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A theoretical approach to understand spatial organization in complex ecologies. (English) Zbl 1343.92549

Summary: Predicting the fate of ecologies is a daunting, albeit extremely important, task. As part of this task one needs to develop an understanding of the organization, hierarchies, and correlations among the species forming the ecology. Focusing on complex food networks we present a theoretical method that allows to achieve this understanding. Starting from the adjacency matrix the method derives specific matrices that encode the various inter-species relationships. The full potential of the method is achieved in a spatial setting where one obtains detailed predictions for the emerging space-time patterns. For a variety of cases these theoretical predictions are verified through numerical simulations.

MSC:

92D40 Ecology
92D25 Population dynamics (general)
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[1] Adamson, M. W.; Morozov, A. Y., Revising the role of species mobility in maintaining biodiversity in communities with cyclic competition, Bull. Math. Biol., 74, 2004-2031 (2012) · Zbl 1329.92097
[2] Avelino, P. P.; Bazeia, D.; Losano, L.; Menezes, J., von Neumann’s and related scaling laws in rock-paper-scissors-type games, Phys. Rev. E, 86, 031119 (2012)
[3] Avelino, P. P.; Bazeia, D.; Losano, L.; Menezes, J.; Oliveira, B. F., Junctions and spiral patterns in generalized rock-paper-scissors models, Phys. Rev. E, 86, 036112 (2012)
[4] Avelino, P. P.; Bazeia, D.; Menezes, J.; Oliveira, B. F., String networks in \(Z_N\) Lotka-Volterra competition models, Phys. Lett. A, 378, 393-397 (2014) · Zbl 1396.92063
[5] Avelino, P. P.; Bazeia, D.; Losano, L.; Menezes, J.; Oliveira, B. F., Interfaces with internal structures in generalized rock-paper-scissors models, Phys. Rev. E, 89, 042710 (2014)
[6] Cheng, H.; Yao, N.; Huang, Z. G.; Park, J.; Do, Y.; Lai, Y.-C., Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions, Sci. Rep., 4, 7486 (2014)
[7] Daly, A. J.; Baetens, J. M.; De Baets, B., The impact of initial evenness on biodiversity maintenance for a four-species in silico bacterial community, J. Theor. Biol., 387, 189-205 (2015) · Zbl 1343.92536
[8] Dobrinevski, A.; Alava, M.; Reichenbach, T.; Frey, E., Mobility-dependent selection of competing strategy associations, Phys. Rev. E, 89, 012721 (2014)
[9] Frey, E., Evolutionary game theorytheoretical concepts and applications to microbial communities, Physica A, 389, 4265-4298 (2010) · Zbl 1225.91010
[10] Gokhale, C. S.; Traulsen, A., Evolutionary multiplayer games, Dyn. Games Appl., 4, 468-488 (2014) · Zbl 1314.91033
[11] Grošelj, D.; Jenko, F.; Frey, E., How turbulence regulates biodiversity in systems with cyclic competition, Phys. Rev. E, 91, 033009 (2015)
[12] Guisoni, N. C.; Loscar, E. S.; Girardi, M., Phase diagram of a cyclic predator-prey model with neutral-pair exchange, Phys. Rev. E, 88, 022133 (2013)
[13] Hauert, C.; De Monte, S.; Hofbauer, J.; Sigmund, K., Volunteering as red queen mechanism for cooperation in public goods game, Science, 296, 1129-1132 (2002)
[14] He, Q.; Mobilia, M.; Täuber, U. C., Spatial rock-paper-scissors models with inhomogeneous reaction rates, Phys. Rev. E, 82, 051909 (2010)
[15] He, Q.; Mobilia, M.; Täuber, U. C., Coexistence in the two-dimensional may-leonard model with random rates, Eur. Phys. J. B, 82, 97-105 (2011)
[16] He, Q.; Täuber, U. C.; Zia, R. K.P., On the relationship between cyclic and hierarchical three-species predator-prey systems and the two-species Lotka-Volterra model, Eur. Phys. J. B, 85, 141 (2012) · Zbl 1515.92056
