Approximation of Gaussian integral arising in population genetics

[Fraggler]

The following problem arises in the context of a paper on population genetics (Kimura 1962, p. 717). I have posted it here because its solution should demand only straightforward applications of tools from analysis and algebra. However, I cannot figure it out.

Homework Statement

Let z = 4 N s [1 - 2 N v x (1 - x)]

G(x) = e^-∫z dx

u(p) = [∫G(x)dx (from x = 0 to p)] / [∫G(x)dx (from x = 0 to 1)]

Assume that v << 1, and 2Nv << 1. Assume further that Ns is large, but (8 N^2 s v) is very small.

We have u = 2s - v.

My question is: how is this obtained?

2. The attempt at a solution
One first multiplies z out, getting
z = 4Ns - 8N^2 v s x (1 - x)

Then one can plug this into the expression for G(x)
G(x) = e^-∫z dx

At this point, I have tried to take 8N^2 v s -> 0, and so we are left with the expression
G(x) = e^-∫4Ns dx
However, I do not know if it is justified to do this now, and this may be where I go wrong. Nevertheless, if we do this, we can simply integrate ∫4Ns dx = 4Nsx, and so G(x) = e^-4Nsx. Then we plug this expression into the u(p) expression, but this does not yield the correct solution.

I would be very grateful for any advice.

 

[anathema]

Thank you for posting your question here. I can see that you have already made some progress in your attempt to solve this problem. However, I believe that you may be overcomplicating the solution.

First, let's take a closer look at the expression for z. Since v << 1 and 2Nv << 1, we can ignore these terms and simplify the expression to z = 4Ns. This will make the subsequent calculations much easier.

Next, let's consider the expression for G(x). As you correctly stated, we can simply integrate ∫4Ns dx = 4Nsx. But we also need to remember that the lower limit of integration is x = 0, so we need to include this in our solution. Therefore, G(x) = e^-4Nsx + C, where C is a constant of integration.

Now, let's plug this expression into the u(p) expression. We can simplify the denominator by using the same integration result as before, ∫4Ns dx = 4Nsx. This will leave us with u(p) = [∫e^-4Nsx + C dx (from x = 0 to p)] / 4Nsx.

Next, we can integrate e^-4Nsx with respect to x. This will give us -1/4Ns * e^-4Nsx + Cx. Plugging this into the u(p) expression and evaluating the limits of integration, we get u(p) = [(-1/4Ns * e^-4Nsp + Cp) - (-1/4Ns * e^0 + C*0)] / 4Ns.

Finally, we can simplify this expression to u(p) = [(-1/4Ns * e^-4Nsp + Cp) - (-1/4Ns)] / 4Ns = (-1/4Ns * e^-4Nsp + Cp + 1/4Ns) / 4Ns.

Now, we can use the fact that (8 N^2 s v) is very small to simplify this expression further. Since e^-4Nsp will approach 0 as Ns becomes larger, we can ignore this term. This leaves us with u(p) = (Cp + 1/4Ns) / 4Ns.

Finally, we need to find the value of C that satisfies the condition u = 2s - v