[SOLVED] Evolutionary-Dynamics Homework 6

Exercises marked with a ” ” are programming exercises. These can be solved in a programming lan- guage of your choice. Please make sure to hand in your code along with your answers to these exercises.

Problem 1: One-dimensional Fokker-Planck equation

Consider the one-dimensional Fokker-Planck equation with constant coefficients, ∂tψ(p,t) = −m∂pψ(p,t)+ 2v∂p2ψ(p,t),

with p∈R,v>0.

(a)

(b)

(c)

Show that for vanishing selection, m = 0,
ψ(p,t) = √2πvt exp −2vt

(1)

(1 point)

(2)

Simulate a random walk, starting at 0 where at each time step the position is either increased by 1 with probability 21 or decreased by 1 otherwise. Simulate many walks for N = 10, N = 100 and N = 1000 steps. Compare the results to (2). Find the relationship between the means and variances. (1 point)

Construct a solution for constant selection, m ̸= 0, by substituting z = p − mt for p in (1). What is the mean and variance? (1 point)

1 􏰃p2􏰄
solves the Fokker-Planck equation. To which initial condition does this solution correspond?

Problem 2: Diffusion approximation of the Moran process

Derive a diffusion approximation for the Moran process of two species. Assume the first species has a small selective advantage s.

1. (a)  The general definition for the drift coefficient is
   M(p) = E[X(t)−X(t −1) | X(t −1) = i]/N,

   where p = i/N and X(t) denotes the abundance of the first allele. Evaluate this expression for the Moran process with selection. Show that this yields the result for the Wright-Fisher process from the lecture, divided by N. (tutorial exercise)

2. (b)  By a similar argument calculate the diffusion coefficient V(p). Use your findings to set up the diffusion equation for the Moran model. (1 point)
3. (c)  Nowassumethats≪1.Approximateyourresultsfrom(a)and(b)andusethegeneralexpression for the fixation probability ρ(p0) to show that the fixation probability is given by: (1 point)

   1−e−s
   ρ(p0 =1/N)= 1−e−Ns. (3)

1

(d) Take the limit to derive a result for the fixation probability of a neutral allele, s = 0. Evaluate (3) for N = 10 and N = 1000 for both positive, s = 1%, and negative selection, s = −1%, respectively. Compare your results with ρ1 of the exact Moran process. (1 point)

Problem 3: Absorption time in the diffusion approximation

In the diffusion approximation of a process with absorbing states 0 and 1 the expected fixation time, conditioned on absorption in state 1, reads:

,

􏰅p 􏰅p whereρ(p)denotesthefixationprobability,A(p)= 2M(p)/V(p)dp,andS(p)= exp(−A(p))dp.

􏰃􏰅 1 ρ(p)(1−ρ(p)) 1−ρ(p0) 􏰅 p0 ρ(p)2 􏰄 τ1(p0) = 2(S(1)−S(0)) −A(p) dp+ −A(p) dp

(a) (b)

(c)

00
Calculate the conditional expected waiting time for fixation, τ1(p0), of an allele of frequency p0

in the neutral Wright-Fisher process. Approximate the result for small p0. (2 points)

Compute τ0, the conditional expected waiting time until extinction (absorption in state 0) in the neutral Wright-Fisher process. Also derive the unconditioned expected waiting time τ ̄ until either fixation or extinction. (1 point)

Compare your analytical results for the absorption times τ1,τ0, and τ ̄ with those from numerical simulations of the neutral WF-process. Use N = 100 individuals and initial frequencies of p0 = 0.5, as well as p0 = 1/N. Do 1,000 simulations each (or more) and remember to use a suitably

long simulation time.

(1 point)

p0eV(p) ρ(p0)0eV(p)

2