The behavior of a model that generalizes the Lotka-Volterra problem into three dimensions is presented. The results show the analytic derivation of stability diagrams that describe the system's qualitative features. In particular, we show that for a certain value of the bifurcation parameter the system instantly jumps out of a steady state solution into a chaotic solution that portrays a fractal torus in the three-dimensional phase space. This scenario, is referred to as the explosive route to chaos and is attributed to the non-transversal saddle connection type bifurcation. The stability diagrams also present a region in which the Hopf type bifurcation leads to periodic and chaotic solutions. In addition, the bifurcation diagrams reveal a qualitative similarity to the data obtained in the Texas and Bordeaux experiments on the Belousov-Zhabotinskii chemical reaction. The paper is concluded by showing that the model can be useful for representing dynamics associated with biological and chemical phenomena.
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Abraham, R. H. and C. D. Shaw. 1985.Dynamics—The Geometry of Behavior, Part 3. Aerial Press, Inc., P. O. Box 1360, Sanata Cruz, California 95061.
Argoul, F., A. Arnéodo, P. Richetti and J. C. Roux. 1987 “From Quasiperiodicity to Chaos in the Belousov-Zhabotinskii Reaction I. Experiment.”,J. Chem. Phys. 86 (6), 3325.
Arnéodo, A., P. Coullet and C. Tresser. 1980. “Occurrence of Strange Attractors in Three-dimensional Volterra Equation.”Phys. Lett. 79A, 259–263.
Arnéodo, A., P. Coullet, J. Peyraud and C. Tresser. 1982. “Strange Attractors in Volterra Equations for Species in Competition”.J. math. Biol. 14, 153–157.
Arnold, V. I. 1984.Catastrophe Theory. Springer-Verlag.
Davis, H. T. 1962.Introduction to Nonlinear Differential and Integral Equations. Dover.
Devaney, R. L. 1987. “Chaotic Bursts in Nonlinear Dynamical Systems”.Science 235, 342.
Feinberg, M. 1980. “Chemical Oscillations, Multiple Equilibria, and Reaction Networks.” InDynamics and Modeling of Reactive Systems. Ray, Stewart and Conley (Eds), Academic Press.
Gardini, L., C. Mammana and M. G. Messia. 1986. “Bifurcations in Three-Dimensional Lotka-Volterra Competitive Models”.Adv. Modelling and Simulation 5 (3), 7–14.
Gear, C. W. 1971.Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, NJ.
Gilpin, M. E. 1979. “Spiral Chaos in a Predator-Prey Model”,Amer. Naturalist 113, 301–306.
Gleick, J. 1987.CHAOS—Making a New Science. Viking Penguin, Inc.
Guckenheimer, J. and P. Holmes. 1983.Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag.
Hale, J. K.Topics in Dynamic Bifurcation Theory. American Math. Soc., Providence, Rhode Island, No. 47.
IMSL Library Fortran Subroutines, IMSL Inc., Houston, Texas.
Lorenz, E. N. 1963. “Deterministic Non-Periodic Flows”.J. atmos. Sci. 20, 130–141.
Lotka, A.J. 1920. “Undamped Oscillations Derived from the Law of Mass Action”.J. Am. Chem. Soc. 42, 1595.
May, R. W., and W. J. Leonard. 1975. “Nonlinear Aspects of Competition between Three Species”.SIAM J. Appl. Math. 29(2), 243–253.
Nicolis, G. and I. Prigogine. 1977.Self-Organization in Nonequilibrium Systems—From Dissipative Structures to Order through Fluctuations. John Wiley.
Richetti, P., J. C. Roux, F. Argoul and A. Arnéodo. 1987. “From Quasiperiodicity to Chaos in the Belousov-Zhabotinskii Reaction II—Modeling and Theory”.J. chem. Phys. 86 (6), 3339.
Rössler, O. E. 1976. “Chaotic Behavior in Simple Reaction System”.Z Naturforschung 31a, 259–264.
Samardzija, N. 1983. “Stability Properties of Autonomous Homogeneous Polynomial Differential Systems”.J. Diff. Equations 48 (1), 60–70.
Schaffer, W. M. 1985. “Order and Chaos in Ecological Systems”.Ecology 66 (1), 93–106.
Shaw, C. D.The Dripping Faucet as a Model Chaotic System. Aerial Press, Inc. P.O. Box 1360, Santa Cruz, California 95061.
Smale, S. 1966. “Structurally Stable Systems are not Dense”.Amer. J. Math. 88, 491–496.
Smale, S. 1976. “On the Differential Equations of Species in Competition”.J. math. Biol. 3, 5–7.
Sparrow, C. 1982.The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer-Verlag.
Turner, J.S., J. C. Roux, W. D. McCormick and H. L. Swinney. 1981. “Alternating Periodic and Chaotic Regimes in a Chemical Reaction—Experiment and Theory”.Phys. Lett. 85A, 9–12.
Tyson, J. J., and J. C. Light. 1973. “Properties of Two-component Bimolecular and Trimolecular Chemical Reaction System.J. chem. Phys. 59 (8), 4164.
Volterra, V. 1931. “Leçons sur la Théorie Mathématiques de la Lutte pour la Vie.” Paris.
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Samardzija, N., Greller, L.D. Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model. Bltn Mathcal Biology 50, 465–491 (1988). https://doi.org/10.1007/BF02458847
- Steady State Solution
- Chaotic Attractor
- Strange Attractor
- Interarrival Time
- Stability Diagram