Page 35 - Applied Probability

1. Basic Principles of Population Genetics
18
(b) Solve the two equations in (a).
(c) When t x = 1, show that the fraction of affected males represent-
1
(1 − t y ). This fraction does not depend
ing new mutations is
3
on the mutation rate.
(d) If t x = 1 and t y = 0, then prove that p ∞x ≈ 4µ and p ∞y ≈ 3µ.
12. In the selection model of Section 1.5, it of some interest to deter-
mine the number of generations n it takes for allele A 1 to go from
frequency p 0 to frequency p n . This is a rather difficult problem to
treat in the context of difference equations. However, for slow selec-
tion, considerable progress can be made by passing to a differential
equation approximation. This entails replacing p n by a function p(t)
of the continuous time variable t. If we treat one generation as our
unit of time, then the analog of difference equation (1.4) is
dp pq[s − (r + s)p]
= ,
dt ¯ w
2
2
where q =1 − p and ¯ w =1 − rp − sq . If we take this approximation
seriously, then
n p n ¯ w
n ≈ dt = dp.
0 p 0 pq[s − (r + s)p]
Show that this leads to

1 p n 1 1 − p n
n ≈ − 1 ln + − 1 ln
s p 0 r 1 − p 0

1 1 |s − (r + s)p n |
− + − 1 ln
r s |s − (r + s)p 0 |
when p n and p 0 are both on the same side of the internal equilibrium
point and neither r nor s is 0. Derive a similar approximation when
s =0 or r = 0. Why is necessary to postulate that p n and p 0 be
on the same side of the internal equilibrium point? Is it possible to
calculate a negative value of n? If so, what does it mean?
13. Let f(p) be a continuously differentiable map from the interval [a, b]
into itself, and let p ∞ = f(p ∞ ) be an equilibrium (fixed) point of the

iteration scheme p n+1 = f(p n). If |f (p ∞ )| < 1, then show that p ∞
is a locally stable equilibrium in the sense that lim n→∞ p n = p ∞ for
p 0 sufficiently close to p ∞ . How fast does p n converge to p ∞ ? Apply
this general result to determine the speed of convergence to linkage
equilibrium for an autosomal locus.

14. To explore the impact of genetic screening for carriers, consider a
lethal recessive disease with two alleles, the normal allele A 2 and the