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

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.

arXiv comment21 pages, 12 figures

References in corpus (6)

• Predator-prey cycles from resonant amplification of demographic stochasticity
• Markovian Dynamics on Complex Reaction Networks
• Limit cycles, complex Floquet multipliers and intrinsic noise
• Enhanced stochastic oscillations in autocatalytic reactions
• Intrinsic noise and discrete-time processes
• The theory of individual based discrete-time processes

Cited by in corpus (6)

• Stochastic Maps, Continuous Approximation and Stable Distribution
• Network stochastic resonance under higher-order interactions
• Critical fluctuations of noisy period-doubling maps
• Modeling of Sensory Characteristics Based on the Growth of Food Spoilage Bacteria
• Suppressing escape events in maps of the unit interval with demographic noise
• Stochastic quasi-cycles as a simple explanation for the time evolution of the Cape Rodney-Okakari Point Marine ecological reserve