A simple spatiotemporal chaotic Lotka–Volterra model

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.

[1]  F. Takens,et al.  On the nature of turbulence , 1971 .

[2]  J. Sprott.  Chaos and time-series analysis , 2001 .

[3]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[4]  S. Gould,et al.  Punctuated equilibria: the tempo and mode of evolution reconsidered , 1977, Paleobiology.

[5]  J. Sprott.  Strange Attractors: Creating Patterns in Chaos , 1993 .

[6]  Julien Clinton Sprott,et al.  Competition with evolution in ecology and finance , 2004 .

[7]  K. Brading,et al.  Symmetries in physics: philosophical reflections , 2003, quant-ph/0301097.

[8]  H. Meinhardt.  Models of biological pattern formation , 1982 .

[9]  J. Yorke,et al.  Attractors on an N-torus: Quasiperiodicity versus chaos , 1985 .

[10]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[11]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

[12]  Julien Clinton Sprott,et al.  Coexistence and chaos in complex ecologies , 2005 .

[13]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[14]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[15]  D. J. Albers,et al.  Routes to Chaos in Neural Networks with Random Weights , 1998 .

[16]  Celso Grebogi,et al.  Are three-frequency quasiperiodic orbits to be expected in typical nonlinear dynamical systems , 1983 .

[17]  D. Aronson,et al.  Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study , 1982 .

[18]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[19]  Hiroo Yonezu,et al.  Analog integrated circuits for the Lotka-Volterra competitive neural networks , 1999, IEEE Trans. Neural Networks.

[20]  H. Peitgen,et al.  Functional Differential Equations and Approximation of Fixed Points , 1979 .

[21]  P. Verhulst.  Recherches mathématiques sur la loi d’accroissement de la population , 1845, Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Bruxelles.

[22]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[23]  R. Macarthur.  Species packing and competitive equilibrium for many species. , 1970, Theoretical population biology.