# Approximate probabilistic cellular automata for the dynamics of single-species populations under discrete logisticlike growth with and without weak Allee effects

## TL;DR

This paper develops and analyzes probabilistic cellular automata models that approximate logistic-like growth for single-species populations, including effects of weak Allee dynamics, providing insights into population behavior through theoretical and numerical methods.

## Contribution

It introduces a six-parameter PCA framework for modeling population dynamics with logistic growth and weak Allee effects, classifies valid models, and explores simplified one-parameter versions.

## Key findings

- Certain PCA models exhibit negative cubic terms indicating weak Allee effects.
- Constraints on transition probabilities ensure realistic population dynamics.
- Numerical simulations demonstrate diverse population behaviors.

## Abstract

We investigate one-dimensional elementary probabilistic cellular automata (PCA) whose dynamics in first-order mean-field approximation yields discrete logisticlike growth models for a single-species unstructured population with nonoverlapping generations. Beginning with a general six-parameter model, we find constraints on the transition probabilities of the PCA that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof. Several possible models display a negative cubic term that can be interpreted as a weak Allee factor. We also investigate the conditions under which a one-parameter PCA derived from the more general six-parameter model can generate valid population growth dynamics. Numerical simulations illustrate the behavior of some of the PCA found.