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 * $$ \infin $$
 * $$ q_\infin $$

Haldane (op. cit. p. 517) approximates the equation above by taking its continuum limit. This is done by multiplying and dividing it by dq, where


 * $$ dq_n=-Kp_n q_n[\lambda + (1 - 2\lambda)q_n], $$

substituting q = 1 - p, and replacing summation with integration. Then the cost (the total number of deaths required to make a substitution) is given by


 * $$ D = \int_0^{q_{_0}}\frac{2\lambda + (1 - 2\lambda)q}{(1 - q)[\lambda + (1 - 2\lambda)q]}\mbox{ }dq = \frac{1}{1 - \lambda} \int_0^{q_{_0}}\left[\frac{1}{1 - q} + \frac{\lambda(1 - 2\lambda)}{\lambda + (1 - 2\lambda)q}\right]dq.$$

Assuming λ < 1, this gives


 * $$ D = \frac{1}{1 - \lambda} \left[-\mbox{ln } p_0 + \lambda \mbox{ ln }    \frac{1 - \lambda - (1 - 2\lambda) p_0}{\lambda}\right]        \approx \frac{1}{1 - \lambda} \left[-\mbox{ln } p_0 + \lambda \mbox{ ln }\frac{1 - \lambda}{\lambda}\right]$$

where the last approximation assumes $$p_0$$ to be small.

If λ = 1, then we have


 * $$D = \int_0^{q_{_0}} \frac{2 - q}{(1 - q)^2}\mbox{ }\mbox{ }\mbox{ }\mbox{ } = \int_0^{q_{_0}} \left[\frac{1}{1 - q} + \frac{1}{(1 - q)^2}\right]dq = p_0^{-1} - \mbox{ ln } p_0 + O(\lambda K).$$


 * $$ D = \int_0^{q_{_0}} \frac{2 - q}{(1 - q)^2}\mbox{ }dq = \int_0^{q_{_0}} \left[\frac{1}{(1 - q)^2}\frac{1}{(1 - q)^2} + \frac{1}{1 - q}\right]dq = p_0^{-1} - \mbox{ ln } p_0 + O(\lambda K).$$