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

Authored by Tania Tome, Carvalho Kelly C de

Date Published: 2007

DOI: 10.1088/1751-8113/40/43/005

Sponsors: Brazilian National Council for Scientific and Technological Development (CNPq)

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

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 by 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.

Tags

Simulation individual-based models Dynamics Coexistence population systems Discrete Pattern-formation Threshold Lotka-volterra model