Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

Abstract

We study the general properties of stochastic two-species models for predator-prey competition and coexistence with Lotka-Volterra type interactions defined on a d-dimensional lattice. Introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical, deterministic mean-field picture. Already within mean-field theory, however, spatial constraints, modeling locally limited resources, lead to the emergence of a continuous active-toabsorbing state phase transition. Field-theoretic arguments, supported by Monte Carlo simulation results, indicate that this transition, which represents an extinction threshold for the predator population, is governed by the directed percolation universality class. In the active state, where predators and prey coexist, the classical center singularities with associated population cycles are replaced by either nodes or foci. In the vicinity of the stable nodes, the system is characterized by essentially stationary localized clusters of predators in a sea of prey. Near the stable foci, however, the stochastic lattice Lotka-Volterra system displays complex, correlated spatio-temporal patterns of competing activity fronts. Correspondingly, the population densities in our numerical simulations turn out to oscillate irregularly in time, with amplitudes that tend to zero in the thermodynamic limit. Yet in finite systems these oscillatory fluctuations are quite persistent, and their features are determined by the intrinsic interaction rates rather than the initial conditions. We emphasize the robustness of this scenario with respect to various model perturbations.

Influence of local carrying capacity restrictions on stochastic predator–prey models

“…This suggests the form C AA (x) ∝ C BB (x) ≈ F e −|x|/ξ , with equal correlation lengths ξ for the predators and prey; in addition, both ξ and the amplitude F appear to depend only weakly on the predation rate. Qualitatively similar (but quantitatively different) to simulations with restriced site occupation numbers (compare figure 4 in [24]), in figure 3(c) we observe local anti-correlations between the A and B species for 0 ≤ x ≤ 2 which are of course caused by the predation reaction. There are pronounced positive correlations up to about 20 lattice constants, which indicates the width of the rather diffuse predator-prey activity fronts seen in figure 2. reduces the amplitude of the population oscillations, as evident in figure 4(b).…”

“…Neither criticism however pertains to stochastic spatial predator-prey models [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], for which the meanfield approximation recovers the original Lotka-Volterra differential equations. In stark contrast with the deterministic Lotka-Volterra rate equations, such stochastic lattice predator-prey models in fact display remarkably robust features (for a recent overview, see [24]): sufficiently deep in the species coexistence phase, the population densities oscillate in an irregular manner, however with characteristic periods and amplitudes that vanish in the thermodynamic limit [13,15,16,17,20,21,23,24]; these erratic oscillations are induced by recurrent activity waves that initially form concentric rings, and upon merging produce complex spatiotemporal structures [11,19,23,24].…”

Renormalization Group: Applications in Statistical Physics

“…Stochastic models in population dynamics and ecology are naturally formulated in a chemical reaction language, and hence amenable to these field-theoretic tools [28,29].…”

“…Stochastic fluctuations as well as reaction-induced noise and correlations are thus crucial ingredients to properly describe the large-scale features of spatially extended Lotka-Volterra systems even and especially far away from the extinction threshold (for recent overviews, see Refs. [28,29]). …”

Field Theoretic Methods

“…[33]): The predator-prey coexistence phase is governed, for sufficiently large values of the predation rate, by an incessant sequence of 'pursuit and evasion' wave fronts that form quite complex dynamical patterns, as depicted in Figure 1, which shows snapshots taken in a two-dimensional lattice Monte Carlo simulation where each site could at most be occupied by a single particle. In finite systems, these correlated structures induce erratic population oscillations whose features are independent of the initial configuration.…”