What we would like to do now is to find a simple equation for the cost of natural selection. Following Haldane, we find a differential equation for the cost in a single generation, and then integrate that over all generations.
We can find the differential change in q with respect to time. Using the generation for the unit of time,
   dq/dt = [-s*q*p]/[1 - s*q].

Haldane chose to ignore the denominator for small s. For instance, if s is .01 and q is .99, the denominator would be 1 - 0.01*.99 which equals 0.9901. Ignoring the denominator introduces an error of 1 - 0.9901  = 0.0099 in one generation. This is an error of less than 1% per generation, and it will go down as q is reduced. Haldane felt this could be ignored, but others have disagreed with Haldane on this point.

Ignoring the denominator gives us:
   dq/dt = -s*q*p
Remembering that p = 1 - q,
   dq/dt = -s*q*(1 - q)

Also, by rearrangement, we see that dt = -dq/(s*q*(1-q).

Integrating gives us:

   D = s * ò₀^¥ q*dt    (That is, s multiplied by the integral from t= infinity to t=0 of q*dt.)
      Because dt = -dq/(s*q*(1-q), q at time infinity = 0 (the a allele is disappearing from the population), and q at time 0 is defined as q₀ we can replace dt and put the range of the integral in terms of q as follows:
    D  = s * ò_q0⁰q*[-dq/s*q*(1-q)]
    D  = s * ò_q0⁰q*[-dq/s*q*(1-q)]   (The s and q terms are both canceled out.)
    D  = ò_q0⁰-dq/(1-q)
    D = ò₀^q0dq/(1-q)                       (Switching the order of integration canceled the negative sign.)
Evaluating the integral:

    D = -ln(1 - q₀) + O(s)

O(s) is a non specified (but small) correction factor that accounts for the fact that the a alleles are never truly eliminated from the population because new copies will be created due to new mutations (back mutations). I am aware of no reason to consider this correction factor further and will henceforth omit it.

   Remembering that p₀ = 1 - q₀ and dropping the correction factor, we have

   D = -ln(p₀)
This equation tells us that the fraction of selective deaths for a substitution to occur is given by the natural logarithm of the initial frequency of the allele that will be fixed.