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

Mowlaei, Shahir; Roman, Ahmed; Pleimling, Michel

doi:10.1088/1751-8113/47/16/165001

Cited by 26 publications

(32 citation statements)

References 83 publications

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“…This treatment was then extended to the case of dominance-removal and dominancereplacement cyclic competitions with linear mobility and no mutations (p, z, γ > 0, µ = 0) [46,143]. Recently, a class of RPS-like models with more than three species (with dominance-removal and hopping but no mutations) has also been considered by generalizing this approach [151].…”

Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%

Cyclic dominance in evolutionary games: a review

Szolnoki

Mobilia

Jiang

et al. 2014

J. R. Soc. Interface.

447

338

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Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rockpaper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

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Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%

Cyclic dominance in evolutionary games: a review

Szolnoki

Mobilia

Jiang

et al. 2014

J. R. Soc. Interface.

447

338

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“…[15] for a variant of the model considered here with only dominance-removal competition (ζ = μ = 0 and δ D = δ E ). The treatment was then extended to also include dominance-replacement competition (with μ = 0 and δ D = δ E ) [ 17,23] and has recently been generalized to more than three species [38]. In all these works, the derivation of the CGLE relies on the fact that the underlying mean-field dynamics quickly settles on a two-dimensional manifold on which the flows approach the absorbing boundaries forming heteroclinic cycles [13,22].…”

Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%

“…[15,17,18,23,38]. In particular, the functional dependence of the CGLE parameter (7) differs from that used in Refs.…”

Section: Complex Ginzburg-landau Equationmentioning

confidence: 99%

Characterization of spiraling patterns in spatial rock-paper-scissors games

2014

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The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.

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“…In agreement with the impressive implications fundamental research on cyclical interactions has, it is little surprising that the classical rock-paper-scissors game-the workhorse for research on cyclic dominance-has been studied so extensively, not least by methods of statistical physics [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58], which are indispensable for a comprehensive treatment of the game and its extensions in structured populations. Although the rules of the game can be written down in a short sentence, the complexity of spatial patterns that emerge spontaneously as a consequence of the simple microscopic rules is unparalleled.…”

Section: Introductionmentioning

confidence: 96%

Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers

Szolnoki

Perc

2015

New J. Phys.

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If rock beats scissors and scissors beat paper, one might assume that rock beats paper too. But this is not the case for intransitive relationships that make up the famous rock-paper-scissors game. However, the sole presence of paper might prevent rock from beating scissors, simply because paper beats rock. This is the blueprint for the rock-paper-scissors game with protection spillovers, which has recently been introduced as a new paradigm for biodiversity in well-mixed microbial populations.Here we study the game in structured populations, demonstrating that protection spillovers give rise to spatial patterns that are impossible to observe in the classical rock-paper-scissors game. We show that the spatiotemporal dynamics of the system is determined by the density of stable vortices, which may ultimately transform to frozen states, to propagating waves, or to target waves with reversed propagation direction, depending further on the degree and type of randomness in the interactions among the species. If vortices are rare, the fixation to waves and complex oscillatory solutions is likelier. Moreover, annealed randomness in interactions favors the emergence of target waves, while quenched randomness favors collective synchronization. Our results demonstrate that protection spillovers may fundamentally change the dynamics of cyclic dominance in structured populations, and they outline the possibility of programming pattern formation in microbial populations.

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

Cited by 26 publications

References 83 publications

Cyclic dominance in evolutionary games: a review

Cyclic dominance in evolutionary games: a review

Characterization of spiraling patterns in spatial rock-paper-scissors games

Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers

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