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Journal of Mathematical Biology

, Volume 2, Issue 3, pp 241–249 | Cite as

An application of Wazewski's method to a non-linear boundary value problem which arises in population genetics

  • C. Conley
Article

Summary

A non-linear boundary value problem is treated using the principle of T. Wazewski. The equation is d2/d, x2p (x)+s (x) p (1−p)=0 where s (x) is non zero near ±∞. The boundary condition on p at ±∞ is 0 and 1 according as sgn s (±∞) is −1 or +1. Two essentially different cases are treated, namely sgn s (+ ∞)= ±sgn s (-∞). A radially symmetric problem with x ε R2 is also discussed. The Wazewski principle allows one to describe the sets of initial data which satisfy the boundary conditions at + ∞ and at -∞ and to show how they intersect. The problem arises in the study of clines in population genetics theory.

Keywords

Boundary Condition Initial Data Stochastic Process Probability Theory Population Genetic 
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]
    Haldane, J. B. S.: The theory of the cline. J. Genet. 48, 277–284 (1948).Google Scholar
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    Crow, J. F., Kimura, M.: An Introduction to Population Genetics Theory. New York: Harper and Row 1970.Google Scholar
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    Slatkin, M.: Gene flow and selection in a cline. Genetics 75, 733–756 (1973).Google Scholar
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    Hanson, W. D.: Effects of partial isolation (distance) migration, and fitness requirements among environmental pockets upon steady state gene frequencies. Biometrics 22, 453–468 (1966).Google Scholar
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    Wazewski, T.: Sur une méthode topologique de l'examen de l'allure asymptotique des integrales des équations differentielles. Proc. Internat. Congress Math. (Amsterdam, 1954), Vol. III., pp. 132–139 MR 19, 272. Groningen: Nordhoff; Amsterdam: North Holland.Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • C. Conley
    • 1
  1. 1.Mathematics Research CenterUniversity of WisconsinMadisonUSA

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