Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model

Abstract

Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see, e.g., B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan ͓Nature 418, 171 ͑2002͔͒ and B. Kirkup and M. A. Riley ͓Nature 428, 412 ͑2004͔͒. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely, the "rock-paper-scissors" or cyclic Lotka-Volterra model. While the deterministic approach ͑rate equations͒ predicts the coexistence of the species resulting in regular ͑yet neutrally stable͒ oscillations of the population densities, we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction: After long enough time, two of the three species die out. As main findings we provide analytic estimates and numerical computation of the extinction probability at a given time. We also discuss the implications of our results for a broad class of competing population systems.

Evolutionary and population dynamics: A coupled approach

“…Differences in birth and death rates Eqs. (13) and (26) are negligible for small population size. The arrest of the dynamics only becomes effective at later times where an increase in population size implies a significantly declining birth rate.…”

“…This comprises, for example, the prisoner's dilemma, the snowdrift game and other games in well-mixed populations [11][12][13]15]. It further ranges from the role of spatial arrangements and network interactions [16][17][18][19][20][21][22][23][24][25] via cyclic dominance [7,[26][27][28][29][30][31][32][33], structured populations [34,35], modified update rules [36,37], multiplayer games [38], and evolutionary algorithms [39] to the influence of internal and external fluctuations [40][41][42][43][44][45]. While these models consider a wide range of evolutionary aspects, they mostly rely on one key assumption, a decoupled, constant population size.…”

From pairwise to group interactions in games of cyclic dominance

“…In experiments with Escherichia coli, for example, it has been shown that arranging the bacteria on a Petri dish is crucial for keeping all three competing strains alive [8,36]. Accordingly, simulations of spatial rock-paper-scissors and related games of cyclic dominance have a long and fruitful history [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56], which is firmly rooted in methods of statistical physics. In general, if the mobility in the population is sufficiently small [7], the spatial rock-paper-scissors game leads to the stable coexistence of all three competing strategies, whereby the coexistence is maintained by the spontaneous formation of complex spatial patterns.…”

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

“…Perhaps suprisingly this includes even long-range processes [16,17], generalizations to 'smart' predators and prey who respectively follow / evade the other species [11,14], and a variation where the predation reaction is split up into two separate and independent processes [23]. Indeed, already a zero-dimensional stochastic predator-prey model exhibits prominent population oscillations driven by the inevitable internal reaction noise [22]; such finitesize fluctuations also drastically affect the properties of, e.g., the three-species cyclic Lotka-Volterra model [25].…”