Skip to main content
SpringerLink
Log in

Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Analytic and numerical solutions to two coupled nonlinear diffusion equations are studied. They are the modified equations of Volterra and Lotka for the spatially stratified predatorprey population model. In a bounded domain with the reflecting boundary, equilibrium, stability, and transition to time-periodic solutions are analyzed. For a wide class of initial states, the solutions to the initial boundary-value problem evolve into their corresponding stable, space-homogeneous, periodic oscillations. In an unbounded domain, a family of traveling wave solutions is found for certain exponential, initial distributions in the limit as the diffusion coefficientv 1 of the prey tends to zero. In the presence of both diffusions, the results of a numerical simulation to an initial-value problem showed the rapid formation of the Pursuit-Evasion Waves whose speed of propagation and amplitudes increase with the diffusion coefficientv 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Hadeler, K. P., U. an der Heiden and F. Rothe. 1974. “Nonhomogeneous Spatial Distribution of Populations.”J. Math. Biol.,1, 165–176.

    Article  MATH  MathSciNet  Google Scholar 

  • Hirsch, M. W. and S. Smale. 1974.Differential Equations, Dynamical Systems and Linear Algebra. New York: Academic Press.

    MATH  Google Scholar 

  • Kolmogorov, A. N., I. Petrovsky and N. Piskunov. 1937. “Etude de l'equation de la chaleur avec croissance … et son application à un probleme biologique.”Bull. U. État Moscou,6, 1–25.

    MATH  Google Scholar 

  • Kopell, N. and L. N. Howard. 1973. “Plane Wave Solutions to Reaction-Diffusion Equations.”Stud. Appl. Math.,52, 291–328.

    MATH  MathSciNet  Google Scholar 

  • Montroll, E. W. 1968. “Lectures on Nonlinear Rate Equations, Especially Those with Quadratic Nonlinearities.” In:Lectures in Theoretical Physics, 10A. (Ed. A. Barut and W. Brittin) pp. 531–573. New York: Gordon and Breach.

    Google Scholar 

  • Rinzel, J. and J. B. Keller, 1973. “Travelling Wave Solutions of a Nerve Conduction Equation.”Biophys. J.,13, 1313–1337.

    Article  Google Scholar 

  • Rosen, G. 1975. “Solutions to Systems of Nonlinear Reaction-Diffusion Equations.”Bull. Math. Biol. 37, 277–289.

    Article  MATH  MathSciNet  Google Scholar 

  • Stakgold, I., D. D. Joseph and D. H. Sattinger, eds. 1973.Nonlinear Problems in the Physical Sciences and Biology. New York: Springer-Verlag.

    MATH  Google Scholar 

  • Williams, S. A. and P. L. Chow. 1976. “Nonlinear Reaction-Diffusion Equations for Interacting Populations,”J. Math. Analy. Appl, to appear.

  • Zubov, V. 1964.Methods of A. M. Liapunov and Their Applications. Gronigen: Noordhoff.

    Google Scholar 

Download references

Author information

Author notes

  1. W. C. Tam

    Present address: Harvey Mudd College, 91711, Claremont, California

Authors and Affiliations

  1. Department of Mathematics, Wayne State University, 48202, Michigan, Detroit, U.S.A.

    P. L. Chow & W. C. Tam

Authors
  1. P. L. Chow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. C. Tam
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chow, P.L., Tam, W.C. Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion. Bltn Mathcal Biology 38, 643–658 (1976). https://doi.org/10.1007/BF02458639

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02458639

Keywords

Search

Navigation