Generalized exponential function and discrete growth models

TL;DR

This paper introduces a one-parameter generalization of the exponential function that unifies various discrete population models, providing a new physical interpretation and analyzing properties of the resulting models.

Contribution

It presents a novel exponential generalization that unifies multiple population dynamics models and extends the Richards and Ricker models with new analytical insights.

Findings

Unified population models via a new exponential generalization

Derived fixed points and stability conditions for generalized models

Connected scramble and contest models through a single framework

Abstract

Here we show that a particular one-parameter generalization of the exponential function is suitable to unify most of the popular one-species discrete population dynamics models into a simple formula. A physical interpretation is given to this new introduced parameter in the context of the continuous Richards model, which remains valid for the discrete case. From the discretization of the continuous Richards' model (generalization of the Gompertz and Verhuslt models), one obtains a generalized logistic map and we briefly study its properties. Notice, however that the physical interpretation for the introduced parameter persists valid for the discrete case. Next, we generalize the (scramble competition) $\theta$-Ricker discrete model and analytically calculate the fixed points as well as their stability. In contrast to previous generalizations, from the generalized $\theta$-Ricker model one is able to retrieve either scramble or contest models.