# MATH 564: Evolutionary Dynamics

**Meeting Times:**Monday, 10am-12pm & Wednesday, 10-11am

**First Class:**Wednesday, September 7th

**Location:**Buchannon, Room B316

## Instructor

Christoph Hauert

**Office:** Mathematics, Room 234

**Hours:** by appointment

**Email:** hauert@math.ubc.ca (please indicate 'math 564' in subject line)

## Announcements

- Homework 1 is available. Due Wednesday, September 28, 2016.
- Homework 2 is available. Due Wednesday, October 5, 2016.
- Wednesday October 12, 2016: Short presentations of project ideas (4-5mins plus 2-3mins discussion).
- Homework 3 is available. Due Wednesday, October 26, 2016.

## 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.

### Goals

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 small 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 |
Introduction | |

Week 2, Sept. 12 |
Constant selection (deterministic dynamics in infinite populations; stochastic dynamics in finite populations) | |

Week 3, Sept. 19 |
Frequency dependent selection (introduction to evolutionary game theory) | |

Week 4, Sept. 26 |
Evolutionary graph theory | Discussions of presentations |

Week 5, Oct. 3 |
Games on graphs | |

Week 6, Oct. 10 |
Classical versus evolutionary game theory | Presentations |

Week 7, Oct. 17 |
Repeated interactions | Discussions of projects |

Week 8, Oct. 24 |
From finite to infinite populations, replicator dynamics | |

Week 9, Oct. 31 |
Structured populations, pair approximation | |

Week 10, Nov. 7 |
Ecological dynamics & evolutionary games | |

Week 11, Nov. 14 |
Public goods games and applications | |

Week 12, Nov. 21 |
Continuous games (origin of cooperation; adaptive dynamics) | |

Week 13, Nov. 28 |
Project presentations |

## Prerequisites

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.

## Assignments

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

- Homework 1: Moran & replicator dynamics (due September 28
^{th}) - Homework 2: Replicator equation & fixation (due October 5
^{th}) - Homework 3: Pair approximation (due October 26
^{th})

## 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 just a few possible directions:

*Asymmetric cooperation:*- In nature interactions are rarely symmetric because individuals differ e.g. in strength or resources. Explore the dynamics of the prisoner's dilemma, snowdrift game and public goods game in well-mixed populations (analytical and/or simulation based), or on graphs for strong selection (simulation based).
*Fixation probabilities versus fixation times:*- Graph structures can have significant effects on the evolutionary dynamics. While fixation probabilities are well studied determinants of evolutionary success, fixation times are far less understood even though they are equally crucial for the understanding and for characterizing evolutionary changes. Explorations of the relation between graph structures and fixation times; the possible tradeoff between higher fixation probabilities (for nodes or graphs) and longer fixation times; or the effects of different types of updates on the evolutionary dynamics are just a few examples of possible directions.
*Eco-evolutionary dynamics:*- Traditionally, evolutionary models assume infinite populations or finite populations of constant size. Little is known about the stochastic dynamics in dynamically variable populations. Simulation based explorations of fixation probabilities and times in populations of changing size would make another fine project.
*Mutualisms:*- Mutualistic interactions abound in nature and are characterized by exchanges of benefits between two (or more) species/populations. Often the two species differ in size and evolve on different timescales, e.g. plants and microbes. Game theoretical descriptions of the asymmetric interactions and the resulting dynamics represent a largely open field of research.
*Stochastic co-existence:*- In nature many species co-exist and form eco-system communities. In contrast, the stochastic dynamics in finite populations necessitates that eventually every species must go extinct. Hence co-existence can only be a transient phenomena. While much effort has been expended on understanding the probabilities and times it takes for a mutant to take over a resident population, far less is know about the stochastic co-existence of residents and potentially several mutant strains. This is particularly interesting and relevant for interacting species and frequency dependent selection.

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

**Evolutionary Dynamics - general**

*Literature:*

- 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). - Nowak MA, A Sasaki, C Taylor, D Fudenberg (2004).
*Emergence of cooperation and evolutionary stability in finite populations*Nature**428**: 646-650 (PDF). - 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**

*Literature:*

- McAvoy, A. & Hauert, C. (2015) Asymmetric Evolutionary Games PLoS Comp. Biol. 11 (8) e1004349 (PDF).
- 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). - Sigmund, K., Brandt, H., Traulsen, A., & Hauert, C. (2010)
*Social learning promotes institutions for governing the commons*, Nature**466**, 861-863 (PDF). - 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). - Sigmund, K., Hauert, C. & Nowak, M. (2001)
*Reward & Punishment*, Proc. Natl. Acad. Sci. USA**98**, 10757-10762 (PDF).

**Games on graphs**

*Literature:*

- Debarre, F., Hauert, C. & Doebeli, M. (2014)
*Social evolution in structured populations*Nature Communications**5**3409 (PDF). - Ohtsuki, H., Hauert, C., Lieberman, E. & Nowak, M. (2006)
*A simple rule for the evolution of cooperation*, Nature**441**, 502-505 (PDF). - Hauert, C. & Doebeli, M. (2004)
*Spatial structure often inhibits the evolution of cooperation in the Snowdrift game*, Nature**428**, 643-646 (PDF).

**Ecology & Evolution**

*Literature:*

- Huang, W., Hauert, C. & Traulsen, A. (2015)
*Stochastic evolutionary games in dynamic populations*, Proc. Natl. Acad. Sci. USA 112 (29) 9064-9069. (PDF). - Wakano, J., Nowak, M. & Hauert, C. (2009)
*Spatial Dynamics of Ecological Public Goods*, Proc. Natl. Acad. Sci. USA**106**, 7910-7914 (PDF). - Hauert, C., Wakano, J. & Doebeli, M. (2008)
*Ecological Public Goods Games: cooperation and bifurcation*, Theor. Pop. Biol.**73**, 257-263 (PDF).

**Diversification, adaptive dynamics**

*Literature:*

- Doebeli, M.,
*Adaptive Diversification*, Princeton University Press, 2011. - Doebeli, M., Hauert, C. & Killingback, T. (2004)
*The evolutionary origin of cooperators and defectors*, Science**306**, 859-862. (PDF)

### Project timeline

- October 12
^{th}: Presentation of ideas for project (4-5min presentation, 2-3min discussion).

(*Note:*you may give a 'chalk talk' or use slides; slides need to be emailed to me no later than the evening of October 11^{th}.)

### 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.

## 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.