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I am a biology student who has recently started studying population dynamics. I've recently found a model of population growth where the number of organisms at an age class (N) is calculated by:

N(t+1)=Ne^(-m) where m is the natural mortality However the last aye class = Ne^(-m)/1-e^(-m)

The authors says: The inclusion of the (1 – e^(–m)) divisor forces the equation to be the sum of an exponential series.

My question is: Why is it important to make the equation a sum of exponential series?

I will appreciate any guidance. Thanks so much!

Reference: Haddon, Malcolm (2011-10-03). Modelling and Quantitative Methods in Fisheries, Second Edition (Page 341). Taylor and Francis CRC ebook account. Kindle Edition.

Biologist
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The author says "The final component of Eq. 11.3, where $a=a_{max}$, is referred to as the plus group because it combines ages $a_{max}$ and all older ages that are not modelled explicitly". Thus, it is an infinite series,

\begin{eqnarray*} N_{a_{max},1} &=& \sum_{i=a_{max}}^{\infty}{\text{Population of age $i$}} \\ &=& \sum_{i=0}^{\infty}{N_{a_{max}-1,1} (e^{-M}) \left(e^{-M}\right)^{i}} \qquad\text{(using the mortality factor)} \\ &=& N_{a_{max}-1,1} (e^{-M}) \sum_{i=0}^{\infty}{\left(e^{-M}\right)^{i}} \qquad\text{(taking the constant outside the sum)} \\ &=& N_{a_{max}-1,1} \dfrac{e^{-M}}{1 - e^{-M}}. \end{eqnarray*}

That last line uses the formula for a geometric series. If $|r| \lt 1$:

$$\sum_{i=0}^\infty{r^i} = \dfrac{1}{1-r}.$$

Mick A
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