The European Physical Journal Special Topics

, Volume 226, Issue 3, pp 341–351 | Cite as

Relating high dimensional stochastic complex systems to low-dimensional intermittency

Regular Article
Part of the following topical collections:
  1. Nonlinearity, Nonequilibrium and Complexity: Questions and Perspectives in Statistical Physics

Abstract

We evaluate the implication and outlook of an unanticipated simplification in the macroscopic behavior of two high-dimensional sto-chastic models: the Replicator Model with Mutations and the Tangled Nature Model (TaNa) of evolutionary ecology. This simplification consists of the apparent display of low-dimensional dynamics in the non-stationary intermittent time evolution of the model on a coarse-grained scale. Evolution on this time scale spans generations of individuals, rather than single reproduction, death or mutation events. While a local one-dimensional map close to a tangent bifurcation can be derived from a mean-field version of the TaNa model, a nonlinear dynamical model consisting of successive tangent bifurcations generates time evolution patterns resembling those of the full TaNa model. To advance the interpretation of this finding, here we consider parallel results on a game-theoretic version of the TaNa model that in discrete time yields a coupled map lattice. This in turn is represented, a la Langevin, by a one-dimensional nonlinear map. Among various kinds of behaviours we obtain intermittent evolution associated with tangent bifurcations. We discuss our results.

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References

  1. 1.
    K. Christensen, S.A. di Collobiano, M. Hall, H.J. Jensen, Tangled nature model: A model of evolutionary ecology, J. Theor. Biol. 216, 73 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    K. Kunihiko, Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled map lattices – toward a prelude to a field theory of chaos, Prog. Theor. Phys. 72, 480 (1984)ADSCrossRefMATHGoogle Scholar
  3. 3.
    K. Raymond, Pattern formation in two-dimensional arrays of coupled, discrete-time oscillators, Phys. Rev. A 31, 3868 (1986)Google Scholar
  4. 4.
    H. Chaté, P. Manneville, Emergence of effective low-dimensional dynamics in the macroscopic behaviour of coupled map lattices, Europhys. Lett. 17, 291 (1992)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Diaz-Ruelas, H.J. Jensen, D. Piovani, A. Robledo. Tangent map intermittency as an approximate analysis of intermittency in a high dimensional fully stochastic dynamical system: The tangled nature model, Chaos 26, 123105 (2016)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Piovani, J. Grujić, H.J. Jensen. Linear stability theory as an early warning sign for transitions in high dimensional complex systems, J. Phys. A: Math. Theor. 49, 295102 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    H.G. Schuster, Deterministic Chaos, An Introduction (VCH Publishers, 1988)Google Scholar
  8. 8.
    I. Procaccia, H.G. Schuster, Functional renormalization-group theory of universal 1/f noise in dynamical systems, Phys. Rev. A 28, 1210 (1983)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    P.D. Taylor, L.B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci. 40, 145 (1978)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    K. Tokita, A. Yasutomi, Emergence of a complex and stable network in a model ecosystem with extinction and mutation, Theor. Popul. Biol. 63, 131 (2003)CrossRefMATHGoogle Scholar
  11. 11.
    D. Vilone, A. Robledo, A. Sánchez, Chaos and unpredictability in evolutionary dynamics in discrete time, Phys. Rev. Lett. 107, 038101 (2011)ADSCrossRefGoogle Scholar
  12. 12.
    M. Hall, K. Christensen, S.A. di Collobiano, H.J. Jensen, Time-dependent extinction rate and species abundance in a tangled-nature model of biological evolution, Phys. Rev. E 66, 011904 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    P.A. Rikvold, R.K.P. Zia, Punctuated equilibria and 1/f noise in a biological coevolution model with individual-based dynamics, Phys. Rev. E 68, 031913 (2003)ADSCrossRefGoogle Scholar
  14. 14.
    R.K.P. Zia, P.A. Rikvold, Fluctuations and correlations in an individual-based model of evolution, J. Phys. A 37, 5135 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    N. Becker, P. Sibani, Evolution and non-equilibrium physics: A study of the tangled nature model, EPL 105, 18005 (2014)ADSCrossRefGoogle Scholar
  16. 16.
    A.E. Nicholson, P. Sibani, Cultural evolution as a nonstationary stochastic process, Complexity 21, 214 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Vazquez, J.A. del Rio, K.G. Cedano, M. Martinez, H.J. Jensen, An entangled model for sustainability indicators, PLoS ONE 10, e0135250 (2015)CrossRefGoogle Scholar
  18. 18.
    P. Sibani, H.J. Jensen, Stochastic Dynamics of Complex Systems (Imperial College Press, 2013)Google Scholar
  19. 19.
    P.G. Higgs, B. Derrida, Genetic distance and species formation in evolving populations, J. Mol. Evol. 35, 454 (1992)CrossRefGoogle Scholar
  20. 20.
    J.M. Smith, Evolution and the Theory of Games (Cambridge University Press, 1982)Google Scholar
  21. 21.
    E. Ott, T.M. Antonsen, Low dimensional behavior of large systems of globaly coupled oscillators, Chaos 18, 037113 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Instituto de Física, Universidad Nacional Autónoma de México, Ciudad UniversitariaCiudad de MéxicoMexico
  2. 2.Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Ciudad UniversitariaCiudad de MéxicoMexico
  3. 3.Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington CampusSW7 2AZUK
  4. 4.Centre for Advanced Spatial Analysis. University College LondonLondonUK

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