Mathematics Department

MATH 564: Evolutionary Dynamics

Meeting Times: Monday, 10am-12pm & Wednesday, 10-11am
First Class: Wednesday, September 5th
Location: Ponderosa Commons: Oak House (PCOH), room 1302


Christoph Hauert
Office: Mathematics, Room 234
Hours: by appointment
Email: (please indicate 'math 564' in subject line)


Course Outline

Evolution is the unifying theme in biology. Evolutionary processes are responsible for the emergence of the rich variety of species across the planet. Cooperation represents one of the key organizing principles in evolution, and the history of life and of societies could not have unfolded without the repeated cooperative integration of lower level units into higher level entities. Evolutionary theories have attracted increasing attention from other behavioral disciplines including sociology and economics. This has led to the notion of cultural evolution aiming at a better understanding of human cooperation including the emergence of social norms. Cultural evolution follows the same basic selection principle as biological evolution but the lack of the genetic constraints of mutation, recombination and inheritance results in a largely unexplored dynamics governed by the more flexible mechanisms of innovation, learning and imitation.


This course provides a sound introduction into mathematical models of evolution and the theory of games. Modeling techniques that are covered include: stochastic dynamics of invasion and fixation of mutants in a finite population; evolutionary game theory and frequency dependent selection -- each agents' performance is affected by everyone else; adaptive dynamics and the process of diversification and speciation through evolutionary branching; modeling spatially structured populations. In all cases the link to current challenges in research is emphasized by student presentations and discussions of the literature as well as by identifying potential research questions. Each student develops his/her own research project in consultation with the instructor. At the end of the term, all students hand in a written report, present their project to the class and participate in a peer review process assessing the projects of their fellow students.

Tentative Timeline

Week Lecture Topic Notes
Week 1,
Sept. 5
Week 2,
Sept. 10
Constant selection (deterministic dynamics in infinite populations; stochastic dynamics in finite populations)
Week 3,
Sept. 17
Frequency dependent selection (introduction to evolutionary game theory)
Week 4,
Sept. 24
Evolutionary graph theory Discussions of presentations
Week 5,
Oct. 1
Games on graphs
Week 6,
Oct. 8
Classical versus evolutionary game theory Presentations
Week 7,
Oct. 15
Repeated interactions Discussions of projects
Week 8,
Oct. 22
From finite to infinite populations, replicator dynamics
Week 9,
Oct. 29
Structured populations, pair approximation
Week 10,
Nov. 5
Ecological dynamics & evolutionary games
Week 11,
Nov. 12
Public goods games and applications
Week 12,
Nov. 19
Continuous games (origin of cooperation; adaptive dynamics)
Week 13,
Nov. 26
Project presentations


This course combines various topics covered in undergraduate mathematics courses - including differential equations, dynamical systems, stochastic processes, probability, Markov chains, etc. However, committed graduate students from other disciplines that are willing to catch up on mathematical theories they might not be familiar with are encouraged to attend and stimulate discussions with problems from their fields. Knowledge of computer programming and mathematics software such as Maple, Mathematica or MATLAB might be helpful for the project work but are not required.


The homework assignments will be posted below. Late homework is not accepted.

Presentations & Project

Some suggestions for presentations and projects based on recent research results. For some of these topics ideas for manageable projects exist. If you are interested please contact me for more specific information. However, you are free (and encouraged) to choose any other research paper that catches your interest. Ideally, pick a topic that fits with your graduate studies and adds an evolutionary perspective to your research interests. Here are a few possible directions for inspiration:

Rock-Paper-Scissors game:
The cyclic dynamics often serves as a model system to explore the maintenance of biodiversity. Possible extensions include:
  1. Effects of spatial dimensions can be captured by an extension of pair approximation (discussed in class) to three types. Explore the effects of space on frequencies and persistence of diversity.
  2. Dynamics of two or more subpopulations coupled through limited migration. Explore whether this promotes co-existence or results in synchronization. Deterministic and stochastic modelling approaches are both interesting.

Literature suggestions:
- Reichenbach, T., Mobilia, M. and Frey, E. (2007) Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games Nature, 448, 1046-1049.

Asymmetric cooperation:
In nature interactions are rarely symmetric because individuals differ e.g. in strength or resources. Moreover, different strategies may impact their environment and hence change the availability of resources.
  1. Explore the dynamics for constant fitness on graphs (with and/or without feedback).
  2. Explore the stochastic dynamics of social dilemmas in well-mixed finite populations (with or without feedback).
  3. Explore the dynamics of social dilemmas on graphs for strong selection (with or without feedback).

Literature suggestions:
- Maciejewski, W. and Puleo, G. J. (2014) Environmental evolutionary graph theory J. theor. Biol. 360, 117-128.
- Hauser, O., Traulsen, A. and Nowak, M. A. (2014) Heterogeneity in background fitness acts as a suppressor of selection J. theor. Biol. 343, 178-185.
- Hauert, C., Saade, C. and McAvoy, A. (2018) Asymmetric Evolutionary Games with Environmental Feedback arXiv:1807.01735.

