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Journal of Mathematical Biology

, Volume 2, Issue 3, pp 251–263 | Cite as

Travelling fronts in nonlinear diffusion equations

  • K. P. Hadeler
  • F. Rothe
Article

Summary

In Fisher's model for the migration of advantageous genes, in epidemic models and in the theory of combustion similar existence problems for travelling fronts and waves occur. For a general two-dimensional system of ordinary differential equations depending on a parameter the existence of trajectories connecting stationary points is established. For systems derived from diffusion problems these trajectories describe the shape of a travelling front, the corresponding value of the parameter is the propagation speed. The method allows to determine the exact value of the minimal speed in Fisher's model for all interesting choices of selection parameters, i.e. for intermediate heterozygotes and for inferior heterozygotes.

Keywords

Ordinary Differential Equation Stationary Point Mathematical Biology Matrix Theory Propagation Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion, and nerve propagation, Proceedings of the Tulane Program in partial differential equations. (Lecture Notes in Mathematics.) Berlin-Heidelberg-New York: Springer 1975.Google Scholar
  2. [2]
    Conley, C.: An application of Wazewski's method to a nonlinear boundary value problem which arises in population genetics. J. of Math. Biology (to appear).Google Scholar
  3. [3]
    Crow, J. F., Kimura, M.: An introduction to population genetics, p. 173 ff. New York: Harper and Row 1970.Google Scholar
  4. [4]
    Fife, P. C., McLeod, J. B.: The approach of solutions of nonlinear diffusion equations to travelling wave solutions. (To appear.)Google Scholar
  5. [5]
    Fisher, R. A.: The genetical theory of natural selection. Oxford University Press 1930.Google Scholar
  6. [6]
    Fisher, R. A.: The advance of advantageous genes. Ann. of Eugenics 7, 355–369 (1937).Google Scholar
  7. [7]
    Gelfand, I. M.: Some problems in the theory of quasilinear equations. Uspekhi Math. Nauk (N. S.) 14, 87–158 (1959); American Math. Soc. Transl. (2), 29, 295–381 (1963).Google Scholar
  8. [8]
    Hadeler, K. P.: Mathematik für Biologen. Berlin-Heidelberg-New York: Springer 1974.Google Scholar
  9. [9]
    Hadeler, K. P.: On the equilibrium states in certain selection models. J. of Math. Biology 1, 51–56 (1974).Google Scholar
  10. [10]
    Hoppensteadt, F. C.: Mathematical theories of populations, demographics, genetics, and epidemics. The University of West Florida, Pensacola, Fl. 1974.Google Scholar
  11. [11]
    Kanel', Ja. I.: The behavior of solutions of the Cauchy problem when time tends to infinity, in the case of quasilinear equations arising in the theory of combustion. Dokl. Akad. Nauk SSSR 132, 268–271 (1961); Soviet Math. Dokl. 1, 533–536.Google Scholar
  12. [12]
    Kanel', Ja. I.: Certain problems on equations in the theory of burning. Dokl. Akad. Nauk SSSR 136, 277–280 (1961); Soviet Math. Dokl. 2, 48–51.Google Scholar
  13. [13]
    Kanel', Ja. I.: Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory. Mat. Sbornik (N. S.) 59, 101 (1962), supplement, 245–288.Google Scholar
  14. [14]
    Kanel', Ja. I.: On the stability of solutions of the equation of combustion theory for finite initial functions. Mat. Sbornik (N. S.) 65 (107), 398–413 (1964).Google Scholar
  15. [15]
    Kendall, D. G.: Mathematical models of the spread of infection, Mathematics and computer science in biology and medicine. Medical Research Council 1965.Google Scholar
  16. [16]
    Kolmogoroff, A., Petrovskij, I., Piskunov, N.: Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. Moscou, Ser. Internat., Sec. A, 1 (1937), # 6, 1–25.Google Scholar
  17. [17]
    Rothe, F.: Über das asymptotische Verhalten der Lösungen einer nichtlinearen parabolischen Differentialgleichung aus der Populationsgenetik. Dissertation, Universität Tübingen, 1975.Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • K. P. Hadeler
    • 1
  • F. Rothe
    • 1
  1. 1.Lehrstuhl für BiomathematikUniversität TübingenTübingenFederal Republic of Germany

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