# Solving the Random Pielou Logistic Equation with the Random Variable Transformation Technique: Theory and Applications

## TL;DR

This paper introduces a probabilistic approach to the Pielou logistic equation by randomizing it and applying the transformation of random variables, providing a full stochastic description and demonstrating its application through examples.

## Contribution

It develops a novel probabilistic framework for the randomized Pielou logistic model using the transformation of random variables, including real data application.

## Key findings

- Derived probability density functions for the stochastic process and steady state.
- Provided a full probabilistic description of the model.
- Demonstrated applications with numerical and real data examples.

## Abstract

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.