[17] Hofbauer, J.; Sigmund, K., Evolutionary Games and Population Dynamics (1998), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0914.90287
[18] Intoy, B.; Pleimling, M., Extinction in four species cyclic competition, J. Stat. Mech., P08011 (2013) · Zbl 1456.92119
[19] Intoy, B.; Pleimling, M., Synchronization and extinction in cyclic games with mixed strategies, Phys. Rev. E, 91, 052135 (2015)
[20] Jiang, L. L.; Zhou, T.; Perc, M.; Wang, B. H., Effects of competition on pattern formation in the rock-paper-scissors game, Phys. Rev. E, 84, 021912 (2011)
[21] Jiang, L.-L.; Wang, W.-X.; Lai, Y.-C.; Ni, X., Multi-armed spirals and multi-pairs antispirals in spatial rock-paper-scissors games, Phys. Lett. A, 376, 2292-2297 (2012) · Zbl 1266.91008
[22] Juul, J.; Sneppen, K.; Mathiesen, J., Clonal selection prevents tragedy of the commons when neighbors compete in a rock-paper-scissors game, Phys. Rev. E, 85, 061924 (2012)
[23] Juul, J.; Sneppen, K.; Mathiesen, J., Labyrinthine clustering in a spatial rock-paper-scissors ecosystem, Phys. Rev. E, 87, 042702 (2013)
[24] Kang, Y. B.; Pan, Q. H.; Wang, X. T.; Me, H. F., A golden point rule in rock-paper-scissors-lizard-spock game, Physica A, 392, 2652-2659 (2013) · Zbl 1402.91045
[25] Knebel, J.; Krüger, T.; Weber, M. F.; Frey, E., Coexistence and survival in conservative Lotka-Volterra networks, Phys. Rev. Lett., 110, 168106 (2013)
[26] Lamouroux, D.; Eule, S.; Geisel, T.; Nagler, J., Discriminating the effects of spatial extent and population size in cyclic competition among species, Phys. Rev. E, 86, 021911 (2012)
[27] Lütz, A. F.; Risau-Gusman, S.; Arenzon, J. J., Intransitivity and coexistence in four species cyclic games, J. Theor. Biol., 317, 286-292 (2013) · Zbl 1368.92150
[28] May, R. M., Stability and Complexity in Model Ecosystems (1974), Cambridge University Press: Cambridge University Press Cambridge, England
[29] Maynard Smith, J., Models in Ecology (1974), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0312.92001
[30] Maynard Smith, J., Evolution and the Theory of Games (1982), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0526.90102
[31] Mowlaei, S.; Roman, A.; Pleimling, M., Spirals and coarsening patterns in the competition of many speciesa complex Ginzburg-Landau approach, J. Phys. A: Math. Theor., 47, 165001 (2014) · Zbl 1291.92092
[33] Nowak, M. A., Evolutionary Dynamics (2006), Belknap Press: Belknap Press Cambridge, MA · Zbl 1098.92051
[34] Peltomäki, M.; Alava, M., Three- and four-state rock-paper-scissors games with diffusion, Phys. Rev. E, 78, 031906 (2008)
[35] Perc, M.; Szolnoki, A.; Szabó, G., Cyclical interactions with alliance-specific heterogeneous invasion rates, Phys. Rev. E, 75, 052102 (2007)
[36] Provata, A.; Nicolis, G.; Baras, F., Oscillatory dynamics in low-dimensional supportsa lattice Lotka-Volterra model, J. Chem. Phys., 110, 8361-8368 (1999)
[37] Reichenbach, T.; Mobilia, M.; Frey, E., Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games, Nature, 448, 1046-1049 (2007)
[38] Reichenbach, T.; Mobilia, M.; Frey, E., Noise and correlations in a spatial population model with cyclic competition, Phys. Rev. Lett., 99, 238105 (2007)
[39] Reichenbach, T.; Frey, E., Instability of spatial patterns and its ambiguous impact on species diversity, Phys. Rev. Lett., 101, 058102 (2008)
[40] Roman, A.; Konrad, D.; Pleimling, M., Cyclic competition of four speciesdomains and interfaces, J. Stat. Mech., P07014 (2012)
[41] Roman, A.; Dasgupta, D.; Pleimling, M., Interplay between partnership formation and competition in generalized May-Leonard games, Phys. Rev. E, 87, 032148 (2013)
