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

TL;DR

This paper analytically solves various one-species population models, demonstrating a universal ratio through scaling functions that is independent of initial conditions and parameters, and explores the impact of effort rates on extinction and survival.

Contribution

It introduces a general analytical framework for scaling functions in population models, revealing universality and the effects of effort rates on dynamics.

Findings

The ratio of population models with different parameters is universal.

Effort rates influence the transition between extinction and survival.

Analytical solutions for models with finite and infinite carrying capacities are provided.

Abstract

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.