> 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.

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.