# Drummond:PopGen

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Note that the continuous case and the original discrete-generation case agree for all values of <math>t</math>. We can define the ''instantaneous rate of increase'' <math>r = \ln R</math> for convenience. | Note that the continuous case and the original discrete-generation case agree for all values of <math>t</math>. We can define the ''instantaneous rate of increase'' <math>r = \ln R</math> for convenience. | ||

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+ | ==Continuous-time approximation== | ||

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+ | ==Diffusion approximation== |

## Revision as of 09:03, 14 July 2008

## Per-generation and instantaneous growth rates

Let *n*_{i}(*t*) be the number of organisms of type *i* at time *t*, and let *R* be the *per-capita reproductive rate* per generation. If *t* counts generations, then

Now we wish to move to the case where *t* is continuous and real-valued.
As before,

where the last simplification follows from L'Hôpital's rule. Explicitly, let ε = Δ*t*. Then

The solution to the equation

*t*. We can define the

*instantaneous rate of increase*

*r*= ln

*R*for convenience.