Extinction in four species cyclic competition

When four species compete stochastically in a cyclic way, the formation of two teams of mutually neutral partners is observed. In this paper we study through numerical simulations the extinction processes that can take place in this system both in the well mixed case as well as on different types of lattices. The different routes to extinction are revealed by the probability distribution of the domination time, i.e. the time needed for one team to fully occupy the system. If swapping is allowed between neutral partners, then the probability distribution is dominated by very long-lived states where a few very large domains persist, each domain being occupied by a mix of individuals from species that form one of the teams. Many aspects of the possible extinction scenarios are lost when only considering averaged quantities as for example the mean domination time.

[1]  A. Hastings,et al.  Multivariate Moran process with Lotka-Volterra phenomenology. , 2011, Physical review letters.

[2]  Attila Szolnoki,et al.  Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  T. Geisel,et al.  Discriminating the effects of spatial extent and population size in cyclic competition among species. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Erwin Frey,et al.  Extinction in neutrally stable stochastic Lotka-Volterra models. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

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

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

[8]  Kazunori Sato,et al.  Parity law for population dynamics of N-species with cyclic advantage competitions , 1999, Appl. Math. Comput..

[9]  U. Täuber.  Population oscillations in spatial stochastic Lotka–Volterra models: a field-theoretic perturbational analysis , 2012 .

[10]  T. Platkowski,et al.  Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities , 2011 .

[11]  Wen-Xu Wang,et al.  Effect of epidemic spreading on species coexistence in spatial rock-paper-scissors games. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  U. Täuber,et al.  Coexistence in the two-dimensional May-Leonard model with random rates , 2011, 1101.4963.

[13]  Alex Kamenev,et al.  Extinction in the Lotka-Volterra model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  A. Provata,et al.  Fractal properties of the lattice Lotka-Volterra model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  S. Rulands,et al.  Threefold way to extinction in populations of cyclically competing species , 2010, Journal of Statistical Mechanics: Theory and Experiment.

[16]  Andrew G. Glen,et al.  APPL , 2001 .

[17]  Wen-Xu Wang,et al.  Pattern formation, synchronization, and outbreak of biodiversity in cyclically competing games. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Michel Pleimling,et al.  Cyclic competition of four species: domains and interfaces , 2012, 1205.4914.

[19]  R K P Zia,et al.  Saddles, arrows, and spirals: deterministic trajectories in cyclic competition of four species. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Alessandra F. Lütz,et al.  Intransitivity and coexistence in four species cyclic games. , 2013, Journal of theoretical biology.

[21]  U. Täuber,et al.  Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models , 2005, q-bio/0512039.

[22]  Joshua R. Nahum,et al.  Evolution of restraint in a structured rock–paper–scissors community , 2011, Proceedings of the National Academy of Sciences.

[23]  Erwin Frey.  Evolutionary game theory: Theoretical concepts and applications to microbial communities , 2010 .

[24]  Guangcan Yang,et al.  Coevolutionary dynamics with clustering behaviors on cyclic competition , 2012 .

[25]  R. Zia,et al.  Stochastic evolution of four species in cyclic competition: exact and simulation results , 2011, 1205.6394.

[26]  Tao Zhou,et al.  Effects of competition on pattern formation in the rock-paper-scissors game. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  D. Bazeia,et al.  Junctions and spiral patterns in generalized rock-paper-scissors models. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Ben-Naim,et al.  Spatial organization in cyclic Lotka-Volterra systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  THE INFLUENCE OF SPECIES' NUMBER AND THE DENSITY OF VACANT SITES ON THE DEFENSIVE ALLIANCE , 2005 .

[30]  D. Bazeia,et al.  String networks in ZN Lotka–Volterra competition models , 2014 .

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

[32]  G. Szabó,et al.  Competing associations in six-species predator–prey models , 2004, q-bio/0408005.

[33]  D. Bazeia,et al.  von Neummann's and related scaling laws in rock-paper-scissors-type games. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Erwin Frey,et al.  Global attractors and extinction dynamics of cyclically competing species. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[36]  T. Antal,et al.  Fixation of Strategies for an Evolutionary Game in Finite Populations , 2005, Bulletin of mathematical biology.

[37]  Ying-Cheng Lai,et al.  Multi-armed spirals and multi-pairs antispirals in spatial rock-paper-scissors games , 2012 .

[38]  Martin Schottenloher,et al.  Zero-one survival behavior of cyclically competing species. , 2009, Physical review letters.

