Evolutionary dynamics by M.A. Nowak

Martin A. Nowak does some of the best work in evolutionary game theory. I enjoy many of his papers, and his general approach to questions in EGT. Thus, I was very happy when I first heard about his introductory book “Evolutionary Dynamics: Exploring the Equations of Life“. After reading the book, I prepared a reading list for the members of the LNSC. I recomend Nowak’s book as a gentle introduction to the field and include my reading list as a review.

Reading list

Here is what I would recommend to read from the book, along with a brief summary:

Cover of Evolutionary Dynamics by Martin A. Nowak

Chapter 2 (2.0 – 2.3) “What evolution is”

  • Covers the basics of evolution: reproduction (2.1), selection (2.2), and mutation (2.3)
  • Introduces the basic equations and phase space. Explains what the simplex is.
  • Section 2.4 is fun, but does not do justice to sexual evolution (and is never references again), so it can be omitted.

Chapter 4 (4.0 – 4.8) “Evolutionary games”

  • This chapter is essential, it covers the basics of EGT.
  • It is important to stress section 4.5 where the replicator equation is introduced.
  • Section 4.9 was fun for me. When I was first starting EGT, I was unaware of many of the classical results and reproved the equivalence of replicator dynamics and Lotka-Volterra, so seeing it was nostalgic. However, in general this section can be omitted.

Chapter 9 (all) “Spatial games”

  • This is the approach that really founded the idea of doing games in a lattice, and is important for everyone to read

Chapter 10 (Optional) “HIV infection”

  • Although not exactly game theory related. It does show the power of differential equations in biology.
  • Given a bit of thought, it is possible to relate the ideas to game theory. In particular, the asymmetric effect between strains of HIV and their suppressors can be thought of as a game. However, Nowak does not mention that in the book.
  • Mostly, it is a part of Nowak’s research that really hit the spot for theory building in biology.

Chapter 11 (Optional) “Evolution of virulence”

  • Also, much like chapter 10 much of this can be interpreted as games, but Nowak does not make the connection explicit.

Chapter 13 (13.3 – 13.6) “Language evolution”

  • The beginning 13.0 – 13.2 is rather weak. It is better if students have their own background in basic linguistics. I don’t think Nowak does a very careful or rigorous introduction. He drops results on the reader, without really making us appreciate them. For instance, he throws down Godel’s theorem for no real reason and without doing it any justice.
  • The later sections are useful. In particular, they show a really fun application of EGT to language coherence, etc.
  • Some of the conclusions are not particularly breathtaking, but it is interesting to see them reached from evolutionary arguments. It gives you much more insight into both EGT and language.
  • Nowak introduces the replicator-mutator equation here. This is a very general and important framework in EGT.

The ‘further reading’ section is one of the best parts of this book. There are a lot of great sources mentioned there.

Sections to omit

As for the omitted chapters and sections:

Chapter 3 is extremely tangential. The quasi-species equation is briefly mentioned again in chapter 13 (when the replicator-mutator equation is introduced) but chapter 13 can easily be understood and enjoyed without chapter 3. Quasi-species work is extremely interesting and promising, but Nowak’s exposition of it is lacking. Chapter 3 is not really worth reading, and might cause more confusion than good.

I omitted chapter 5, even though it is a whole chapter about prisoner’s dilemma. However, only direct reciprocity (iterated PD) is studied in this chapter. Direct reciprocity is pretty well understood as a mechanism for creating cooperation, hence there is not too much excitement here. It does talk about some of the difficulties of iterated PD, but this is not useful if you are interested in one-shot games.

For people that are more biology inclined, it might be good to read Chapters 6,7, and 8. These 3 chapters deal with finite populations in the way that biologists work with them (i.e. via Moran processes). These are very important for understanding the biological parts of EGT, especially neutral drift and fixation probabilities. In a social context though, there will be no fixation or neutral drift since a non-negligible mutation rate should always be included (and there can be no fixation with a mutation rate).

The reason biologists can ignore mutation rates between strategies and social scientists can’t, I think, is quiet simple. The chance of one strategy mutating into another (via genetic changes) is close to null, and hence can be ignored on any reasonable time scale (most mutations will be neutral or will kill you, a random mutation changing something as substantial as a strategy is nearly impossible). However, in social evolution, a person can always change their strategy, or at least they can do it much easier than in biological evolution. Hence, the Moran processes for social evolution will have no absorbing states and thus those 3 chapters on finite populations (which are only of interest because of absorbing states) become irrelevant. However, for people that read a lot of biology papers it is important to go through those chapters.

I found the chapter on evolutionary dynamics of cancer (chapter 12) lacking. It has no really interesting results and a lot of cancer specific set up. I don’t think Nowak has found a good way to apply evolutionary dynamics to cancer, yet. Personally, I prefer the approach of Axelrod and coauthors. Most of this chapter is a bunch of heavy math (which Nowak usually replaces by heuristics) to conclude pretty obvious results.

