Global aspects of turbulence induced by heteroclinic cycles in competitive diffusion LotkaVolterra equation

Kenji Orihashi*, Yoji Aizawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


This paper presents detailed numerical results of the competitive diffusion LotkaVolterra equation (May-Leonard type). First, we derive the global phase diagrams of attractors in the parameter space including the system size, where transition lines between simple attractors are clearly obtained in accordance with the results of linear stability analysis, but the transition borders become complex when multi-basin structures appear. The complex aspects of the transition borders are studied in the case when the system size decreases. Next, we show the statistical aspects of the turbulence with special attention to the onset of the supercritical Hopf bifurcation. Several characteristic quantities, such as correlation length, correlation time, Lyapunov spectra and Lyapunov dimension, are investigated in detail near the onset of turbulence. Our data show the critical scaling law near the onset only in the restricted parameter domain. However even when the critical indices are not determined accurately, it is shown that the empirical scaling relations are obtained in a wide parameter domain far from the onset point and those scaling indices satisfy several relations. These scaling relations are discussed in comparison with the result derived by the phase reduction method. Lastly, we make a conjecture about the stability of an ecosystem based on the bifurcation diagram: the ecosystem obeying the LotkaVolterra equation in the case of May-Leonard type is stabilized more as the system size increases.

Original languageEnglish
Pages (from-to)1853-1862
Number of pages10
JournalPhysica D: Nonlinear Phenomena
Issue number23
Publication statusPublished - 2011 Nov 15


  • Correlation length and time
  • Diffusion LotkaVolterra equation
  • Lyapunov dimension
  • Lyapunov spectra
  • Spatio-temporal chaos
  • Turbulence

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics


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