Scaling Function, Universality and Analytical Solutions of Generalized One-Species Population Dynamics Models

We consider several one-species population dynamics model with finite and infinite carrying capacity, time dependent growth and effort rates and solve them analytically. We show that defining suitable scaling functions for a given time, one is able to demonstrate that their ratio with respect to its initial value is universal. This ratio is independent from the initial condition and from the model parameters. Although the effort rate does not break the model universality it produces a transition between the species extinction and survival. A general formula is furnished to obtain the scaling functions.