Quadratic solution of the Migration / Selection
equilibrium
To solve
qi = tqi2 - (m + t)qi + mqM
Recall the quadratic formula: 0
= [-b ![](Plus_or_minus.gif)
(b2 - 4ac)] / 2a
Note that
(1
x)
(
1
(x/2)) [ approximation ]
because (1 + x/2)2 = 1 + x +
(x/4)2
(1
+ x)
iff x << 1,
and qM = 1 (a
constant) in standard model
Then, set terms of quadratic as a
= t b
= - (m + t) = - m - t c = mqM
= [ (m + t) ![](http://www.mun.ca/biology/scarr/Plus_or_minus.gif)
[(-m - t)2 -
4tmqM]
] / 2t
quadratic form
= [ (m + t) ![](Plus_or_minus.gif)
[m2 + t2
+ 2mt - 4tmqM] ] /
2t expand (-(m + t))2
[ (m + t) ![](Plus_or_minus.gif)
[t2 +
2mt - 4tmqM] ] / 2t
if
m < t , then m2 << t2
[
(m + t) ![](Plus_or_minus.gif)
[t2
+ (t2)(2m/t - 4mqM / t)] ] /
2t create common t2 term
[
(m + t)
(t)
[1 + 2m/t - 4mqM / t] ] / 2t
factor out
t2 term as t
[ (m + t)
(t)(1
+ m/t - 2mqM / t) ] /
2t
apply approximation
x = (2m/t
- 4mqM / t) / 2
For which the negative root, if t > 0,
is
![](Qhat.gif)
(m + t) - [t + m - 2mqm] / 2t
t & m terms cancel [ pretty
!]
![](Qhat.gif)
2mqm / 2t = (m / t) (qM)
![](https://www.mun.ca/biology/scarr/Qhat.gif)
= (m / t)
if
qm
= 1 as in standard model