A Carrying Capacity Periodically Variable in Logistic Growth Dynamics

An endogenous extension of the classical logistic equation, in which the carrying capacity κ(t) varies in a periodic manner, is developed within this study. Existing theoretical results characterize the behavior of models of this nature. Nevertheless, to the best of our knowledge, the extant literature does not seem to include contributions with a comparable pragmatic computational purpose. By leveraging the Bernoulli structure of the model alongside a Fourier representation of the periodic forcing, closed-form expressions for the associated periodic solution are derived, and a unified Fourier-series framework is constructed to reconstruct trajectories under general periodic inputs. This approach refines and extends earlier findings on periodic logistic dynamics, resulting in an explicit characterization of the oscillatory regime induced by κ(t), which influences the system’s nonlinear feedback and long-term behavior.