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Application of Hamilton-Jacobi theory to the Lotka-Volterra oscillator

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Abstract

It is known that the Lotka-Volterra coupled nonlinear differential equations for a two-species prey-predator ecosystem possess a periodic solution, although its exact form is not yet obtained analytically. The conventional linearization approximation for solving these nonlinear equations leads to a harmonic oscillator whose frequency depends only on the intraspecific coefficients. We propose here a prescription for obtaining nonlinear correction to the linear frequency by using the Hamilton-Jacobi canonical formalism of classical mechanics. It is found that the first-order correction, which also involves interspecific parameters, exhibits the basic qualitative features of the nonlinearity.

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Literature

  • Dutt, R., P. K. Ghosh and B. B. Karmakar. 1975. “Application of Perturbation Theory to the Nonlinear Volterra-Gause-Witt Model for Prey-Predator Interaction.”Bull. Math. Biol.,37, 139–146.

    Article  MATH  Google Scholar 

  • Frame, J. S. 1974. “Explicit Solutions of Two Species Volterra System.”J. Theor. Biol.,43, 73–81.

    Article  Google Scholar 

  • Goldstein, H. 1962.Classical Mechanics. Addison-Wesley Publishing Co. Inc.

  • Goodwin, B. C. 1963.Temporal Organization in Cells. London: Academic Press.

    Google Scholar 

  • Grasman, J. and E. Veling. 1973. “An Asymptotic Formula for the Period of a Volterra-Lotka System.”Math. Biosci.,18, 185–189.

    Article  MATH  Google Scholar 

  • Lewis Jr., H. R. 1967. “Classical and Quantum Systems with time dependent Harmonic Oscillator type Hamiltonians.”Phys. Rev. Lett.,18, 510–514.

    Article  Google Scholar 

  • Montroll, E. W. 1972. “On Coupled Rate Equations with Quadratic Nonlinearities,”Proc. Natn. Acad. Sci., USA,69, 2532–2536.

    Article  MATH  Google Scholar 

  • Symon, K. R. 1970. “The Adiabatic Invariant of the Linear or Nonlinear Oscillator.”J. Math. Phys.,11, 1320–1330.

    Article  MATH  Google Scholar 

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  1. Department of Physics, Visva-Bharati University, Santiniketan, 731235, West Bengal, India

    Ranabir Dutt

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  1. Ranabir Dutt
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Dutt, R. Application of Hamilton-Jacobi theory to the Lotka-Volterra oscillator. Bltn Mathcal Biology 38, 459–465 (1976). https://doi.org/10.1007/BF02462220

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