Advertisement

Journal of Mathematical Biology

, Volume 17, Issue 1, pp 11–32 | Cite as

Travelling wave solutions of diffusive Lotka-Volterra equations

  • Steven R. Dunbar
Article

Abstract

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c* dividing the waves into two types, waves of speed c < c* being one type, waves of speed c ⩾ c* being of the other type. We present numerical evidence that for this system the wave of speed c* is stable, and that c* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.

Key words

Reaction diffusion equations Travelling waves Diffusive Lotka-Volterra equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bramson, M. D.: Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31, 531–581 (1978)Google Scholar
  2. 2.
    Brown, K. J., Carr, J.: Deterministic epidemic waves of critical velocity. Math. Proc. Camb. Phil. Soc. 81, 431–433 (1977)Google Scholar
  3. 3.
    Chow, P. L., Tam, W. C.: Periodic and travelling wave solutions to Volterra-Lotka equations with diffusion. Bull. Math. Biol. 12, 643–658 (1976)Google Scholar
  4. 4.
    Conley, C.: Isolated invariant sets and the Morse index. BMS Regional Conference Series, No 38, AMS, Providence, RI 1978Google Scholar
  5. 5.
    Conley, C., Gardner, R.: An application of the generalized Morse index to travelling wave solutions of a competitive reaction diffusion model. PreprintGoogle Scholar
  6. 6.
    Dubois, D. M.: A model of patchiness for prey-predator plankton populations. Ecol. Modelling 1, 67–80 (1975)Google Scholar
  7. 7.
    Dunbar, S.: Travelling wave solutions of diffusive Volterra-Lotka interactions equations. Ph.D. Thesis, University of Minnesota 1981Google Scholar
  8. 8.
    Dunbar, S.: Travelling wave solutions of diffusive Lotka-Volterra equations: Heteroclinic orbits in ℝ4. (In preparation)Google Scholar
  9. 9.
    Fife, P. C.: Mathematical aspects of reacting and diffusing systems. Lecture notes in biomathe- matics, vol. 28 Berlin-Heidelberg-New York: Springer 1979Google Scholar
  10. 10.
    Fisher, R. A.: The wave of advance of advantageous genes. Ann. of Eugenics 7, 335–369 (1937)Google Scholar
  11. 11.
    Gantmacher, F. R.: The theory of matrices. Vol. 66 New York: Chelsea Publishing 1964Google Scholar
  12. 12.
    Gardner, R.: Existence and stability of travelling wave solutions of competition models: A degree theoretic approach. J. Diff. Eqns. (in press)Google Scholar
  13. 13.
    Hadeler, K. P., Heiden, U. an der, Rothe, F.: Non homogeneous spatial distributions of populations. J. Math. Biol. 1, 165–174 (1974)Google Scholar
  14. 14.
    Hartman, P.: Ordinary differential equations. Baltimore: Wiley and Sons 1973Google Scholar
  15. 15.
    Hirsch, M., Smale, S.: Differential equations, Dynamical systems and linear algebra. Pure and applied mathematics, vol. 66 New York: Academic Press 1974Google Scholar
  16. 16.
    Kolmogorov, A., Petrovsky, I., Piscounov, N.: Etude de l' equation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Universitet Bull. Math. 1, 1–25 (1937)Google Scholar
  17. 17.
    LaSalle, J. P.: Stability theory for ordinary differential equations. J. Diff. Eqns. 4, 57–65 (1968)Google Scholar
  18. 18.
    McMurtie, R.: Persistence and stability of single species and prey-predator systems in spatially heterogeneous environments. Math. Biosci. 39, 11–51 (1978)Google Scholar
  19. 19.
    Okubo, A.: Diffusion and ecological problems: Mathematical models. Biomathematics, vol. 10. Berlin-Heidelberg-New York: Springer 1980Google Scholar
  20. 20.
    Pauwelussen, J. P.: Heteroclinic waves of the FitzHugh-Nagumo equations. Math. Biosci. 58, 217–242 (1982)Google Scholar
  21. 21.
    Protter, M. H., Weinberger, H. F.: Maximum principles in differential equations. Englewood Cliffs, NJ: Prentice Hall 1967Google Scholar
  22. 22.
    Wyatt, T.: The biology of Oikopleura Dioica and Frittilaria Borealis in the Southern Bight. Mar. Biol. 22, 137 (1973)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Steven R. Dunbar
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA

Personalised recommendations