Summary: This paper provides a two-fold generalization of the logistic population dynamics to a nonautonomous context. First, it is assumed the carrying capacity alone pulses the population behavior changing logistically on its own. In such a way, we get again the model of [P. S. Meyer and J. H. Ausubel, Technological Forecasting and Social Change 61, 209–214 (1999)], numerically computed by them, and we solve it completely through the Gauss hypergeometric function. Furthermore, both the carrying capacity and net growth rate are assumed to change simultaneously following two independent logistic dynamics. The population dynamics is then found in closed form through a more difficult integration, involving a (\(\tau 1, \tau 2\)) extension of the Appell generalized hypergeometric function [A. H. Al-Shammery and S. L. Kalla, Rev. Acad. Canar. Cienc. 12, No. 1–2, 189–196 (2000; Zbl 0986.33008)]; a new analytic continuation theorem has been proved about such an extension.