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Coevolutionary dynamics of a variant of the cyclic Lotka–Volterra model with three-agent interactions

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Abstract

We study a variant of the cyclic Lotka–Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock–Paper–Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two prey plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects turn neutrally stable orbits into erratically diverging trajectories. In particular, we compare analytic predictions for the extinction probability, derived in the Fokker–Planck approximation, with numerical simulations based on the Gillespie stochastic algorithm. We then extend the analysis of the phase portrait to heterogeneous rates. In a well-mixed environment, we observe a continuum of degenerate Hopf bifurcations, generalizing the above one. Neutral stability ensues from a complex equilibrium between different reactions. Remarkably, on a two-dimensional lattice, all bifurcations disappear as a consequence of the spatial locality of the interactions. In the second part of the paper, we investigate the effects of mobility in a lattice metapopulation model with patches hosting several agents. We find that strategies propagate along the arms of rotating spirals, as they usually do in models of cyclic dominance. We observe propagation instabilities in the regime of large wavelengths. We also examine three-agent interactions inducing nonlinear diffusion.“Three at play. That’ll be the day!” (a child in Wings of desire [W. Wenders, 1987])

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References

  1. M.B. Elowitz, S. Leibler, Nature 403, 335 (2000)

    Article  ADS  Google Scholar 

  2. B. Kerr, M.A. Riley, M.W. Feldman, B.J.M. Bohannan, Nature 418, 171 (2002)

    Article  ADS  Google Scholar 

  3. B. Sinervo, C.M. Lively, Nature 380, 240 (1996)

    Article  ADS  Google Scholar 

  4. D.R. Taylor, L.W. Aarssen, Am. Nat. 136, 305 (1990)

    Article  Google Scholar 

  5. D.D. Cameron, A. White, J. Antonovics, J. Ecol. 97, 1311 (2009)

    Article  Google Scholar 

  6. R.A. Lankau, S.Y. Strauss, Science 317, 1561 (2007)

    Article  ADS  Google Scholar 

  7. T. Reichenbach, M. Mobilia, E. Frey, Nature 448, 1046 (2007)

    Article  ADS  Google Scholar 

  8. M. Peltomäki, M. Alava, Phys. Rev. E 78, 031906 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  9. B. Szczesny, M. Mobilia, A.M. Rucklidge, Europhys. Lett. 102, 28012 (2013)

    Article  ADS  Google Scholar 

  10. B. Szczesny, M. Mobilia, A.M. Rucklidge, Phys. Rev. E 90, 032704 (2014)

    Article  ADS  Google Scholar 

  11. S. Rulands, A. Zielinski, E. Frey, Phys. Rev. E 87, 052710 (2013)

    Article  ADS  Google Scholar 

  12. L.L. Jiang, T. Zhou, M. Perc, B.H. Wang, Phys. Rev. E 84, 021912 (2011)

    Article  ADS  Google Scholar 

  13. L.L. Jiang, W.X. Wang, Y.C. Lai, X. Ni, Phys. Lett. A 376, 2292 (2012)

    Article  ADS  Google Scholar 

  14. T. Reichenbach, E. Frey, Phys. Rev. Lett. 101, 058102 (2008)

    Article  ADS  Google Scholar 

  15. T. Reichenbach, M. Mobilia, E. Frey, Phys. Rev. Lett. 99, 238105 (2007)

    Article  ADS  Google Scholar 

  16. H. Shi, W.-X. Wang, R. Yang, Y.-C. Lai, Phys. Rev. E 81, 030901 (2010)

    Article  ADS  Google Scholar 

  17. X. Ni, R. Yang, W.-X. Wang, Y.-C. Lai, C. Grebogi, Chaos 20, 045116 (2010)

    Article  ADS  Google Scholar 

  18. R. Yang, W.-X. Wang, Y.-C. Lai, C. Grebogi, Chaos 20, 023113 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  19. J. Park, Y. Do, B. Jang, Y. Lai, Sci. Rep. 7, 7465 (2017)

    Article  ADS  Google Scholar 

  20. P.P. Avelino, D. Bazeia, L. Losano, J. Menezes, B.F. de Oliveira, M.A. Santos, Phys. Rev. E 97, 032415 (2018)

    Article  ADS  Google Scholar 

  21. A. Szolnoki, M. Mobilia, L.L. Jiang, B. Szczesny, A.M. Rucklidge, M. Perc, J. R. Soc., Interface 11 (2014)

  22. L.D. Mech, L. Boitani (Eds.),Wolves: Behavior, Ecology, and Conservation (University of Chicago Press, Chicago, IL, USA, 2007)

