Moran Processes

Activity (approximately 60 minutes)

Use moran process form and have students play in pairs.

Also requires use of dice, allowing for multiples of 4 and 6. For example:

• 1 d12 (12-sided dice), or
• 1 d6 and 1 d4 (the example used in this page), or
• 1 d6 and 1 d8.

Virtual modification: use breakout rooms of 4. For dice, show how to use the Python random library.

Explain that we will aim to reproduce a Moran process with $n=3$.

We will do this with the Hawk–Dove game:

Recall, this corresponds to sharing of 4 resources:

• two hawks both get nothing;
• a hawk meeting a dove gets 3 out of 4;
• a dove meeting a hawk gets 1 out of 4;
• a dove meeting a dove gets 2 out of 4.

Give students 5 minutes to write out the fitnesses of each type in each possible situation.

Confirm:

Give students 5 minutes to write out the probabilities of selection of each type in each possible situation.

Confirm:

Let students simulate.

Count from groups to obtain mean fixation rate of a hawk.

Show the following code that allows us to simulate the simulation undertaken by the students (this is not the same code as in the notes):

import collections
import matplotlib.pyplot as plt
import numpy as np
import tqdm

def roll_n_sided_dice(n=6):
    """Roll a dice with n sides."""
    return np.random.randint(1, n + 1)

class MoranProcess:
    """
    A class for a Moran process with a population of size n=3 using the
    standard hawk-dove game:

        A = [[0, 3],
             [1, 2]]

    Note that this is a simulation corresponding to an in-class activity
    where students roll dice.
    """
    def __init__(self, number_of_hawks=1, seed=None):
        if seed is not None:
            np.random.seed(seed)

        self.number_of_hawks = number_of_hawks
        self.number_of_doves = 3 - number_of_hawks

        self.dice_and_values_for_hawk_birth = {1: (6, {1, 2, 3}), 2: (4, {1, 2, 3})}
        self.dice_and_values_for_hawk_death = {1: (6, {1, 2}), 2: (6, {1, 2, 3, 4})}

        self.history = [(self.number_of_hawks, self.number_of_doves)]

    def step(self):
        """Select a hawk or a dove for birth and death, then update."""
        birth_dice, birth_values = self.dice_and_values_for_hawk_birth[self.number_of_hawks]
        death_dice, death_values = self.dice_and_values_for_hawk_death[self.number_of_hawks]

        select_hawk_for_birth = self.roll_dice_for_selection(dice=birth_dice, values=birth_values)
        select_hawk_for_death = self.roll_dice_for_selection(dice=death_dice, values=death_values)

        if select_hawk_for_birth:
            self.number_of_hawks += 1
        else:
            self.number_of_doves += 1

        if select_hawk_for_death:
            self.number_of_hawks -= 1
        else:
            self.number_of_doves -= 1

        self.history.append((self.number_of_hawks, self.number_of_doves))

    def roll_dice_for_selection(self, dice, values):
        """Return True if the roll is in the target set."""
        return roll_n_sided_dice(n=dice) in values

    def simulate(self):
        """Run the simulation until the number of hawks is 0 or 3."""
        while self.number_of_hawks in [1, 2]:
            self.step()
        return self.number_of_hawks

    def __len__(self):
        return len(self.history)

This carries out the simulations:

repetitions = 10 ** 5
end_states = []
path_lengths = []

for seed in range(repetitions):
    mp = MoranProcess(seed=seed)
    end_states.append(mp.simulate())
    path_lengths.append(len(mp))

counts = collections.Counter(end_states)
counts[3] / repetitions
# ≈ 0.54666

Discuss obtaining theoretic probabilities of changing state:

Now work through the notes, culminating in the proof of the theorem for the absorption probabilities of a birth–death process.

Discuss and use code from the chapter to show the fixation with the hawk–dove game:

a = np.array([[0, 3], [1, 2]])

Calculate theoretic value using the formula from the theorem:

This gives (for $n=3$):

Thus:

Discussion

Work through the Moran Process chapter.

Discussion Point: After the definition of the moran process, ask how this differs from our example?

Discussion Point: After the definition of fixation probability, ask what the fixation probability of a hawk is based on our simulation?

Discussion Point: After the theoretic fixation in two types ask how this could be used for our example? Leave this as an exercise.