Three- and four-state rock-paper-scissors games with diffusion.

Cyclic dominance of three species is a commonly occurring interaction dynamics, often denoted the rock-paper-scissors (RPS) game. Such a type of interactions is known to promote species coexistence. Here, we generalize recent results of Reichenbach [Nature (London) 448, 1046 (2007)] of a four-state variant of the RPS game. We show that spiral formation takes place only without a conservation law for the total density. Nevertheless, in general, fast diffusion can destroy species coexistence. We also generalize the four-state model to slightly varying reaction rates. This is shown both analytically and numerically not to change pattern formation, or the effective wavelength of the spirals, and therefore not to alter the qualitative properties of the crossover to extinction.

[1]  T. Reichenbach,et al.  Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games , 2007, Nature.

[2]  Three-state Potts model in combination with the rock-scissors-paper game. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Margaret A. Riley,et al.  Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo , 2004, Nature.

[4]  B. Sinervo,et al.  The rock–paper–scissors game and the evolution of alternative male strategies , 1996, Nature.

[5]  Attila Szolnoki,et al.  Vertex dynamics during domain growth in three-state models. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  R. Austin,et al.  Bacterial metapopulations in nanofabricated landscapes , 2006, Proceedings of the National Academy of Sciences.

[7]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[8]  Tainaka.  Stationary pattern of vortices or strings in biological systems: Lattice version of the Lotka-Volterra model. , 1989, Physical review letters.

[9]  R. Dickman,et al.  Nonequilibrium Phase Transitions in Lattice Models , 1999 .

[10]  G Szabó,et al.  Vortex dynamics in a three-state model under cyclic dominance. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Attila Szolnoki,et al.  Rock-scissors-paper game on regular small-world networks , 2004 .

[12]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[13]  Vortex generation in the RSP game on the triangular lattice , 2008, 0801.4859.

[14]  Attila Szolnoki,et al.  Three-state cyclic voter model extended with Potts energy. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M. Milinski,et al.  Volunteering leads to rock–paper–scissors dynamics in a public goods game , 2003, Nature.

[16]  Vladimir P. Zhdanov,et al.  Monte Carlo simulations of oscillations, chaos and pattern formation in heterogeneous catalytic reactions , 2002 .

[17]  Richard V. Solé,et al.  Self-Organization in Complex Ecosystems. , 2006 .

[18]  M. Hoopes,et al.  Stabilizing effects in spatial parasitoid-host and predator-prey models: a review. , 2004, Theoretical population biology.

[19]  Attila Szolnoki,et al.  Phase transitions for rock-scissors-paper game on different networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Erwin Frey,et al.  Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Tainaka.  Vortices and strings in a model ecosystem. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  P. Schwille,et al.  Spatial Regulators for Bacterial Cell Division Self-Organize into Surface Waves in Vitro , 2008, Science.

[23]  M. Feldman,et al.  Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors , 2002, Nature.

[24]  Arne Traulsen,et al.  Cyclic dominance and biodiversity in well-mixed populations. , 2008, Physical review letters.

[25]  Ricard V. Solé,et al.  Self-Organization in Complex Ecosystems. (MPB-42) , 2006 .