[39]  Kim Sneppen,et al.  Clonal selection prevents tragedy of the commons when neighbors compete in a rock-paper-scissors game. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  B. M. Fulk.  MATH , 1992 .

[41]  M. W. Adamson,et al.  Revising the Role of Species Mobility in Maintaining Biodiversity in Communities with Cyclic Competition , 2012, Bulletin of Mathematical Biology.

[42]  Attila Szolnoki,et al.  Self-organizing patterns maintained by competing associations in a six-species predator-prey model. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Kim Sneppen,et al.  Labyrinthine clustering in a spatial rock-paper-scissors ecosystem. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[45]  Grégoire Nicolis,et al.  Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model , 1999 .

[46]  György Szabó,et al.  Competing associations in bacterial warfare with two toxins. , 2007, Journal of theoretical biology.

[47]  Mauro Mobilia,et al.  Oscillatory dynamics in rock-paper-scissors games with mutations. , 2009, Journal of theoretical biology.

[48]  Erwin Frey,et al.  The edge of neutral evolution in social dilemmas , 2009, New Journal of Physics.

[49]  P. Krapivsky,et al.  Fixation in a cyclic Lotka-Volterra model , 1998, cond-mat/9801026.

[50]  Erwin Frey,et al.  Instability of spatial patterns and its ambiguous impact on species diversity. , 2008, Physical review letters.

[51]  Matti Peltomäki,et al.  Three- and four-state rock-paper-scissors games with diffusion. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  G Szabó,et al.  Phase transition in a spatial Lotka-Volterra model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Michel Pleimling,et al.  Cyclic competition of four species: Mean-field theory and stochastic evolution , 2010 .

[54]  Erwin Frey,et al.  Coexistence and survival in conservative Lotka-Volterra networks. , 2013, Physical review letters.

[55]  György Szabó,et al.  Phase transition and selection in a four-species cyclic predator-prey model. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Erwin Frey,et al.  Coexistence in a one-dimensional cyclic dominance process. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[58]  Attila Szolnoki,et al.  Segregation process and phase transition in cyclic predator-prey models with an even number of species. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  Erwin Frey,et al.  Entropy production of cyclic population dynamics. , 2010, Physical review letters.

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

[61]  Kei-ichi Tainaka,et al.  Critical Phenomena in Cyclic Ecosystems: Parity Law and Selfstructuring Extinction Pattern , 1997 .

[62]  G Szabó,et al.  Defensive alliances in spatial models of cyclical population interactions. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[63]  Ben-Naim,et al.  Segregation in a One-Dimensional Model of Interacting Species. , 1996, Physical review letters.

[64]  Mauro Mobilia,et al.  When does cyclic dominance lead to stable spiral waves? , 2012, 1210.8376.

[65]  J. Vandermeer,et al.  Self-organized spatial pattern determines biodiversity in spatial competition. , 2012, Journal of theoretical biology.

[66]  Attila Szolnoki,et al.  Cyclical interactions with alliance-specific heterogeneous invasion rates. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  Ying-Cheng Lai,et al.  Basins of attraction for species extinction and coexistence in spatial rock-paper-scissors games. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[68]  S. Jørgensen.  Models in Ecology , 1975 .

[69]  Thilo Gross,et al.  Cyclic dominance in adaptive networks , 2011 .

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

[71]  G. Szabó,et al.  Evolutionary games on graphs , 2006, cond-mat/0607344.

[72]  General Properties of a System of $S$ Species Competing Pairwise , 2010, 1101.0018.

[73]  Mauro Mobilia,et al.  Spatial rock-paper-scissors models with inhomogeneous reaction rates. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[74]  Michel Pleimling,et al.  Mobility and asymmetry effects in one-dimensional rock-paper-scissors games. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[75]  A J McKane,et al.  Predator-prey cycles from resonant amplification of demographic stochasticity. , 2005, Physical review letters.

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

[77]  Mauro Mobilia,et al.  Fluctuations and correlations in lattice models for predator-prey interaction. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[78]  U. Täuber,et al.  On the relationship between cyclic and hierarchical three-species predator-prey systems and the two-species Lotka-Volterra model , 2011, 1111.1674.

[79]  Erwin Frey,et al.  Noise and correlations in a spatial population model with cyclic competition. , 2007, Physical review letters.

[80]  Michel Pleimling,et al.  Interplay between partnership formation and competition in generalized May-Leonard games , 2013, 1303.3139.