# Does carrying capacity have the ability to act at a distance?

This question concerns the generalized logistic function, discussed in Wikipedia here:

http://en.wikipedia.org/wiki/Generalised_logistic_function

If we take the lower asymptote to be 0, then the upper asymptote is called the carrying capacity. Fine, and for sake of this question, let’s take the lower asymptote to be 0 (not that it matters for the mathematics of this question, but that gives us the convenient handle of “carrying capacity” to use in this question.) The problem is that in the differential form (Y’(t) = …), it appears that the rate of population growth can be increased simply by increasing the carrying capacity. In other words, the distance of the boundary of the constraint is somehow being “telegraphed” to the population. This seems counter-intuitive to me. Until the population starts to feel the effects of the constraint, a relaxing of the constraint should be imperceptible to the population. Obviously, I’m missing something, but what is it?

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The logistics model is an approximation. For values of $y$ far from the carrying capacity, the growth is close to exponential, but it is not exactly exponential; there are extra terms. While those terms don't matter much, they do matter somewhat. The further away you are, the less they matter. – Arturo Magidin Mar 21 '12 at 4:31
In the logistic model, the growth rate is modeled to be proportional to the "room left for growth". This room left for growth is the difference between the carrying capacity and the population. So the model inherently "telegraphs" the carrying capacity down to the population. As Arturo explains, the effect is small though when the room left for growth is large. – alex.jordan Mar 21 '12 at 4:59

Changing the capacity $K$ changes the growth process for the population $P(t)$, not only the rate at which the process takes place. In other words, the differential equation for $P$ is not independent of $K$.