Intrinsic noise and two-dimensional maps: Quasicycles, quasiperiodicity, and chaos

Authored by Alan J McKane, Cesar Parra-Rojas, Joseph D Challenger, Duccio Fanelli

Date Published: 2014

DOI: 10.1103/physreve.90.032135

Sponsors: Italian Ministries Chilean National Commission for Scientific and Technological Research (CONICYT)

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

Abstract

We develop a formalism to describe the discrete-time dynamics of systems containing an arbitrary number of interacting species. The individual-based model, which forms our starting point, is described by a Markov chain, which in the limit of large system sizes is shown to be very well-approximated by a Fokker-Planck-like equation, or equivalently by a set of stochastic difference equations. This formalism is applied to the specific case of two species: one predator species and its prey species. Quasicycles, stochastic cycles sustained and amplified by the demographic noise, previously found in continuous-time predator-prey models are shown to exist, and their behavior predicted from a linear noise analysis is shown to be in very good agreement with simulations. The effects of the noise on other attractors in the corresponding deterministic map, such as periodic cycles, quasiperiodicity, and chaos, are also investigated.

Tags

Complexity stability Fluctuations Populations Ecological models Predator-prey models Direct dynamical test 2-species competition Deterministic chaos