Stochastic population dynamics in spatially extended predator–prey systems

Abstract

Abstract. Spatially extended population dynamics models that incorporate demographic noise serve as case studies for the crucial role of fluctuations and correlations in biological systems. Numerical and analytic tools from non-equilibrium statistical physics capture the stochastic kinetics of these complex interacting manyparticle systems beyond rate equation approximations. Including spatial structure and stochastic noise in models for predator-prey competition invalidates the neutral Lotka-Volterra population cycles. Stochastic models yield long-lived erratic oscillations stemming from a resonant amplification mechanism. Spatially extended predator-prey systems display noise-stabilized activity fronts that generate persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively via a Doi-Peliti field theory mapping of the master equation; related tools allow detailed characterization of extinction pathways. The critical steadystate and non-equilibrium relaxation dynamics at the predator extinction threshold are governed by the directed percolation universality class. Spatial predation rate variability results in more localized clusters, enhancing both competing species' population densities. Affixing variable interaction rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of three-species competition with 'rock-paper-scissors' interactions metaphorically describe cyclic dominance. These models illustrate intimate connections between population dynamics and evolutionary game theory, underscore the role of fluctuations to drive populations toward extinction, and demonstrate how space can support species diversity. Two-dimensional cyclic three-species May-Leonard models are characterized by the emergence of spiraling patterns whose properties are elucidated by a mapping onto a complex GinzburgLandau equation. Multiple-species extensions to general 'food networks' can be classified on the mean-field level, providing both fundamental understanding of ensuing cooperativity and profound insight into the rich spatio-temporal features and coarsening kinetics in the corresponding spatially extended systems. Novel space-time Stochastic population dynamics in spatially extended predator-prey systems 2 patterns emerge as a result of the formation of competing alliances; e.g., coarsening domains that each incorporate rock-paper-scissors competition games.

The effect of habitats and fitness on species coexistence in systems with cyclic dominance

“…a Petri dish) three cyclically competing strains of bacteria coexist for a long time, whereas in the well-mixed environment of shaken flasks two species go extinct quickly. Both theoretical (Reichenbach et al, 2007a(Reichenbach et al, ,b, 2008He et al, 2011;Lamouroux et al, 2012;Rulands et al, 2013;Szczesny et al, 2013Szczesny et al, , 2014Szolnoki et al, 2014;Dobramysl et al, 2018) and experimental (Siegert and Weijer, 1995;Igoshin et al, 2004) studies have revealed that spatio-temporal patterns, as for example propagating fronts or spiral waves, often accompany long-lasting coexistence between different species.…”

Fluctuations and Correlations in Chemical Reaction Kinetics and Population Dynamics

“…The language and also numerical and mathematical tools developed for the investigation of chemical kinetics may be directly transferred to spatially extended stochastic population dynamics. 5,35,36 Aside from the logistic population growth with finite carrying capacity (7), another classical and paradigmatic model in ecology concerns the competition and coexistence of prey with their predator species, as first independently constructed by Lotka and Volterra. 1,3 Let B indicate the prey species, who left on their own merely undergo (asexual) reproduction B → B + B with rate σ.…”

Ecological drift and competitive interactions predict unique patterns in temporal fluctuations of population size

“…Simultaneous interactions within multi-species communities and their effects on temporal population variability have received relatively less attention than pairwise interactions, and have sometimes generated conflicting results (Godoy et al 2017, Grilli et al 2017. Models developed in this context have used a generalized Lotka-Volterra framework of first-order differential equations (Kokkoris et al 2002, Dobramysl et al 2018, or stochastic particle systems (Foley 1994) either with static (e.g., Cattin et al 2004) or dynamic parameters (e.g., Cohen et al 1990). In most of them, long-term stability requires specific parameter combinations and/or variable interaction strength (Taylor 1988, Allesina and Tang 2012, Serv an et al 2018).…”