Evolutionary Dynamics Exercises 6 -Solved

Problem 1: One-dimensional Fokker-Planck equation 
Consider the one-dimensional Fokker-Planck equation with constant coeffificients, 
∂tψ(p,t) = −m∂pψ(p,t) + 
v
2
∂p2ψ(p,t), 
(1) 
with p ∈ R, v > 0. 
(a) Show that for vanishing selection, m = 0, 
ψ(p,t) = 
1 
√2πvt exp − p
2 
2vt 
 
(2) 
solves the Fokker-Planck equation. To which initial condition does this solution correspond? 
(b) § 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. 

(c) Construct a solution for constant selection, m 
6 
= 
0, by substituting z = p−mt for p in (1). What is 
the mean and variance? 
 
Problem 2: Diffusion approximation of the Moran process 
Derive a diffusion approximation for the Moran process of two species. Assume the fifirst species has a 
small selective advantage s. 
(a) The general defifinition for the drift coeffificient 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 fifirst 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) 
(b) By a similar argument calculate the diffusion coeffificient V(p). Use your fifindings to set up the 
diffusion equation for the Moran model. 
(c) Now assume that s  1. Approximate your results from (a) and (b) and use the general expression 
for the fifixation probability 
ρ(p0) to show that the fifixation probability is given by: 
 
ρ(p0 = 1/N) = 
1
−e−
s 
1−e−
Ns 
. 
(3) 
1(d) Take the limit to derive a result for the fifixation 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. 

Problem 3: Absorption time in the diffusion approximation 
In the diffusion approximation of a process with absorbing states 0 and 1 the expected fifixation time, 
conditioned on absorption in state 1, reads: 
τ1(p0) = 2(S(1)−S(0)) Z 
1
p0 
ρ(
p
)(
1
−ρ(p
)) 
e
−
A(
p)V(p
) 
dp+ 
1−
ρ
(
p
0) 
ρ(
p
0
) 
Z 0p0 
ρ
(p
)2 
e−A
(p
)V
(p)
dp , 
where ρ(p) denotes the fifixation probability, A(p) = Z 0p 
2M(p)/V(p)dp, and S(p) = Z 0p 
exp(−A(p))dp. 
(a) Calculate the conditional expected waiting time for fifixation, τ1(p0), of an allele of frequency p0 
in the neutral Wright-Fisher process. Approximate the result for small p0. 

(b) 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 
fifixation or extinction. 

(c) § 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. 

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