• Marc Harper, Relative entropy in evolutionary dynamics, *Azimuth*, 22 January 2014.

• Marc Harper, Stationary stability in finite populations, *Azimuth*, 24 March 2015.

Here are the slides and video of his talk:

• Marc Harper, Information transport and evolutionary dynamics.

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I have a simple idea, but I don’t know if it is possible the complete applicability.

There is a simple class of equation with only integer solution (I am thinking the Fibonacci series, for example), and the dynamic of these system can be solved with the finite difference equation.

So that the Moran process is not applicable in this case, but it is possible to write equations with integer solution using real population, and if the continuous equation have solution, then it is probable to obtain integer solutions (if the classical solution is too complex, then the integer solution is too complex).

Then it is possible an extension of gradient in the discrete equation (for a potential with discrete values, with an arbitrary assignment of integer values in a grid of i-j, and with an analogy with the classical field, but without scale invariance).

Perhaps it might be possible even an Hamiltonian, and Lagrangian, description with integer solution, and the asymptotic behavior could be a simple potential (quadratic, or linear integer potential). ]]>

In connection with this simple case “For a moment consider a population of just two types A and B.” This is very similar to the following:

Suppose you have a stochastic Petri net with two species A and B and the following transitions

A+B -> 2A at rate u

A+B -> 2B at rate u

A -> B at rate v

B -> A at rate v

The total number of A’a and B’s is a constant N.This system is simple enough to find the equilibrium distribution, which is a beta-binomial distribution (http://en.wikipedia.org/wiki/Beta-binomial_distribution) with α=β=v/u. If v>u this is unimodal, with a maximum at N/2, and if u>v, this is U-shaped.

It uses continuous time and rates not transistion probabilities, and I made it symmetric, but I think its nearly the same thing. And I guess it extends to more than two types with a Dirichlet-multinomial distribution equilibrium distribution.

A couple of typos.

After “that for a very similar model with fitness landscape given by” f_b not f_B.

After “selection ratio above, which now becomes (for two population types):” raw latex.