A Simple Spatiotemporal Chaotic Lotka-Volterra Model
J. C. Sprott and J. C.
Departments of Physics, University
of Wisconsin, 1150 University Avenue, Madison, WI 53706, USA
Institute for Advanced Studies in
Basic Sciences, Zanjan, Iran
A mathematically simple example of a high-dimensional (many-species)
Lotka-Volterra model that exhibits spatiotemporal chaos in one spatial
dimension is described. The model consists of a closed ring of
identical agents, each competing for fixed finite resources with two of
its four nearest neighbors. The model is prototypical of more
complicated models in its quasiperiodic route to chaos (including
attracting 3-tori), bifurcations, spontaneous symmetry breaking, and
spatial pattern formation.
Ref: J. C. Sprott, J. C.
Wildenberg, and Y. Azizi, Chaos Solitons & Fractals 26, 1035-1043 (2005)
The complete paper is available in PDF
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Fig. 1. Eigenvalues of the coexisting equilibrium with N = 100 and s = 1.
Fig. 2. A one-dimensional ring of species interacting according to
Fig. 3. Largest Lyapunov exponent and Kaplan-Yorke dimension for Eq.
(2) with s = 1.
Fig. 4. Largest Lyapunov exponent for
Eq. (2) with N
= 100 showing
the quasiperiodic route to chaos that is typical of high-dimensional
Fig. 5. Spatiotemporal plot of |xi+1
| from Eq. (2) with N
= 480 and s
Fig. 6. Spatiotemporal
cross-correlation function with N
= 100 and s
= 1 showing
propagation and dispersion.
Fig. 7. Total biomass,
biodiversity, and a typical x
) versus time for Eq. (2) with N
= 100 and s