Chaos in Low-Dimensional Lotka-Volterra Models of Competition

J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K Noel and J. C. Sprott
Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA

Received 5 September 2005, in final form 18 August 2006
Published 15 September 2006

ABSTRACT

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 1

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 2

Figure 3. Attractor projected onto x1x2x3 space.
Figure 3

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

Figure 5. Homoclinic connection projected onto the x1x2 plane.
Figure 5

Figure 6. (a) Bifurcation diagram showing successive maxima of x1 as the coupling variable s is increased and (b) the corresponding largest LE.
Figure 6

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 Q124 is stable in the window s = 1.04 to s = 1.12, while Q34 becomes stable at s = 1.08 and is the only attractor for 1.31 < s < 1.4. Hysteresis occurs as the orbit remains at Q34 even if s is lowered below 1.31 until Q34 becomes unstable. At s = 1.06875 there are coexisting limit cycles and at s = 1.2375 the strange attractor coexists with a limit cycle, indicated by the single points (*).
Figure 7

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

Figure 9. The plot of Lambdan showing convergence to 0.0178, a close approximation to the ASE for the symbolic dynamics.
Figure 9

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

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