[42] Rulands, S.; Reichenbach, T.; Frey, E., Threefold way to extinction in populations of cyclically competing species, J. Stat. Mech., L01003 (2011)
[43] Rulands, S.; Zielinski, A.; Frey, E., Global attractors and extinction dynamics of cyclically competing species, Phys. Rev. E, 87, 052710 (2013)
[44] Rulquin, C.; Arenzon, J. J., Globally synchronized oscillations in complex cyclic games, Phys. Rev. E, 89, 032133 (2014)
[45] Shi, H.; Wang, W.-X.; Yang, R.; Lai, Y.-C., Basins of attraction for species extinction and coexistence in spatial rock-paper-scissors games, Phys. Rev. E, 81, 030901(R) (2010)
[46] Schreiber, S. J.; Killingback, T. P., Spatial heterogeneity promotes coexistence of rock-paper-scissors metacommunities, Theor. Popul. Biol., 86, 1-11 (2013) · Zbl 1296.92219
[47] Sole, R. V.; Basecompte, J., Self-Organization in Complex Ecosystems (2006), Princeton University Press: Princeton University Press Princeton, NJ
[48] Szabó, G.; Czárán, T., Phase transition in a spatial Lotka-Volterra model, Phys. Rev. E, 63, 061904 (2001)
[49] Szabó, G.; Czárán, T., Defensive alliances in spatial models of cyclical population interactions, Phys. Rev. E, 64, 042902 (2001)
[50] Szabó, G.; Sznaider, G. A., Phase transition and selection in a four-species cyclic predator-prey model, Phys. Rev. E, 69, 031911 (2004)
[51] Szabó, G., Competing associations in six-species predator-prey models, J. Phys. A: Math. Gen., 38, 6689-6702 (2005) · Zbl 1069.92030
[52] Szabó, G.; Fáth, G., Evolutionary games on graphs, Phys. Rep., 446, 97-216 (2007)
[53] Szabó, G.; Szolnoki, A., Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species, Phys. Rev. E, 77, 011906 (2008)
[54] Szabó, G.; Szolnoki, A.; Sznaider, G. A., Segregation process and phase transition in cyclic predator-prey models with an even number of species, Phys. Rev. E, 76, 051921 (2007)
[55] Szabó, P.; Czárán, T.; Szabó, G., Competing associations in bacterial warfare with two toxins, J. Theor. Biol., 248, 736-744 (2007) · Zbl 1451.92204
[56] Szabó, G.; Szolnoki, A.; Borsos, I., Self-organizing patterns maintained by competing associations in a six-species predator-prey model, Phys. Rev. E, 77, 041919 (2008)
[57] Szabó, G.; Bodó, K. S.; Allen, B.; Nowak, M. A., Four classes of interactions for evolutionary games, Phys. Rev. E, 92, 022820 (2015)
[58] Szczesny, B.; Mobilia, M.; Rucklidge, A. M., When does cyclic dominance lead to stable spiral waves?, Europhys. Lett., 102, 28012 (2013)
[59] Szczesny, B.; Mobilia, M.; Rucklidge, A. M., Characterization of spiraling patterns in spatial rock-paper-scissors games, Phys. Rev. E, 90, 032704 (2014)
[60] Szolnoki, A.; Perc, M., Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers, New J. Phys., 17, 113033 (2015)
[61] Vandermeer, J.; Yitbarek, S., Self-organized spatial pattern determines biodiversity in spatial competition, J. Theor. Biol., 300, 48-56 (2012) · Zbl 1397.92742
[62] Varga, L.; Vukov, J.; Szabó, G., Self-organizing patterns in an evolutionary rock-paper-scissors game for stochastic synchronized strategy updates, Phys. Rev. E, 90, 042920 (2014)
[63] Venkat, S.; Pleimling, M., Mobility and asymmetry effects in one-dimensional rock-paper-scissors games, Phys. Rev. E, 81, 021917 (2010)
[64] Vukov, J.; Szolnoki, A.; Szabó, G., Diverging fluctuations in a spatial five-species cyclic dominance game, Phys. Rev. E, 88, 022123 (2013)
[65] Wang, W.-X.; Lai, Y.-C.; Grebogi, C., Effect of epidemic spreading on species coexistence in spatial rock-paper-scissors games, Phys. Rev. E, 81, 046113 (2010)
[68] Winkler, A. A.; Reichenbach, T.; Frey, E., Coexistence in a one-dimensional cyclic dominance process, Phys. Rev. E, 81, 060901(R) (2010)
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