# Stable oscillations of a predator-prey probabilistic cellular automaton: a mean-field approach

## TL;DR

This paper investigates a probabilistic cellular automaton model of predator-prey dynamics, revealing stable oscillations through mean-field approximations, which are more biologically realistic than classical models.

## Contribution

It introduces a mean-field approach to a predator-prey cellular automaton, demonstrating stable oscillations via a Hopf bifurcation, unlike traditional Lotka-Volterra models.

## Key findings

- Stable population oscillations are identified.
- Oscillations are associated with limit cycles.
- Model is more biologically realistic than Lotka-Volterra.

## Abstract

We analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator-prey system. The individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired on the processes of the Lotka-Volterra model. Two levels of mean-field approximations are set up. The simple approximation is equivalent to an extended patch model, a simple metapopulation model with patches colonized by prey, patches colonized by predators and empty patches. This approximation is capable of describing the limited available space for species occupancy. The pair approximation is moreover able to describe two types of coexistence of prey and predators: one where population densities are constant in time and another displaying self-sustained time-oscillations of the population densities. The oscillations are associated with limit cycles and arise through a Hopf bifurcation. They are stable against changes in the initial conditions and, in this sense, they differ from the Lotka-Volterra cycles which depend on initial conditions. In this respect, the present model is biologically more realistic than the Lotka-Volterra model.