have the equivalent of four semesters of calculus (including ordinary differential equations and multivariable calculus), a working knowledge of elementary linear algebra and probability, and computer literacy appropriate for mathematically-minded science or engineering students. Some homework problems will involve numerical computations or simulations. If you are not certain as to whether you meet the course prerequisites, contact Prof. King to inquire.
You must attend lectures and complete the assigned homework exercises, which will be collected & graded. Your final course grade will be determined from the sum of points earned on the homework (40%), on the midterm examination (20%), and on the final examination (40%).
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, 16:00–17:20.
The nature of the course is such that attendance at lectures is required. Lectures will be recorded so that those forced to miss the occasional lecture will be able to catch up. Viewing lecture recordings will not prove to be an adequate substitute for attendance.
To be announced. Office hours will be conducted via Zoom: the Zoom links are available via the course Canvas.
We will have a dedicated Slack channel, which will use this for all online communication about the course, including:
To contact the instructor, use DM over this Slack channel in preference to email.
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.
Homework Exercises will be assigned from time to time during the lectures. Each Exercise will be posted to the course Canvas immediately after the lecture in which it was assigned. It will be due approximately one week after it is assigned. These should be uploaded to Gradescope on time: The late penalty is 25% per day or portion thereof.
There will be two timed take-home examinations. Each will be made available at an announced time and will have a fixed deadline for submission. The exams will be downloaded from Gradescope, completed within a specified amount of time, and uploaded again to Gradescope. The goal of the exams is to probe depth of understanding, thoroughness of reasoning, and clarity of expression rather than speed. Accordingly, a generous amount of time will be allotted for completion of the exam.
In the past, we have used the text, Evolutionary Games and Population Dynamics by Josef Hofbauer and Karl Sigmund (Cambridge University Press, 1998, DOI: 10.1017/CBO9781139173179), and the exposition follows that of those authors to some extent, but this text will not be required. Occasionally, select readings will be provided on the course Canvas, but the course lecture notes are meant to be self-contained.
The following are some reference materials that you may find useful.
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