Coexistence in a one-dimensional cyclic dominance process

Abstract

Cyclic ͑rock-paper-scissors-type͒ population models serve to mimic complex species interactions. Focusing on a paradigmatic three-species model with mutations in one dimension, we observe an interplay between equilibrium and nonequilibrium processes in the stationary state. We exploit these insights to obtain asymptotically exact descriptions of the emerging reactive steady state in the regimes of high and low mutation rates. The results are compared to stochastic lattice simulations. Our methods and findings are potentially relevant for the spatiotemporal evolution of other nonequilibrium stochastic processes. Stochastic interacting particle systems are a fruitful testing ground for understanding generic principles in nonequilibrium dynamics. Unfortunately, the treatment of such processes is marred by the absence of detailed balance so that the insight one has gained by analytical means is not yet satisfactory and only few systems have been solved exactly ͓1,2͔. Some of them serve as a paradigm for very complex biological and sociological systems. An example is the contact process, which describes the outbreak of an epidemic, displaying a phase transition from an absorbing to an active state as the rate of infection is increased ͓3͔. Another famous process is the voter model, caricaturing opinion making. It is proven rigorously that on a regular lattice there is a stationary state where the two "opinions" coexist, so long as the dimension is larger than two, such that the random walk is not recurrent ͓2,4͔. Extensive studies have also been conducted on the coarsening dynamics of coalescing or annihilating particles, both for diffusional motion and ballistic motion of the particles ͓5-8͔. In this context, much work was devoted to the long time behavior of the average domain size, which as a function of time typically displays scaling.Frachebourg et al. ͓6,7͔ studied the coarsening dynamics of a model known as the rock-paper-scissors ͑RPS͒ game, one of the most widely studied prototype models for biodiversity ͓9-11͔, displaying cyclic dominance between its three agents. In this Rapid Communication we study the influence of mutations on this model. An integral part of evolution, mutations have been posited to promote biodiversity in microbial communities ͓12͔. We will argue that the RPS is a natural framework for a nonequilibrium version of the IsingGlauber model, which at zero temperature amounts to an annihilating random walk. While previous studies have addressed coarsening and the transition to an absorbing state, we focus on the description of the stationary reactive state at finite "temperature," i.e., interfaces between domains are created at finite mutation rate. In the Ising-Glauber model the interfaces perform a random walk, whereas for the RPS they drift left or right and even move ballistically in a certain regime. Since the coarsening dynamics is counteracted by the creation of interfaces, the system evolves into a nontrivial stationary state. For very large and very low mutation rates, equilibrium tu...

Evolutionary game theory: Theoretical concepts and applications to microbial communities

“…For a one-dimensional version of the the rock-paper-scissors game Frachebourg and collaborators [134,135] have found that starting from some random distribution, the species organize into domains that undergo (power law) coarsening until finally one species takes over the whole lattice. Including mutation the coarsening process is counteracted and by an interesting interplay between equilibrium and non-equilibrium processes a reactive stationary state emerges [136]. The list of interesting examples, of course, continues and one may hope that in the future there will be an even more fruitful interaction between biologically relevant processes and basic research in non-equilibrium dynamics.…”

Spirals and coarsening patterns in the competition of many species: a complex Ginzburg–Landau approach

“…The simplest few species models, like the three-species rock-paper-scissors model or its four species variant, have been the subject of a range of in-depth studies that have unveiled many generic properties of systems with cyclic competition [6,7,9,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,…”

Cyclic competition of four species: domains and interfaces

“…Recent years saw a flurry of studies of systems composed of multiple species that interact in a cyclic way. Most of these studies focused on the case of three species [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39], a special situation where every species interact with every other species. Only rather few papers, however, dealt with more realistic cases where a given species interacts with only a subgroup of all species living in the same ecological environment [6,8,9,29,40,41,42,43,44,45,46,47,48,50,51,…”