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# Blog - stationary stability in finite populations

It's a followup to his earlier one, Blog - relative entropy in evolutionary dynamics. Time to finish this mini-series!

Give it a read and ask questions if there's stuff that you don't understand!

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1.

Hi John, thanks for the reminder. I think that I need to add some more background on the Moran process and give the the entire text another pass. Please give me a chance to update it, and I'll post here when it's ready for more eyes on it.

Comment Source:Hi John, thanks for the reminder. I think that I need to add some more background on the Moran process and give the the entire text another pass. Please give me a chance to update it, and I'll post here when it's ready for more eyes on it.
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2.

So sorry for the delay! I've added a lot of background material to the article, particularly for the Moran process, and tried to connect it to other recent Azimuth articles better, referencing to Manoj Gopalkrishnan's article on Lyapunov functions and Matteo Smerlak's fluctuation theorem article. I also cleaned up the text and linked out to some papers available on the ArXiv for some more background and examples.

There is also a link to one of my pages that has many plots for various fitness landscapes -- one for each of the 47 possible phase plots of the replicator equation on three types (http://people.mbi.ucla.edu/marcharper/stationary_stable/3x3/incentive.html). I very briefly mentioned that the main results also apply to the Wright-Fisher process, for which there are the same 47 examples here: (http://people.mbi.ucla.edu/marcharper/stationary_stable/3x3/wright_fisher.html). If people are interested in Wright-Fisher then I could elaborate a bit.

Please let me know if there are places that could use some more background, especially the first part summarizing the previous article I wrote and the Lyapunov theorem as you discussed in your information geometry series.

Comment Source:So sorry for the delay! I've added a lot of background material to the article, particularly for the Moran process, and tried to connect it to other recent Azimuth articles better, referencing to Manoj Gopalkrishnan's article on Lyapunov functions and Matteo Smerlak's fluctuation theorem article. I also cleaned up the text and linked out to some papers available on the ArXiv for some more background and examples. There is also a link to one of my pages that has many plots for various fitness landscapes -- one for each of the 47 possible phase plots of the replicator equation on three types (http://people.mbi.ucla.edu/marcharper/stationary_stable/3x3/incentive.html). I very briefly mentioned that the main results also apply to the Wright-Fisher process, for which there are the same 47 examples here: (http://people.mbi.ucla.edu/marcharper/stationary_stable/3x3/wright_fisher.html). If people are interested in Wright-Fisher then I could elaborate a bit. Please let me know if there are places that could use some more background, especially the first part summarizing the previous article I wrote and the Lyapunov theorem as you discussed in your information geometry series.
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3.

Okay, it's time for us to polish and publish this blog article, now that Jan Galkowski's is done!

Comment Source:Okay, it's time for us to polish and publish this blog article, now that [[Jan Galkowski]]'s is done!
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4.
edited June 2014

I made a first crack at polishing the style, grammar and formatting.

There is an undefined term "fixation states" which need to be explained - or even better, replaced by something everyone can understand. (It comes up only twice, so better to avoid introducing it in the first place.) "Fixation" sounds a bit like "absorption", which is another concept used here.

Comment Source:I made a first crack at polishing the style, grammar and formatting. There is an undefined term "fixation states" which need to be explained - or even better, replaced by something everyone can understand. (It comes up only twice, so better to avoid introducing it in the first place.) "Fixation" sounds a bit like "absorption", which is another concept used here.
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5.
I changed them both to absorbing states (the proper Markov process term). The states that the population (without mutation) can fixate are just the boundary states (N, 0, ..., 0) and permutations.
Comment Source:I changed them both to absorbing states (the proper Markov process term). The states that the population (without mutation) can fixate are just the boundary states (N, 0, ..., 0) and permutations.
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6.
edited June 2014

Thanks.

1) At some point the article says:

Hopefully by the end of this post, you’ll see how all of these diagrams:

which is not a sentence.

2) You write:

Incentive proportionate selection is

$$\frac{\varphi_A(a,b)}{\varphi_A(a,b) + \varphi_B(a,b)},$$

This sentence is of the general form

Some undefined term is

$$\int^{ome} c_{omplicated} {f(ormula)}$$

So, it's not very clear, especially because you don't say what $\varphi_A(a,b)$ and $\varphi_B(a,b)$ are.

Since I'm experienced at reading mathematical prose, I can make guesses. I can guess that maybe "incentive proportionate selection" is the name for some particular form of "incentive function".

Are $\varphi_A(a,b)$ and $\varphi_B(a,b)$ just arbitrary functions of $a$ and $b$?

3) A bit before this, you said an incentive function is a function that "controls the selective process". I don't know what "controls the selective process" means. It would probably be good to explain that an incentive function is a real-valued function of the types $a$ and $b$ (if that's correct), and then say a bit about what it does.

Comment Source:Thanks. 1) At some point the article says: > Hopefully by the end of this post, you’ll see how all of these diagrams: which is not a sentence. 2) You write: > Incentive proportionate selection is > $$\frac{\varphi_A(a,b)}{\varphi_A(a,b) + \varphi_B(a,b)},$$ This sentence is of the general form > Some undefined term is > $$\int^{ome} c_{omplicated} {f(ormula)}$$ So, it's not very clear, especially because you don't say what $\varphi_A(a,b)$ and $\varphi_B(a,b)$ are. Since I'm experienced at reading mathematical prose, I can make guesses. I can guess that maybe "incentive proportionate selection" is the name for some particular _form_ of "incentive function". Are $\varphi_A(a,b)$ and $\varphi_B(a,b)$ just arbitrary functions of $a$ and $b$? 3) A bit before this, you said an incentive function is a function that "controls the selective process". I don't know what "controls the selective process" means. It would probably be good to explain that an incentive function is a real-valued function of the types $a$ and $b$ (if that's correct), and then say a bit about what it does.
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7.
I fixed the fragment, and made another pass over the whole post. I also expanded the sections around 2 and 3 for clarity.
Comment Source:I fixed the fragment, and made another pass over the whole post. I also expanded the sections around 2 and 3 for clarity.
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8.
edited June 2014

Thanks! I'll have another go at it today.

Comment Source:Thanks! I'll have another go at it today.
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9.
edited March 2015

I finally completed work on this article - sorry for the huge delay. It will appear soon:

Comment Source:I finally completed work on this article - sorry for the huge delay. It will appear soon: * Marc Harper, [Stationary stability in finite populations](https://johncarlosbaez.wordpress.com/2015/03/25/stationary-stability-in-finite-populations/), Azimuth Blog, 25 March 2015.