Specialization & division of labour
In nature, individuals rarely only interact in a single game at a time and rather engage in multiple interactions simultaneously with one or several partners. For example, microbial populations produce and secrete various substances, which may be costly to produce and benefit (or harm) others. Consider two (or more) games where the fitness of individuals depends on their performance in each game. For example, fitness could be the sum of payoffs in each game or their product to further emphasize that all interactions are important. Escalating production costs (reduced efficiency) or saturating costs (exploting synergies) could be particulalry interesting both in terms of investing in a single or multiple resources. Explore the dynamics effort allocation into the different games.
  1. Consider the dynamics of discrete strategies that invest in one and/or another resource (or none). Determine conditions under which specialization can occur.
  2. Track evolutionary changes in continuous strategies, which represent the effort allocation in two (or more) games. In particular consider two games and a single trait that determines whether to invest in the first or second game.

Literature suggestions (adaptive dynamics):
- Doebeli, M., Hauert, C. & Killingback, T. (2004) The evolutionary origin of cooperators and defectors Science 306, 859-862.

Eco-evolutionary dynamics:
Formalizing and analyzing the interplay between ecological and evolutionary processes is a challenging and active area of research.
  1. Effects of variable population sizes in spatially structured populations can be captured by extending pair approximation for two types to include empty sites as a third type. Note: in terms of methods this is related to pair approximation in the rock-paper-scissors game but the dynamics and results are distinct.
  2. Explore whether ecological processes promote or suppress polymorphisms.

Literature suggestions:
- Huang, W., Hauert, C. & Traulsen, A. (2015) Stochastic evolutionary games in dynamic populations Proc. Natl. Acad. Sci. USA 112 (29) 9064-9069.
- Huang, W., Haubold, B., Hauert, C. & Traulsen, A. (2012) Emergence of stable polymorphism driven by evolutionary games between mutants Nature Communications 3 919.

And some research articles that might be interesting to explore further:

  • Evolutionary Dynamics - general
    1. Hauert, Ch. (2008) Evolutionary Dynamics in Proceedings of the NATO Advanced Study Institute on Evolution from Cellular to Social Scales, eds. Skjeltorp, A. T. & Belushkin, A. V., Springer, Dordrecht, The Netherlands, pp. 11-44 (PDF).
    2. Nowak MA, A Sasaki, C Taylor, D Fudenberg (2004). Emergence of cooperation and evolutionary stability in finite populations Nature 428: 646-650 (PDF).
    3. Traulsen, A., Hauert, C., De Silva, H., Nowak, MA & Sigmund, K. (2009) Exploration dynamics in evolutionary games Proc. Natl. Acad. Sci. USA 106 709-712 (PDF).
  • Cooperation
    1. Hilbe, C., Šimsa, Š., Chatterjee, K. & Nowak, M. A. (2018) Evolution of cooperation in stochastic games. Nature 559, 246-249 (DOI)
    2. McAvoy, A. & Hauert, C. (2015) Asymmetric Evolutionary Games. PLoS Comp. Biol. 11 (8) e1004349 (PDF).
    3. Hilbe, C., Nowak, MA. & Sigmund, K. (2013) Evolution of extortion in Iterated Prisoner's Dilemma games, Proc Natl. Acad. Sci. USA 110 6913-6918 (PDF).
    4. Sigmund, K., Brandt, H., Traulsen, A., & Hauert, C. (2010) Social learning promotes institutions for governing the commons, Nature 466, 861-863 (PDF).
    5. Hauert, C, Traulsen, A., Brandt, H., Nowak, M. A. & Sigmund, K. (2007) Via freedom to coercion: the emergence of costly punishment, Science 316, 1905-1907 (PDF).
    6. Sigmund, K., Hauert, C. & Nowak, M. (2001) Reward & Punishment, Proc. Natl. Acad. Sci. USA 98, 10757-10762 (PDF).
  • Games on graphs
    1. Allen B, Lippner, G, Chen, Y-T., Fotouhi, B., Momeni, N., Yau, S-T., & Nowak, M. A. (2017). Evolutionary dynamics on any population structure. Nature 544: 227-230. (DOI)
    2. Debarre, F., Hauert, C. & Doebeli, M. (2014) Social evolution in structured populations Nature Communications 5 3409 (PDF).
    3. Ohtsuki, H., Hauert, C., Lieberman, E. & Nowak, M. (2006) A simple rule for the evolution of cooperation, Nature 441, 502-505 (PDF).
    4. Hauert, C. & Doebeli, M. (2004) Spatial structure often inhibits the evolution of cooperation in the Snowdrift game, Nature 428, 643-646 (PDF).
  • Ecology & Evolution
    1. Constable, G. W. A, Rogers, T., McKane, A. J. & Tarnita C. E. (2016) Demographic noise can reverse the direction of deterministic selection, Proc. Natl. Acad. Sci. USA 113 (32): E4745–E4754 (PDF).
    2. Huang, W., Hauert, C. & Traulsen, A. (2015) Stochastic evolutionary games in dynamic populations, Proc. Natl. Acad. Sci. USA 112 (29) 9064-9069 (PDF).
    3. Wakano, J., Nowak, M. & Hauert, C. (2009) Spatial Dynamics of Ecological Public Goods, Proc. Natl. Acad. Sci. USA 106, 7910-7914 (PDF).
    4. Hauert, C., Wakano, J. & Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, Theor. Pop. Biol. 73, 257-263 (PDF).
  • Diversification, adaptive dynamics
    1. Doebeli, M., Adaptive Diversification, Princeton University Press, 2011.
    2. Doebeli, M., Hauert, C. & Killingback, T. (2004) The evolutionary origin of cooperators and defectors, Science 306, 859-862. (PDF)

