# Moran Processes

### Notes

### Videos

Modelling evolution in a discrete population: motivating the Moran process - YouTube - Private

The definition of the Moran process - YouTube - Private

Moran processes and games: fitness functions from games. - YouTube - Private

Selection probabilities for the Moran process on a game - YouTube - Private

The Moran process with mutation - YouTube - Private

The Moran process on 2 types: a mathematically tractable model. - YouTube - Private

A formula for the fixation probability for Moran processes with 2 types. - YouTube - Private

Using Python to simulate the Moran Process with Nashpy - YouTube - Private

### Class meeting notes

## Typical Programming Exercises

For each of the following matrices \(A\) and initial populations \(x_0\), for the corresponding normal form game:

- create a variable
`initial_population`

which has value the corresponding `nashpy`

initial population.
- create a variable
`probabilities`

which has value the fixation probabilities of the simulated Moran process (using the given repetitions and random seed).

a. \(A = \begin{pmatrix}1 & 2 \\ 2 & 1\end{pmatrix}
\qquad
x_0 = \begin{pmatrix}10 & 0\end{pmatrix}\)

`repetitions=500`

and `seed=0`

b. \(A = \begin{pmatrix}1 & 2 \\ 2 & 1\end{pmatrix}
\qquad
x_0 = \begin{pmatrix}3 & 0\end{pmatrix}\)

`repetitions=350`

and `seed=4`

c. \(A = \begin{pmatrix}1 & 2 & 3 \\ 2 & 1 & 4 \\ 2 & 3 & 1\end{pmatrix}
\qquad
x_0 = \begin{pmatrix}3 & 1 & 2\end{pmatrix}\)

`repetitions=1`

and `seed=2`

d. \(A = \begin{pmatrix}1 & 2 & 3 \\ 2 & 1 & 4 \\ 2 & 3 & 1\end{pmatrix}
\qquad
x_0 = \begin{pmatrix}6 & 2 & 4\end{pmatrix}\)

`repetitions=350`

and `seed=0`

Solutions

## Log of past relevant classes

In class today we looked at the Moran
Process. We did this by
considering the Hawk Dove game.

In class today we revisited the Moran
Process and specifically we
looked at calculating so called fixation probabilities analytically.

In class today we looked at the Moran
Process. We did this by
considering the Hawk Dove game.