Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process

TL;DR

This paper introduces a new deterministic growth model inspired by Gompertz and Korf laws, analyzes its properties, and explores a stochastic birth-death process counterpart, including transition probabilities and extinction probabilities.

Contribution

The paper presents a novel growth model combining features of Gompertz and Korf laws and studies its stochastic birth-death process equivalent.

Findings

The deterministic model captures key growth features like inflection point and lag time.

The stochastic process's transition probabilities and extinction probabilities are derived.

The special case of a simple birth process provides insights into the model's behavior.

Abstract

We propose a new deterministic growth model which captures certain features of both the Gompertz and Korf laws. We investigate its main properties, with special attention to the correction factor, the relative growth rate, the inflection point, the maximum specific growth rate, the lag time and the threshold crossing problem. Some data analytic examples and their performance are also considered. Furthermore, we study a stochastic counterpart of the proposed model, that is a linear time-inhomogeneous birth-death process whose mean behaves as the deterministic one. We obtain the transition probabilities, the moments and the population ultimate extinction probability for this process. We finally treat the special case of a simple birth process, which better mimics the proposed growth model.