The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the $lim_{n\rightarrow \infty} (1+1/n)^n$ formula, a growth that is infinitely divided into smaller and smaller exponential increases.

I wonder how was this matched in the following equation, the model of population size for cases with continuous breeding:

$$ \frac{\mathrm{d}N}{\mathrm{d}t} = rN\\ N(t)=N(0)e^{rt} $$

My doubt is: organisms in actual world do not approach $\infty$ in the $e$ limit formula while they bread, i.e. there necessairly are non zero gaps between population increases, and the fact that the increases occur randomly (i.e. that ages overlap) doesn't change the fact, that $n$ is not approaching $\infty$, and there are discrete steps in the growth. Why is it possible to scale the result with $r$? Is it approximation, or does $e$ and/or exponential growth in general have an explicit mathematical property, that could be perceived as an ability to trade off the "density" of the infinitely divided time unit over the modified exponent $rt$, which is like stretching the infinitely dense $e$ growth by spreading it over time? This is maybe vague description, but what is clear here, is are there any guarantees about how close the result will be to the original? To just mention, there are discrete population growth models created for cases with quite unclearly defined "generation based" (i.e. "discrete-like") breedings, described by equations like e.g.:

$$ N_t = \lambda^tN_0 $$

Where $\lambda$ is the geometric growth factor, what appears to be explicit modification of $e$ performed instead of scaling $t$.

Appendix 1. The parameter $r$ is collected per capita birth and death rates $r = b - d$ and is called the intrinsic rate of increase or exponential growth rate. The model is being applied to organisms like Homo sapiens or the bacteria in a culture flask, with continuous breeding and overlapping generations, where all ages are be present simultaneously, and population size change steadily in small increments with the birth and death of individuals at any time.

Appendix 2. To not just state bold assertions about how $e$ formula "works" – $1+1/1$ means 100% increase in 1 unit of time, $(1+1/2)(1+1/2)$ means first increasing by half in the middle of the unit of time, and them from that increased amount, repeating the same at the end of the unit of time, etc.

  • $\begingroup$ That's a very long question. You're more likely to get answers if you make it shorter, as then people with short attention spans like me would read it. $\endgroup$ – GFauxPas Feb 5 '15 at 2:42

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