A probabilistic cellular automata model for the dynamics of a population driven by logistic growth and weak Allee effect

TL;DR

This paper introduces a probabilistic cellular automata model for population dynamics that incorporates logistic growth and weak Allee effects, revealing phase transitions in population survival.

Contribution

It develops a novel probabilistic cellular automata framework that captures complex population behaviors including phase transitions and critical phenomena.

Findings

The model exhibits a phase transition in the directed percolation universality class.

The mean field approximation yields a cubic map with logistic and Allee effects.

A critical probability separates population extinction and survival phases.

Abstract

We propose and investigate a one-parameter probabilistic mixture of one-dimensional elementary cellular automata under the guise of a model for the dynamics of a single-species unstructured population with nonoverlapping generations in which individuals have smaller probability of reproducing and surviving in a crowded neighbourhood but also suffer from isolation and dispersal. Remarkably, the first-order mean field approximation to the dynamics of the model yields a cubic map containing terms representing both logistic and weak Allee effects. The model has a single absorbing state devoid of individuals, but depending on the reproduction and survival probabilities can achieve a stable population. We determine the critical probability separating these two phases and find that the phase transition between them is in the directed percolation universality class of critical behaviour.