Global aspects of turbulence induced by heteroclinic cycles in competitive diffusion Lotka–Volterra equation

• Published in: Physica. D Vol. 240; no. 23; pp. 1853 - 1862
• Main Authors: Orihashi, Kenji, Aizawa, Yoji
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15-11-2011; Elsevier
• Subjects: Computational fluid dynamics; Correlation length and time; Diffusion Lotka–Volterra equation; Exact sciences and technology; Fluid flow; Lotka-Volterra equations; Lyapunov dimension; Lyapunov spectra; Mathematical analysis; Mathematical models; Physics; Predator-prey simulation; Spatio-temporal chaos; Turbulence; Turbulent flow; Correlation length and time; Spatio-temporal chaos; Turbulence; Lyapunov spectra; Lyapunov dimension; Diffusion Lotka–Volterra equation; Chaos; Phase diagrams; Diffusion Lotka-Volterra equation; Time correlation; Empirical relation; Attractors; Non linear phenomenon; Scaling laws; Linear stability; Correlation time; Diffusion equation; Hopf bifurcation; Correlation length; Volterra integral equations
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2011.01.001

This paper presents detailed numerical results of the competitive diffusion Lotka–Volterra 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 Lotka–Volterra equation in the case of May-Leonard type is stabilized more as the system size increases.
► We study the turbulence of the diffusion Lotka–Volterra equation by numerically. ► Detailed phase diagrams of the system for parameters and system size are derived. ► Our data show some statistics have scaling relations in some domain. ► Relational expressions are realized by using the phase reduction method.