Received 5 September 2005, in final form 18 August 2006

Published 15 September 2006

The occurrence of chaos in basic Lotka-Volterra models of four
competing species is studied. A brute-force numerical search
conditioned on the largest Lyapunov exponent (LE) indicates that chaos
occurs in a narrow region of parameter space but is robust to
perturbations. The dynamics of the attractor for a maximally chaotic
case are studied using symbolic dynamics, and the question of
self-organized critical behaviour (scale-invariance) of the solution is
considered.

Ref: J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel, and J. C. Sprott, Nonlinearity 19, 2391-2404 (2006)

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Figure 1. Carrying simplex for equation (1) with parameters in equation (3).

Figure 2. The dynamics on the boundary of the carrying simplex in figure 1. The tetrahedron has been unfolded and laid flat for better viewing.

Figure 3. Attractor projected onto x

Figure 4. Time series for each species (vertical scale is 0 to 1).

Figure 5. Homoclinic connection projected onto the x

Figure 6. (a) Bifurcation diagram showing successive maxima of x

Figure 7. This Monte Carlo scan over the space of initial conditions attempted to locate coexisting attractors in the range 0.8 < s < 1.4. The average variance of each variable was calculated along every orbit and summed. Significant differences in the variances for a single value of the bifurcation parameter s indicated multiple attractors. The fixed point Q

Figure 8. Graph of symbolic dynamics of the attractor. The observed probabilities of the transitions are also indicated.

Figure 9. The plot of Lambda

Figure 10. Probability distribution function of volatility showing a power-law scaling (arbitrary scales).

Figure 11. Probability of the largest LE, showing the rarity of chaos.