Introduction to Complex Systems

Practical Problem 14: Evolutionary Dynamics

In this problem we will design a models that capture the key elements of evolutionary processes

  1. replication
  2. variation (mutation)

The idea is to reimplement models similar to the explorables explorables:

  1. “Maggots in the Wiggle Room”
  2. “A Patchwork Darwinge”

Part 1: The Moran process

This is a turtle based simulation.

Write a model in which turtles move around randomly like we did in many previous simulations.

Each turtle should have a turtle specific property called fitness which you can implement using the turtles-own command

Initially the turtles should have uniform fitness, say a value of $1$. The should also have another variable named species that identifies the species label

turtles-own [fitness species]

Intitially all turtles should belong to the same species, so we give them a number 1.

ask species [set species 1]

In the paint function we can just color them according to the species index:

ask turtles [set color species]

Now two things can happen:

  1. Replication
  2. Mutation

Replication

Replication should happen at a higher basic rate than mutation. For each rate you should have a slider parameter, say replicationrate and mutationrate.

When a turtle replicates it should do this with a rate that is a product of the basic replicationrate and the turtles fitness relative to the mean fitness of the entire population:

$$ r_i = R_0 \times f_i / \overline f $$

where $r_i$ is turtle i’s replication rate $R_0$ is the overall replicationrate, $f_i$ is the fitness parameter and $\overline f$ the mean fitness of the population.

Initially the population has uniform fitness, to the replication is also uniform.

When a turtle replicates

  1. it hatches a baby
  2. a randomly chosen other turtle is killed

This keeps the number of turtles constant.

Mutation

Now let the turtles mutate. With a small mutation rate, turtles mutate when replication occurs. That means when a baby is hatched, a new species emerges with a fitness different from the parent and a species index one larger than the maximum species index in the population

let ms max-of [species] of turtles
let parentfitness fitness
hatch 1 [
	set species ms + 1 
	set fitness parmentfitness + (random-float df - df / 2)
]

In this example the fitness of the mutant is the parent’s fitness plut a random increment that can be positive or negativ and a range of df. That typical fitness change range should be a parameter.

Plotting fitness

You can add a plot widget to your simulation and plot the mean fitness of the population as a function of time.

Part 2: Competitive Fitness

This is a similar model based on patches.

  • Like above we assign a fitness value to a patch using the patches-own command and initialize the fitness value of each patch.

  • In this simulation patches can turn neighboring patches into their own type (color and fitness) with a probability function that looks like this

$$ P = \frac{1}{1+\exp\left( - \sigma (f_1 - f_2) \right)} $$

  • where $f_1$ and $f_2$ are the fitness parameters of the competing patches and $\sigma$ is a steepness parameter (slider), you can start with a value of $1$.

  • Implement mutations just like in simulation 1.