  23. C.B. Stanford,Chimpanzee and Red Colobus: The Ecology of Predator and Prey (Harvard University Press, Cambridge, MA, USA, 2001)

  24. J.B. Samuels,Dolphins: Ecology, Behavior, and Conservation Strategies (Nova Science Publishers, Incorporated, Hauppauge, NY, USA, 2014)

  25. G.B. Schaller,The Serengeti Lion: A Study of Predator–Prey Relations (University of Chicago Press, Chicago,IL, USA, 2009)

  26. B. Hölldobler, E.O. Wilson,The Ants (Belknap Press of Harvard University Press, Cambridge, MA, USA, 1990)

  27. J. Pérez, A. Moraleda-Muñoz, F.J. Marcos-Torres, J. Muñoz Dorado, Environ. Microbiol. 18, 766 (2016)

    Article  Google Scholar 

  28. J.E. Berleman, J.R. Kirby, FEMS Microbiol. Rev. 33, 942 (2009)

    Article  Google Scholar 

  29. J. Muñoz Dorado, F.J. Marcos-Torres, E. García-Bravo, A. Moraleda-Muñoz, J. Pérez, Front. Microbiol. 7, 781 (2016)

    Article  Google Scholar 

  30. E. Jurkevitch, Microbe Mag. 2 (2007)

  31. E.D. Kelsic, J. Zhao, K. Vetsigian, R. Kishony, Nature 521, 516 (2015)

    Article  ADS  Google Scholar 

  32. C.T. Bergstrom, B. Kerr, Nature 521, 431 (2015)

    Article  ADS  Google Scholar 

  33. A. Szolnoki, M. Perc, New J. Phys. 17, 113033 (2015)

    Article  ADS  Google Scholar 

  34. A. Szolnoki, J. Vukov, M. Perc, Phys. Rev. E 89, 062125 (2014)

    Article  ADS  Google Scholar 

  35. H. Cheng, N. Yao, Z.-G. Huang, J. Park, Y. Do, Y.-C. Lai, Sci. Rep. 4, 7486 (2014)

    Article  Google Scholar 

  36. A. Cazaubiel, A.F. Lütz, J.J. Arenzon, J. Theor. Biol. 430, 45 (2017)

    Article  Google Scholar 

  37. A.F. Lütz, A. Cazaubiel, J.J. Arenzon, Games 8 (2017)

  38. T. Toffoli, N. Margolus,Cellular Automata Machines: A New Environment for Modeling (MIT Press, Cambridge, MA, USA, 1987)

  39. C. Lett, P. Auger, J.-M. Gaillard, Theor. Popul. Biol. 65, 263 (2004)

    Article  Google Scholar 

  40. T. Reichenbach, M. Mobilia, E. Frey, Phys. Rev. E 74, 051907 (2006)

    Article  ADS  Google Scholar 

  41. L. Frachebourg, P.L. Krapivsky, E. Ben-Naim, Phys. Rev. E 54, 6186 (1996)

    Article  ADS  Google Scholar 

  42. L. Frachebourg, P.L. Krapivsky, E. Ben-Naim, Phys. Rev. Lett. 77, 2125 (1996)

    Article  ADS  Google Scholar 

  43. L. Frachebourg, P.L. Krapivsky, J. Phys. A: Math. Gen. 31, L287 (1998)

    Article  ADS  Google Scholar 

  44. A. Provata, G. Nicolis, F. Baras, J. Chem. Phys. 110, 8361 (1999)

    Article  ADS  Google Scholar 

  45. G.A. Tsekouras, A. Provata, Phys. Rev. E 65, 016204 (2001)

    Article  ADS  Google Scholar 

  46. G. Szabó, T. Czárán, Phys. Rev. E 63, 061904 (2001)

    Article  ADS  Google Scholar 

  47. G. Szabó, G. Arial Sznaider, Phys. Rev. E 69, 031911 (2004)

    Article  ADS  Google Scholar 

  48. A. Szolnoki, M. Perc, Phys. Rev. E 93, 062307 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  49. A. Szolnoki, M. Perc, Sci. Rep. 6, 38608 (2016)

