Chaos in Low-Dimensional Lotka-Volterra Models of Competition
J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K Noel and J. C.
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
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
Ref: J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel, and J. C. Sprott,
Nonlinearity 19, 2391-2404
The complete paper is available in PDF
Return to Sprott's Books and Publications.
Figure 1. Carrying simplex for equation (1) with parameters in equation
Figure 2. The dynamics on the boundary of the carrying simplex in
figure 1. The tetrahedron has been unfolded and laid flat for better
Figure 3. Attractor projected onto x1x2x3
Figure 4. Time series for each species (vertical scale is 0 to 1).
Figure 5. Homoclinic connection projected onto the x1x2
Figure 6. (a
diagram showing successive maxima of x1
as the coupling variable s
increased and (b
corresponding largest LE.
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
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
Hysteresis occurs as the orbit remains at Q34
even if s
is lowered below 1.31 until Q34
becomes unstable. At
= 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 8. Graph of symbolic dynamics of the attractor. The observed
probabilities of the transitions are also indicated.
Figure 9. The plot of Lambdan
showing convergence to 0.0178, a close approximation to the ASE for the
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.