Question

Moran process In this exercise we will implement a simplified version of a Moran process. A Moran process is a "stochastic process" used in biology to describe finite populations. It was first proposed by statistician Patrick Moran in 1958

1
. Assume we have a population of N individuals. The number N of individuals is assumed to be constant throughout the whole process. Each individual can be of type 0 (thought of as the "standard" type) or of type 1 (thought of as the "mutant" type). Initially there will be N−1 individuals of type 0 and 1 individual of type 1 . The process consists of a sequence of "generations". In each generation, one individual, chosen randomly, "dies" and is "replaced" by a copy of another individual, also chosen randomly. This will continue until either the mutant individuals disappear or the population is taken over by the mutants. (a) Write a function in R called simulate moran. It should receive as input n (which is assumed to be a positive integer). This function starts by creating a vector of size n containing (n - 1) 0-entries and 1 1-entry: population <- c(rep(0,n−1),1) This vector will represent the population (of size n ). So the population initially has n−1 individuals of type 1 and 1 individual of type 1 . This function will then run a simulation according to the description above. Notice this will involve a while loop that will run while the population still contains at least one type 0 individual and at least one type 1 individual. This function then returns a vector containing two entries: the first one should be the number of generations it took for the process to stop, and the second one should be the type that took over the whole population (either 0 or 1 ). Make sure you test the function is working properly by calling it with small values of n (for instance 5,10,50). (b) Write a for loop that calls the function above 1000 times with n=100. What percentage of the time was the population taken over by the mutants? On average, how many generations did each simulation last?

Answer 1

(a) The simulate_moran function in R simulates the Moran process, returning the number of generations and the dominant type.

(b) The for loop in R calls simulate_moran 1000 times with n = 100, calculating the percentage of mutant takeovers and the average generations per simulation.

Step-by-step explanation:

(a) The `simulate_moran` function is developed in R to model the Moran process. Initially, a population vector is constructed, reflecting n-1 type 0 individuals and one type 1 individual. The function iterates through a while loop simulating each generation until either type 0 or type 1 dominates the entire population. The function then returns information about the duration of the process (number of generations) and which type eventually takes over the population.

(b) A for loop is utilized to run the `simulate_moran` function 1000 times with n = 100. The percentage of times the population was taken over by the mutants is computed by counting the occurrences where type 1 dominates and dividing it by the total simulations (1000). To determine the average number of generations per simulation, the total number of generations across all simulations is summed up and divided by the total simulations.

This process offers insights into the behavior of the Moran process by simulating numerous scenarios and assessing the frequency and duration of each outcome, providing a statistical perspective on the population dynamics.