    Article  ADS  Google Scholar 

  50. D. Bazeia, B.F. de Oliveira, A. Szolnoki, EPL 124, 68001 (2018)

    Article  ADS  Google Scholar 

  51. D. Bazeia, B.F. de Oliveira, A. Szolnoki, Phys. Rev. E 98, 052408 (2019)

    Article  ADS  Google Scholar 

  52. D. Walker, G. Walker,The Official Rock Paper Scissors Strategy Guide (Simon & Schuster, New York, NY, USA, 2004)

  53. S. Wiggins,Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer Science & Business Media, New York, NY, USA, 2013)

  54. S.H. Strogatz,Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, Boca Raton, FL, USA, 2018)

  55. R. May, W. Leonard, SIAM J. Appl. Math. 29, 243 (1975)

    Article  MathSciNet  Google Scholar 

  56. N.G. Van KampenStochastic Processes in Physics and Chemistry (North-Holland Publishing Company, Amsterdam, NL, 1983)

  57. D.T. Gillespie, J. Comput. Phys. 22, 403 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  58. D.T. Gillespie, J. Phys. Chem. 81, 2340 (1977)

    Article  Google Scholar 

  59. M. Berr, T. Reichenbach, M. Schottenloher, E. Frey, Phys. Rev. Lett. 102, 048102 (2009)

    Article  ADS  Google Scholar 

  60. M. Mobilia, J. Theor. Biol. 264, 1 (2010)

    Article  Google Scholar 

  61. A.J. McKane, T.J. Newman, Phys. Rev. E 70, 041902 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  62. C.A. Lugo, A.J. McKane, Phys. Rev. E 78, 051911 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  63. D. Lamouroux, S. Eule, T. Geisel, J. Nagler, Phys. Rev. E 86, 021911 (2012)

    Article  ADS  Google Scholar 

  64. B. Szczesny, Coevolutionary dynamics in structured populations of three species, Ph.D. Thesis, 2014

  65. S.M. Cox, P.C. Matthews, J. Comput. Phys. 176, 430 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  66. E. Frey, Physica A 389, 4265 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  67. W. van Saarloos, Phys. Rep. 386, 29 (2003)

    Article  ADS  Google Scholar 

  68. B.H. Gilding, Differ. Integral Equ. 9, 919 (1996)

    MathSciNet  Google Scholar 

  69. B.H. Gilding, R. Kersner, J. Differ. Equ. 124, 27 (1996)

    Article  ADS  Google Scholar 

  70. B.H. Gilding, H. Kersner, J. Phys. A: Math. Gen. 38, 3367 (2005)

    Article  ADS  Google Scholar 

  71. B.H. Gilding, H. Kersner,Travelling Waves in Nonlinear Diffusion–Convection–Reaction (Birkhäuser, Basel, CH, 2004)

  72. J.L. Vázquez, Commun. Contemp. Math. 09, 731 (2007)

    Article  Google Scholar 

  73. G. Ponti et al., The role of medium size facilities in the HPC ecosystem: the case of the new CRESCO4 cluster integrated in the ENEAGRID infrastructure, inProceedings ofthe 2014 International Conference on High Performance Computing and Simulation (HPCS2014) (2014), art. 6903807, pp. 1030–1033

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Authors and Affiliations

  1. ENEA—Italian National Agency for New Technologies, Energy and Sustainable Economic Development Via Enrico Fermi 45, 00044, Frascati, Italy

    Filippo Palombi

  2. ENEA – Italian National Agency for New Technologies, Energy and Sustainable Economic Development Via Martiri di Monte Sole 4, 40129, Bologna, Italy

    Stefano Ferriani

  3. ISTAT – Italian National Institute of Statistics Via Cesare Balbo 16, 00184, Rome, Italy

    Simona Toti

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  1. Filippo Palombi
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  2. Stefano Ferriani
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  3. Simona Toti
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Correspondence to Filippo Palombi.

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Palombi, F., Ferriani, S. & Toti, S. Coevolutionary dynamics of a variant of the cyclic Lotka–Volterra model with three-agent interactions. Eur. Phys. J. B 93, 194 (2020). https://doi.org/10.1140/epjb/e2020-100552-5

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