Page 27 - Applied Probability

1. Basic Principles of Population Genetics
10
At a fixed point p ∞ ∈ [0, 1], we have ∆p ∞ = 0. In view of equation
s
. The third
(1.4), this can occur only when p ∞ equals 0, 1, or possibly
r+s
point is a legitimate fixed point if and only if r and s have the same sign.
In the case r> 0 and s ≤ 0, the linear function g(p)= s − (r + s)p satisfies
g(0) ≤ 0 and g(1) < 0. It is therefore negative throughout the open interval
(0, 1), and equation (1.4) implies that ∆p n < 0 for all p n ∈ (0, 1). It follows
that the decreasing sequence p n has a limit p ∞ < 1 when p 0 < 1. Equation
(1.4) shows that p ∞ > 0 is inconsistent with ∆p ∞ = lim n→∞ ∆p n =0.
Hence, we arrive at the intuitively obvious conclusion that the A 1 allele is
driven to extinction. In the opposite case r ≤ 0 and s> 0, the A 2 allele is
driven to extinction.
When r and s have the same sign, it is helpful to consider the difference
s s
p n+1 − =∆p n + p n −
r + s r + s
s

(r + s)p n q n p n − s
r+s
= − 2 2 + p n −
1 − rp − sq r + s
n n
2 2
1 − rp − sq − (r + s)p n q n s
n
n
= p n −
2
1 − rp − sq 2 r + s
n n
s
1 − rp n − sq n
= p n − .
2
1 − rp − sq 2 n r + s
n
If both r and s are negative, then the factor
1 − rp n − sq n
λ(p n )=
2
1 − rp − sq 2 n
n
> 1,
s
and p n − has constant sign and grows in magnitude. Therefore, ar-
r+s
guments similar to those given in the r> 0 and s ≤ 0 case imply that
s
s
lim n→∞ p n = 0 for p 0 < r+s and lim n→∞ p n = 1 for p 0 > r+s . The point
s is an unstable equilibrium.
r+s
s
If both r and s are positive, then 0 ≤ λ(p n ) < 1, and p n − has
r+s
constant sign and declines in magnitude. In this case, lim n→∞ p n = s ,
r+s
and the point s is a stable equilibrium.For p 0 ≈ s ,
r+s r+s
s

λ(p n ) ≈ λ
r + s
r + s − 2rs
= ,
r + s − rs
s s n s
and p n − ≈ λ( ) (p 0 − ). In other words, p n approaches its
r+s r+s r+s
equilibrium value locally at the geometric rate λ( s ).
r+s
The rate of convergence of p n to 0 or 1 depends on whether there
is selection against the heterozygous genotype A 1 /A 2 . Consider the case