Predator-prey quasicycles from a path-integral formalism

Abstract

The existence of beyond mean-field quasicycle oscillations in a simple spatial model of predator-prey interactions is derived from a path-integral formalism. The results agree substantially with those obtained from analysis of similar models using system size expansions of the master equation. In all of these analyses, the discrete nature of predator-prey populations and finite-size effects lead to persistent oscillations in time, but spatial patterns fail to form. The path-integral formalism goes beyond mean-field theory and provides a focus on individual realizations of the stochastic time evolution of population not captured in the standard master-equation approach.

Robust ecological pattern formation induced by demographic noise

“…To analyze this Lagrangian directly is difficult, due to exponential terms and diffusive noise. To make progress, we derive a systematic expansion and mean field theory (MFT) in powers of √ V motivated by the Ω-expansion [18,29]. We assume the formŝ…”

“…Surprisingly, however, systematic treatments of individual-level models (ILMs) of predator-prey dynamics show that the population fluctuations become amplified [15], and lead to time-dependent oscillations (quasi-cycles) that can be distinguished from deterministic limit cycle behavior [16]. Disappointingly, to date, no novel spatial effects of demographic noise have been identified, despite several attempts [17,18].In this Rapid Communication, we demonstrate that noise-induced pattern formation arises in a simple but biologically-relevant predator-prey model, and show that if it is analyzed as an ILM, patterns occur over a much larger range of ecologically relevant parameters than predicted by MFT, even in the thermodynamic limit. We accomplish this by calculating the phase diagram and power spectrum of the model analytically.…”

“…Surprisingly, however, systematic treatments of individual-level models (ILMs) of predator-prey dynamics show that the population fluctuations become amplified [15], and lead to time-dependent oscillations (quasi-cycles) that can be distinguished from deterministic limit cycle behavior [16]. Disappointingly, to date, no novel spatial effects of demographic noise have been identified, despite several attempts [17,18].…”

Fluctuation-driven Turing patterns

“…Appropriately rescaled [28], the continuum limit and mapping to the functional integral formulation yields the Lagrangian…”

Effect of demographic noise in a phytoplankton-zooplankton model of bloom dynamics

“…Let us therefore partition the domain in volumes of horizontal size ∆x, each one containing instantaneously N P , N Z individuals of the P and Z groups. We can introduce instantaneous concentrations P, Z = Ω −1 P,Z N P,Z , coarse grained at horizontal scale ∆x, with Following the strategy utilized in [33,34,42], we adopt as modelling assumption that the birth and death rates B P,Z and D P,Z at the population scale, reflect birth and death rates at the individual level, conditioned to the instantaneous values of the concentration fields P (x, t) and Z(x, t). An hypothesis of independence and Markovianity of the birth-death events, at scales of interest, underlies this assumption.…”