Homework April 21, 2015 (to be returned on April 28)
We want to determine mean and variance of the allele frequency variation
∆p in situations where the environment is fluctuating very
rapidly. Viabilities
(t)
(t)
(t)
(t)
(t)
(t)
(t)
are w11 = 1 + Y1 ; w22 = 1 + Y2 and w12 = 1 + Y1 + Y2 /2 for the
homozygous A1 A1 , A2 A2 and the heterozygous A1 A2 or A2 A1 , respectively.
The Y ’s are random variables supposed all independent and the index t refers
to the generations.
1) From the general formula that we derived during the lectures for the
variation ∆p of the frequency p of allele A1 under random mating and selection with the previous viabilities, show that :
(t)
∆p
(t)
(t)
Y1 − Y2
p(t) q (t)
=
.
2 1 + p(t) Y1(t) + q (t) Y2(t)
(1)
where ∆p(t) indicates the difference between the values at the generations
t + 1 and t.
2) Suppose that the Y ’s are small and expand (1) to the second order.
(t)
(t)
3) Mean and variances of the Y ’s are : hY1 i = µ1 ; hY2 i = µ2 and
(t)
(t)
Var Y1 = Var Y2 = σ 2 , with the means µi of the same order as the variance
σ 2 . Determine the expression of the mean h∆p(t) i and Var ∆p(t) .
The previous calculation was done under the hypothesis of additivity.
To generalize to non-additive situations, one can suppose that
viabilities are
(t)
(t)
w11 = W 1 + Y1
(t)
(t)
; w22 = W 1 + Y2
(t)
(t)
and w12 = W 1 +
(t)
Y1 +Y2
2
,
where the Y ’s are again random variables, supposed all independent and
the function W (x) is a convex function, e.g. the Michaelis-Menten curve
discussed during the course W (x) = x(1 + a)/(a + x).
1) Supposing the Y ’s small and having means and variances as above,
Q
1/t
g
t−1 i
calculate the geometric means, e.g. w11
(t) =
, and show that
i=0 w11
the condition for the heterozygote to be fitter than both homozygotes is
|µ1 − µ2 | <
σ 2 02
00
W
(1)
−
W
(1)
,
2W 0 (1)
00
(2)
where W 0 (1) and W (1) are the first and second derivatives of W (x) at x = 1.
2) Obtain the condition (2) of coexistence for the explicit form of W given
above, recover the additive limit and discuss whether the general case gives
a range for coexistence larger or smaller than in the additive case.
1