Conclusion and open problems

As I stated earlier, I think the book is a great introduction. However, I don’t think it equips you with the skills to start doing research. I am not sure how easy the book is to use as a textbook, but Nowak uses it in both his undergraduate and graduate course on mathematical biology.

Do you think this is a good introduction to the field? Is there another text you prefer? Would you like to take or teach a course from this book? Would you like to see a specific chapter reviewed in more depth?

About Artem Kaznatcheev
From the ivory tower of the School of Computer Science and Department of Psychology at McGill University, I marvel at the world through algorithmic lenses. My specific interests are in quantum computing, evolutionary game theory, modern evolutionary synthesis, and theoretical cognitive science. Previously I was at the Institute for Quantum Computing and Department of Combinatorics & Optimization at the University of Waterloo and a visitor to the Centre for Quantum Technologies at the National University of Singapore.

8 Responses to Evolutionary dynamics by M.A. Nowak

  1. Thomas Shultz says:

    Artem’s review of Nowak’s book makes a great start to this blog-post site. I agree with much of what Artem says about this excellent book. The book is a superb and readable introduction to the field of evolutionary game theory.

    Artem makes a subtle but important point about the differential importance of mutation in biological vs social evolution, although I’m not sure about some of the terminology here. Presumably the distinction is between evolution of physical vs social characteristics, both of which could be subject to biological evolution. A strategy for behavior would presumably be a social characteristic, but possibly influenced by biological evolution. If not, this should be clarified in further discussion. Social (or cultural) evolution can be quite different than biological evolution in both mechanisms (memes vs genes) and rate of change, although admittedly with considerable similarity in the mathematics of algorithms.

    For readers with less available time, I believe it is possible to catch the main drift of Nowak’s presentation merely by processing the excellent figures that grace the chapters. These figures are extremely well designed and thus highly informative.

    • For my point between the difference in mutation between biological and social evolution, I really meant the difference between physical/biological evolution and social evolution like imitation, etc. In many models these produce identical, or very similar dynamics (like in NowakMay1992 for instance), and in models that are less abstract they sometimes produce more distinct results.

      The rate of change part doesn’t matter too much, since most simulations have a very abstract notion of time that could correspond to 1 second or 20 years without changing the model at all. Thus a relabeling of the time variable is not really important. The difference in mechanism between memes and genes is much more important. For one, memes can be modified by learning during the individuals life producing a sort Lamarcksitic evolution instead of the standard Darwinian kind.

      I think this is definitely something we should explore in a future post.

  2. Julian Z X says:

    Yeah, like a lot of people I learned a lot of EGT from Nowak. I didn’t use it as an introduction, (I had read EGT from miscellaneous papers), but having this book to summarize and to clarify ideas (especially between what is well known and what is interesting) was invaluable.

    The extremely easy math, presented in a handwaving manner, by someone who is capable of high formalism is what I would consider a great strength of this book. Because Nowak understands well the underlying rigor, he is able to present the mathematics with great heuristics that really get to the core intuitions, without the danger of actually being wrong. Each of his heuristic statements can be tightened to waterproofness, but not doing so for this book was, in my opinion, a wise choice. Any undergrad with a background in calculus should be able to read the book without excessive head-bludgeoning or outside reading.

    A great intro, but as Artem said, not enough for research into this lively field.

  3. kylerjbrown says:

    This book is a great introduction, and it is an excellent place to start for undergrads. I’ve used the first half of this book as reference numerous times.

    Nowak has a new book out, Supercooperators http://www.amazon.com/gp/product/1439100187/ which I have not yet read but is intended for a lay audience. I wonder if this book might also serve as an introduction.

  4. Pingback: Slides for Roca, Cuesta & Sanchez’s EGT: Temporal and spatial effects beyond replicator dynamics « Theory, Evolution, and Games Group

  5. Pingback: Micro-vs-macro evolution is a purely methodological distinction | Theory, Evolution, and Games Group

  6. John Kennedy says:

    Bit late to the party here so doubt I’ll get any response, but either way, thanks for this outline of the book. It served as a gentle introduction to EGT. That being said, I’ve worked through most the material you referenced above and am now looking for something to build upon that material. What would you recommend for a second “course” on the subject for someone interested in EGT as it applies to cancer? Any thoughts on the Hoffbauer and Sigmund text that is referenced occasionally by Nowak? Thanks again!

    • It totally depends on your math background. Hofbauer and Sigmund is a solid book, but if you have a good math background then it is not necessary to get into EGT and if you don’t have a good math background then it is better off to spend time acquiring that background (Strogatz’s Nonlinear Dynamics And Chaos might be a good starting point there). If you want personal experience then when I learnt EGT, I did so by just jumping right into the current research literature and reading as widely as I could.

      I am sure that you know this already, but this blog has lots of posts on mathematical oncology. Most of those are EGT related. You might be interested in taking a look at them. If something specific doesn’t make sense then ask right on the relevant post and I will try to explain it or point you further.

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