Earlier quoted context omitted.
A hockey stick is the initial part of an S-curve. A point to consider is that all complex systems are functions of multiple variables and more often than not, non-linear. A complete view of the past and the present is not a guarantee of the future. At best, it can be a guide for having cautious behavior.
(It's a day ending in y, so here I go again...) It depends on the sign of the exponent. Negative exponential growth, as we see in wild animal populations, is not guaranteed to turn into an S-curve. As the passenger pigeon would attest, were any of them left. Meanwhile, to consider a positive exponent case, there's a real question of what will actually cause the bend in the curve of human population. Will it be gettin…
Sigmoid functions are defined over real input values (not integers). If you were to model the decline in (integer) numbers of passenger pigeons with reals, you would get a sigmoid function were the number at the end approximates, but never reaches zero.