Complex Spatiotemporal Dynamics in Lotka-Volterra Ring Systems
J. C.
Wildenberg, J. A. Vano, and J. C. Sprott
Departments of Physics, University
of Wisconsin, Madison, Wisconsin 53706, USA
ABSTRACT
Mathematical models in ecology often need to incorporate spatial
dependence to accurately model real-world systems. We consider
competitive Lotka-Volterra models modified to include this spatial
dependence through organization of the competing species into a
one-dimensional ring by an appropriate choice of the interaction
matrix. We show that these systems can exhibit complex dynamics,
spatiotemporal chaos, and spontaneous symmetry breaking. A
high-dimensional, spatially homogeneous, nearest-neighbor example with
interaction strengths decreasing with distance is characterized
including an analysis of how the dynamics of the system vary with
dimension. We also show the existence of Lyapunov functions that arise
from this method of including spatial dependence and how they prohibit
complex dynamics for certain regions of the parameter space. We utilize
these Lyapunov functions to reduce the required calculation time in
brute-force searches of parameter space. A short comparison is given to
derived line systems including a contrast between the eigenvalues of
the two systems.
Ref: J. C.
Wildenberg, J. A. Vano, and J. C. Sprott,
Ecological Complexity 3,
140-147 (2006)
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