Bulletin of Mathematical Biology

, Volume 60, Issue 3, pp 435–448 | Cite as

The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model

  • Yuzo Hosono


This paper concerns the minimal speed of traveling wave fronts for a two-species diffusion-competition model of the Lotka-Volterra type. An earlier paper used this model to discuss the speed of invasion of the gray squirrel by estimating the model parameters from field data, and predicted its speed by the use of a heuristic analytical argument. We discuss the conditions which assure the validity of their argument and show numerically the existence of the realistic range of parameter values for which their heuristic argument does not hold. Especially for the case of the strong interaction of two competing species compared with the intraspecific competition, we show that all parameters appearing in the system affect the minimal speed of invasion.


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  1. Aronson, D. G. (1975). Nonlinear diffusion in population genetics, combustion and nerve propagation. Lect. Notes Math. 446, 5–49.MATHMathSciNetCrossRefGoogle Scholar
  2. Aronson, D. G. and H. F. Weinberger (1978). Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76.MathSciNetCrossRefMATHGoogle Scholar
  3. Berestycki, H. and L. Nirenberg (1992). Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéare 9, 497–572.MathSciNetMATHGoogle Scholar
  4. Bramson, M. (1983). The convergence of solutions of the Kolmogorov nonlinear diffusion equation to traveling waves. Mem. Amer. Math. Soc. 44.Google Scholar
  5. Fisher, R. (1937). The wave of advance of advantageous genes. Ann. of Eugenics 7, 335–369.Google Scholar
  6. Hadeler, K. P. and F. Rothe (1975). Traveling fronts in nonlinear diffusion equations. J. Math. Biology 2, 251–263.MathSciNetCrossRefMATHGoogle Scholar
  7. Hosono, Y. (1989). Singular perturbation analysis of traveling waves for diffusive Lotka-Volterra competing models. Num. Appl. Math. 2, 687–692.MathSciNetGoogle Scholar
  8. Hosono, Y. (1995). Traveling waves for diffusive Lotka-Volterra competition models II: a geometric approach. Forma. 10, 235–257.MATHMathSciNetGoogle Scholar
  9. Kanel J. I. and L. Zhou (1996). Existence of wave front solutions and estimates of wave speed for a competition-diffusion system. Nonlin. Anal. TMA 27, 579–587.MathSciNetCrossRefMATHGoogle Scholar
  10. Kan-on, Y. (1997). Fisher wave fronts for the Lotka-Volterra competition model with diffusion. Nonlin. Anal. TMA 28, 145–164.MATHMathSciNetCrossRefGoogle Scholar
  11. Kolmogorov, A. N., I. Petrovsky and N. Piscounoff. (1937). Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. Moskow, Ser. Int. A1, 1–25.Google Scholar
  12. Murray, J. D. (1989). Mathematical Biology, New York, Springer.MATHGoogle Scholar
  13. Okubo, A., P. K. Maini, M. H. Williamson and J. D. Murray (1989). On the spatial spread of the gray squirrel in Britain. Proc. R. Soc. Lond. B238, 113–125.CrossRefGoogle Scholar
  14. Stokes, A. N. (1976). On two types of moving front in quasilinear diffusion. Math. Biosci. 31, 307–315.MATHMathSciNetCrossRefGoogle Scholar
  15. Stokes, A. N. (1977). Nonlinear diffusion waveshapes generated by possibly finite initial disturbances. J. Math. Anal. Appl. 61, 370–381.MATHMathSciNetCrossRefGoogle Scholar
  16. Tang M. M. and P. C. Fife (1980). Propagating fronts for competing species equations with diffusion. Arch. Rat. Mech. Anal. 73, 69–77.MathSciNetCrossRefMATHGoogle Scholar
  17. Uchiyama, K. (1978). The behavior of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ. 18, 453–508.MATHMathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 1998

Authors and Affiliations

  • Yuzo Hosono
    • 1
  1. 1.Department of Information and Communication SciencesKyoto Sangyo UniversityKyotoJapan

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