Networks with dispersed degrees save stable coexistence of species in cyclic competition.

Coexistence of individuals with different species or phenotypes is often found in nature in spite of competition between them. Stable coexistence of multiple types of individuals have implications for maintenance of ecological biodiversity and emergence of altruism in society, to name a few. Various mechanisms of coexistence including spatial structure of populations, heterogeneous individuals, and heterogeneous environments, have been proposed. In reality, individuals disperse and interact on complex networks. We examine how heterogeneous degree distributions of networks influence coexistence, focusing on models of cyclically competing species. We show analytically and numerically that heterogeneity in degree distributions promotes stable coexistence.

[1]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[2]  S. Levin.  Dispersion and Population Interactions , 1974, The American Naturalist.

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

[4]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[5]  J. Connell.  Diversity in tropical rain forests and coral reefs. , 1978, Science.

[6]  N. Shigesada,et al.  Spatial segregation of interacting species. , 1979, Journal of theoretical biology.

[7]  A. Hastings.  Disturbance, coexistence, history, and competition for space , 1980 .

[8]  L W Buss,et al.  Competitive intransitivity and size-frequency distributions of interacting populations. , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[9]  D. Tilman.  Resource competition and community structure. , 1983, Monographs in population biology.

[10]  Herbert W. Hethcote,et al.  Background and Basic Concepts , 1984 .

[11]  W. Hamilton,et al.  The Evolution of Cooperation , 1984 .

[12]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[13]  R. May,et al.  Host–parasitoid associations in patchy environments , 1990, Nature.

[14]  Michael P. Hassell,et al.  Spatial structure and chaos in insect population dynamics , 1991, Nature.

[15]  Allan Stewart-Oaten,et al.  Aggregation and Stability in Metapopulation Models , 1992, The American Naturalist.

[16]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[17]  Kei-ichi Tainaka,et al.  Paradoxical effect in a three-candidate voter model , 1993 .

[18]  R. Durrett,et al.  The Importance of Being Discrete (and Spatial) , 1994 .

[19]  D. Tilman.  Competition and Biodiversity in Spatially Structured Habitats , 1994 .

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

[21]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[22]  R. Durrett,et al.  Spatial aspects of interspecific competition. , 1998, Theoretical population biology.

[23]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[24]  P. Chesson.  Mechanisms of Maintenance of Species Diversity , 2000 .

[25]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[26]  Marcus Frean,et al.  Rock–scissors–paper and the survival of the weakest , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[27]  György Szabó,et al.  Phase transitions and volunteering in spatial public goods games. , 2002, Physical review letters.

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

[29]  C. Hauert,et al.  Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games , 2002, Science.

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

[31]  Ilkka Hanski,et al.  Cyclic Dynamics in a Simple Vertebrate Predator-Prey Community , 2003, Science.

[32]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[33]  Alessandro Vespignani,et al.  Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. , 2003, Physical review letters.

[34]  N. Konno,et al.  Multi-state epidemic processes on complex networks. , 2005, Journal of theoretical biology.