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

TL;DR

This paper develops a formalism to analyze stochastic dynamics in multi-species systems, demonstrating how demographic noise influences predator-prey interactions and complex behaviors like quasicycles, quasiperiodicity, and chaos.

Contribution

It introduces a formalism connecting individual-based Markov models to stochastic difference equations for multi-species systems, and applies it to predator-prey dynamics to analyze noise effects.

Findings

Quasi-cycles are confirmed to exist and match predictions from linear noise analysis.

Demographic noise influences deterministic attractors such as periodic cycles, quasiperiodicity, and chaos.

The formalism accurately predicts stochastic behaviors in predator-prey models.

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. Quasi-cycles --- 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.