    Project timeline

    • October 10th: Presentation of ideas for project (4-5min presentation, 2-3min discussion).
      Provide brief biological/social motivation of your project together with some ideas of what you are planning to investigate in greater detail and possibly how (analytics, numerical, simulation based).
      (Note: you may give a 'chalk talk' or use slides; slides need to be emailed to me no later than the evening of October 9th. Microsoft PowerPoint, Apple Keynote and pdf are all acceptable. However, PowerPoint can have issues with fonts on Macs...)
    • November 23rd 30th December 2nd: Project reports due (electronically by midnight), see below for guidelines on structure and length.
    • November 24th December 1st3rd: Assignement of three projects for peer review.
    • November 26th & 28th November 30: Presentation of projects in ANGU 437 at 11:15am (~15min presentation, ~5min discussion).
    • November 30th December 7th: Peer reviews due (electronically by midnight). Reviews are anonymous, see below for guidelines.
    • December 1st 8th: Anonimized peer reviews distributed to project authors.
    • December 7th 12th: Final project report due (electronically by midnight).

    Project guidelines

    The project paper should be a short, concise summary of your mini-research project with at most 12 pages (12pt font size, double spaced, excluding title page and references). As a target audience it might be best to think of your peers in class - do not expect familiarity with the specifics of your project but you can rely on mathematical knowledge and some exposure to dynamical systems and evolutionary game theory. The paper should start with a brief introduction (~2 pages) that sketches the problem and puts it in a wider context and concludes with a discussion (~2 pages) that highlights the results and relates them to the broader context stated in the beginning. The model and results must be the central piece and should be described with sufficient detail that the reader can easily follow your line of argument but there is no need to show every step e.g. in your mathematical derivations. The emphasis must lie on an clear, intuitive and consistent presentation.

    If necessary, an appendix can be added that does not count towards your page limit. The appendix could, for example, contain detailed calculations, proofs and/or simulation details. However, the main text must remain self-contained and clear without consulting the supplement.

    The final report must include a brief (point-to-point) response to the reviewers comments and how they were addressed in the final report.

    Peer review guidelines

    For the peer review you have to write a brief report on the content, presentation and originality of the paper (max. 1 page). The reviewers remain anonymous but the content will be returned to the author - the same as in real life (for some journals the reviews are double-blind but that is unlikely to work in such a small group). Just as some guidance, after or while reading the paper ask yourself questions like: Is the problem well motivated? Is it an interesting and relevant problem? Is the model suitable to address the problem? Is the model well and convincingly presented? Are the derivations of the results clear? Do the conclusions follow from the model? Was it an interesting read? etc.

    Clearly indicate author and title of the project you are reviewing.

    Project reports for peer review (password required to access files).

    Please return the peer reviews for the three projects assigned to you electronically by the end of December 7th. Feel free to check out and comment on any other project as well. All comments will be anonymously forwarded to the author unless you explicitly request to waive your anonimity.

    Grading guidelines

    Your grade for the course will be computed roughly as follows:
    Assignments: (25%) 4-5 problem sets on material discussed in class.
    Presentation: (10%) presentation of research article to the class.
    Term Project, Paper: (30%) small research project.
    Term Project, Presentation: (20%) presentation of research project to the class.
    Term Project, Peer Review: (10%) review the term project papers of your peers.
    Participation: (5%) contributions to discussions in class.

    Useful Resources

    • Martin Nowak, Evolutionary Dynamics, Belknap Press, 2006.
    • Karl Sigmund, The Calculus of Selfishness, Princeton University Press, 2010.
    • Josef Hofbauer & Karl Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
    • Nicholas Christakis & James Fowler, Connected, Little, Brown & Co., 2009.
    • John Maynard Smith & Eörs Szathmary, The Major Transitions in Evolution, W. H. Freeman & Co., 1995.
    • EvoLudo and VirtualLabs: Collections of interactive tutorials on the fascinating dynamical world of evolutionary processes.

    Course webpage: Course schedule.

  • design by Clive Goodinson and adapted by Christoph Hauert.