# MATH 564: Evolutionary Dynamics

**Meeting Times:**Tuesday, Thursday, 11:00 - 12:30

**First Class:**Thursday, September 5th

**Location:**Mathematics Annex, Room 1118 -->

## Instructor

Christoph Hauert

**Office:** Mathematics, Room 234

**Hours:** by appointment

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

## Announcements

## 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. 10 |
Finite populations, constant selection | |

Week 3, Sept. 17 |
Structured populations | |

Week 4, Sept. 24 |
Game theory, cooperation, classical analysis | Discussions of presentations |

Week 5, Oct. 1 |
Finite populations, frequency dependent selection | |

Week 6, Oct. 8 |
From finite to infinite populations, replicator dynamics | Presentations |

Week 7, Oct. 15 |
Structured populations, pair approximation | Discussions of projects |

Week 8, Oct. 22 |
Cooperation, Reward, Punishment & Reputation | |

Week 9, Oct. 29 |
Ecological dynamics & evolutionary games | |

Week 10, Nov. 5 |
Mutant Games | |

Week 11, Nov. 12 |
Continuous games, adaptive dynamics | |

Week 12, Nov. 19 |
Origin of Cooperation | |

Week 13, Nov. 26 |
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.

## Tentative Schedule for Guest Lectures

- Wes Maciejewski
- Evolutionary dynamics in heterogenous populations, November 5
^{th}& 7^{th}. - Alastair Jamieson-Lane
- Evolutionary amplifiers: fixation probabilities on superstars revisited, November 19
^{th}. - Lucas Wardil
- Origin and structure of dynamic cooperative networks, November 21
^{st}.

## Assignments

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

- Homework 1: Replicator dynamics (due September 26
^{th}) - Homework 2: Moran process (due October 8
^{th}) - Homework 3: Pair approximation (due October 29
^{th}) - Homework 4: Classical game theory (due November 5
^{th}, collected only on November 12^{th}) - Homework 5: Evolutionary game theory (due November 12
^{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.

**Evolutionary Dynamics - general**

*Literature:*- 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., Claussen J. C. & Hauert, Ch. (2005)
*Evolutionary dynamics: from finite to infinite populations*Phys. Rev. Lett.**95**, 238701 (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). - Traulsen, A., Claussen J. C. & Hauert, Ch. (2012) Stochastic differential equations for evolutionary dynamics with demographic noise and mutations Phys. Rev. E 85, 041901 (PDF).

- Nowak MA, A Sasaki, C Taylor, D Fudenberg (2004).
**Cooperation**

*Literature:*- Sigmund, K., Hauert, C. & Nowak, M. (2001)
*Reward & Punishment*, Proc. Natl. Acad. Sci. USA**98**, 10757-10762 (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., Brandt, H., Traulsen, A., & Hauert, C. (2010)
*Social learning promotes institutions for governing the commons*, Nature**466**, 861-863 (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., Hauert, C. & Nowak, M. (2001)
**Dynamics of spatial games**

*Literature:***Ecological Public Goods Games**

*Literature:***Diversification**

*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)

- Doebeli, M.,

### Project timeline

- October 10
^{th}: Presentation of ideas for project (~5min presentation, ~5min discussion). - November 26
^{th}& 28^{th}: Presentation of projects (~15min presentation, ~5min discussion). - December 1
^{st}: Project reports due at 11:59pm as pdf-file by email (this is a strict deadline). For guidelines see below. - December 2
^{nd}: Assignment of three project reports to each student for peer review. - December 6
^{th}: Peer reviews due at 11:59pm by email. Reviews are anonymous, see below for guidelines. - December 7
^{th}: Anonimized peer reviews distributed to project authors. - December 11
^{th}: Final project reports due at 11:59pm as pdf-file by email.

### 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:** (20%) 3-4 problem sets on material discussed in class.

**Presentation:** (15%) 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.