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I have in mind things like language learning (native or foreign), and growth to adulthood.

Is acclimatization the same thing as saturation? Can the logistic function be regarded as simply an upside-down version of Newton’s Law of Cooling?


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Do you mean that for $x>>0$, the logistic curve looks similar in shape to exponential decay? This is not a coincidence, as $1/(1+e^{-x})=1-e^{-x}/(1+e^{-x})\approx 1-e^{-x}$, but for many applications, you care about the short term behavior too, and so the long term similarity is immaterial. – Aaron Jun 2 '11 at 21:33
@Aaron: If you'll put this as an answer, I'll accept it. – Mike Jones Oct 5 '11 at 5:57

For Newton's law of cooling, you have exponential decay, $y=A+Be^{-Cx}$, where $A,B,C$ are parameters which control the final temperature, initial temperature, and cooling rate. It satisfies the differential equation $y'=-C(y-A)$. Note that if the initial temperature is less than the final temperature, $B$ will be negative and we will have Newton's law of heating.

In contrast, logistic functions are modifications of $\frac{1}{1+e^{-x}}=1-\frac{e^{-x}}{1+e^{-x}}$. As $x$ gets large, we have $1\gg e^{-x}$ and so $e^{-x}/(1+e^{-x})\approx e^{-x}$, and so for very large $x$, the logistic curve looks a lot like a translated and reflected exponential decay. However, this is only the long term behavior.

Overall, the logistic curve is very different from exponential decay. In particular, when $x\ll 0$, it looks like exponential growth. Indeed, logistic curves have been used to model population growth with a carrying capacity. If we assume a maximum possible population of $M$, then one model for such growth (which has solution a logistic curve) is given by the differential equation

$$ y'=\kappa y (M-y).$$

When $y$ is very small (relative to $M$), this is approximately $y'=\kappa My$. When $y$ is approximately $M$, this is approximately $y'=\kappa M (M-y)$. The first is the equation for exponential growth. The second is Newton's law of cooling. Thus, the logistic curve is interpolating between these two different phenomena.

Unfortunately, for population growth or modeling learning or a host of other applications, you have to care about more than what happens at the extreme ends, as the middle really does matter. For example, for learning (just like with constrained population growth), if you don't know much, you can't learn much, and if you know almost all there is, you can't learn much, but somewhere in the middle is a sweet spot, where you know enough to be able to assimilate a lot (because you have proper context), while at the same time, there are a lot of things still left to learn. If you want to model learning, you want to take into account all 3 of these qualitatively different phases, and not just the last one.

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Thank you for your excellent answer, which I have up-voted and accepted. – Mike Jones Oct 6 '11 at 20:59

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