Skip to main content
SpringerLink
Log in

Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

The spatio-temporal dynamics of three competitive species is considered. Mathematically, the community is described by a system of partial differential equations of Lotka-Volterra type. The properties of the system are investigated both numerically and analytically. We show that for finite initial conditions the dynamics of the system is typically reduced to a succession of travelling diffusive waves, some of which demonstrate rather an unusual behaviour. Particularly, a locally unstable equilibrium can become stable in the wake of a diffusive front. After propagation of the waves, the domain is invaded by irregular spatiotemporal population oscillations that can be classified as spatio-temporal chaos.

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

References

  1. S.R. Dunbar, Travelling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits. SIAM J. Appl. Math.,46 (1986), 1057–1078.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. J.R. Fasham, The statistical and mathematical analysis of plankton patchiness. Oceanogr. Marine Biology Annual Rev.,16 (1978), 43–79.

    Google Scholar 

  3. P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems. Lecture Note in Biomathematics28, Springer-Verlag, Berlin, 1979.

    Google Scholar 

  4. R. Fisher, The wave of advance of advantageous genes. Ann. Eugenics,7 (1937), 255–369.

    Google Scholar 

  5. C. Godfray and M. Hassell, Chaotic beetles. Science,275 (1997), 323–326.

    Article  Google Scholar 

  6. J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems. Cambridge Univ. Press, Cambridge UK, 1988.

    MATH  Google Scholar 

  7. Y. Hosono, The minimal speed of travelling fronts for a diffusive Lotka-Volterra competition model. Bull. Math. Biol.,60 (1998), 435–448.

    Article  MATH  Google Scholar 

  8. K. Kawasaki, A limit cycle in a three competitive species community of Lotka-Volterra type. The Science and Engineering Review of Doshisha University, Kyoto, 2001, in press (in Japanese).

  9. A.N. Kolmogorov, I.G. Petrovsky and N.S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Moscow Univ. Bull. Math.,1 (1937), 1–25.

    Google Scholar 

  10. N. Kopell and L.N. Howard, Plane wave solutions to reaction-diffusion equations. Studies in Appl. Math.,52 (1973), 291–328.

    MATH  MathSciNet  Google Scholar 

  11. S.A. Levin, Physical and biological scales and the modelling of predator-prey interactions in large marine ecosystems. Large Marine Ecosystems: Patterns, Processes and Yields (eds. K. Sherman, L.M. Alexander and B.D. Gold), AAAS, Washington, 1990.

    Google Scholar 

  12. S.A. Levin and L.A. Segel, A hypothesis for the origin of planktonic patchiness. Nature,259 (1976), 659.

    Article  Google Scholar 

  13. H. Malchow, Flow- and locomotion-induced spatial pattern formation in nonlinear population dynamics. Ecological Modelling,82 (1995), 257–264.

    Article  Google Scholar 

  14. H. Malchow and S.V. Petrovskii, Dynamical stabilization of unstable equilibrium in chemical and biological systems. Mathematical and Computer Modelling, (2001), in press.

  15. R.M. May and W. Leonard, Nonlinear aspects of competition between three species. SIAM J. Appl. Math.,29 (1975), 243–252.

    Article  MATH  MathSciNet  Google Scholar 

  16. J.H. Merkin and M.A. Sadiq, The propagation of travelling waves in an open cubic autocatalytic chemical system. IMA J. Appl. Math.,57 (1996), 273–309.

    Article  MATH  MathSciNet  Google Scholar 

  17. J.H. Merkin, V. Petrov, S.K. Scott and K. Showalter, Wave-induced chemical chaos. Phys. Rev. Lett.,76 (1996), 546–549.

    Article  Google Scholar 

  18. J.D. Murray, Mathematical Biology. Springer, Berlin, 1989.

    MATH  Google Scholar 

  19. T. Namba and M. Mimura, Spatial distribution of competing populations. J. Theor. Biol.,87 (1980), 795–814.

    Article  MathSciNet  Google Scholar 

  20. A.H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics. Wiley, New York, 1995.

    Book  MATH  Google Scholar 

  21. A. Okubo, Diffusion and Ecological Problems: Mathematical Models. Springer, Berlin, 1980.

    MATH  Google Scholar 

  22. A. Okubo, P.K. Maini, M.H. Williamson and J.D. Murray, On the spatial spread of the gray squirrel in Britain. Proc. R. Soc. Lond., B238 (1989), 113–125.

    Article  Google Scholar 

  23. M. Pascual, Diffusion-induced chaos in a spatial predator-prey system. Proc. R. Soc. Lond., B251 (1993), 1–7.

    Article  Google Scholar 

  24. S.V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system. Mathematical and Computer Modelling,29 (1999), 49–63.

    Article  MATH  MathSciNet  Google Scholar 

  25. S.V. Petrovskii and H. Malchow, Critical phenomena in plankton communities: KISS model revisited. Nonlinear Analysis: Real World Applications,1 (2000), 37–51.

    Article  MATH  MathSciNet  Google Scholar 

  26. S.V. Petrovskii and H. Malchow, Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol.,59 (2001), 157–174.

    Article  MATH  Google Scholar 

  27. R. Pool, Ecologists flirt with chaos. Science,243 (1989), 310–313.

    Article  Google Scholar 

  28. M. Scheffer, Should we expect strange attractors behind plankton dynamics — and if so, should we bother? J. Plankt. Res.,13 (1991), 1291–1305.

    Article  Google Scholar 

  29. L.A. Segel and J.L. Jackson, Dissipative structure: an explanation and an ecological example. J. Theor. Biol.,37 (1972), 545–559.

    Article  Google Scholar 

  30. J.A. Sherratt, M.A. Lewis and A.C. Fowler, Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA,92 (1995), 2524–2528.

    Article  MATH  Google Scholar 

  31. N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol.,30 (1986), 143–160.

    Article  MATH  MathSciNet  Google Scholar 

  32. N. Shigesada, Mathematical Modelling for Biological Invasions. University of Tokyo Press, Tokyo, 1992 (in Japanese).

  33. N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice. Oxford University Press, Oxford, 1997.

    Google Scholar 

  34. A.M. Turing, On the chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond., B237 (1952), 37–72.

    Article  Google Scholar 

  35. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin, 1985.

    Google Scholar 

  36. V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memor. Accad. Lincei. Ser. 6,2 (1926), 31–113.

    Google Scholar 

  37. S. Yachi, K. Kawasaki, N. Shigesada and E. Teramoto, Spatial patterns of propagating waves of fox rabies. Forma,4 (1989), 3–12.

    Google Scholar 

  38. M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. Dynamical Systems and Stability,8 (1993), 189–217.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shirshov Institute of Oceanology, Nakhimovsky Prospekt 36, 117218, Moscow, Russia

    Sergei Petrovskii

  2. Department of Knowledge Engineering and Computer Sciences, Doshisha University, 610-0321, Kyo-Tanabe, Japan

    Kohkichi Kawasaki

  3. Department of Information and Computer Sciences, Nara Women’s University, Kita-Uoya Nishimachi, 630-8506, Nara, Japan

    Fugo Takasu & Nanako Shigesada

Authors
  1. Sergei Petrovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kohkichi Kawasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fugo Takasu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nanako Shigesada
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Petrovskii, S., Kawasaki, K., Takasu, F. et al. Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species. Japan J. Indust. Appl. Math. 18, 459–481 (2001). https://doi.org/10.1007/BF03168586

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03168586

Key words

Search

Navigation