Advertisement

Journal of Mathematical Biology

, Volume 2, Issue 3, pp 219–233 | Cite as

A selection-migration model in population genetics

  • W. H. Fleming
Article

Summary

We consider a model with two types of genes (alleles) A1 A2. The population lives in a bounded habitat R, contained in r-dimensional space (r= 1, 2, 3). Let u (t, x) denote the frequency of A1 at time t and place x ɛ R. Then u (t, x) is assumed to obey a nonlinear parabolic partial differential equation, describing the effects of population dispersal within R and selective advantages among the three possible genotypes A1A1, A1A2, A2A2. It is assumed that the selection coefficients vary over R, so that a selective advantage at some points x becomes a disadvantage at others. The results concern the existence, stability properties, and bifurcation phenomena for equilibrium solutions.

Keywords

Differential Equation Partial Differential Equation 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion. and nerve propogation. Proc. Tulane Progr. in Partial Differential Eqns. Springer Lecture Notes in Mathematics, 1975.Google Scholar
  2. [2]
    Chafee, N.: Asymptotic behaviour for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Differential Equations.Google Scholar
  3. [3]
    Chafee, N.: Behaviour of solutions leaving the neighborhood of a saddle point for a nonlinear evolution equation, preprint.Google Scholar
  4. [4]
    Conley, C.: An application of Wazewski's method to a nonlinear boundary value problem which arises in population genetics, Univ. of Wisconsin Math. Research Center Tech. Summary Report No. 1444, 1974.Google Scholar
  5. [5]
    Fisher, R. A.: Gene frequencies in a cline determined by selection and diffusion. Biometrics 6, 353–361 (1950).Google Scholar
  6. [6]
    Fleming, W. H.: A nonlinear parabolic equation arising from a selection-migration model in genetics. IRIA Seminars Review, 1974.Google Scholar
  7. [7]
    Haldane, J. B. S.: The theory of a cline. J. Genet. 48, 277–284 (1948).Google Scholar
  8. [8]
    Hestenes, M. R.: Calculus of Variations and Optimal Control Theory. Wiley 1966.Google Scholar
  9. [9]
    Hoppensteadt, F. C.: Analysis of a Stable Polymorphism Arising in a Selection-Migration Model in Population Genetics dispersion and selection. J. Math. Biology 2, 235–240 (1975).Google Scholar
  10. [10a]
    Karlin, S.: Population division and migration-selection interaction. Population Genetics and Ecology. Academic Press 1976.Google Scholar
  11. [10b]
    Karlin, S., Richter-Dyn, N.: Some theoretical analysis of migration-selection interaction in a cline: a generalized 2 range environment. Population Genetics and Ecology. Academic Press 1976.Google Scholar
  12. [11]
    Karlin, S., McGregor J., unpublished.Google Scholar
  13. [12]
    Lions, J. L.: Équations Differentielles Operationelles. Berlin-Göttingen-Heidelberg: Springer, 1961.Google Scholar
  14. [13]
    Lions, J. L.: Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires. Dunod, 1969.Google Scholar
  15. [14]
    Lions, J. L., Magenes, E.: Problèmes aux Limites Non-Homogènes et Applications, vols. I, II. Dunod 1968.Google Scholar
  16. [15]
    Nagylaki, T.: Conditions for the existence of clines. Univ. of Wisconsin Madison Genetics Lab paper No. 1787, 1974.Google Scholar
  17. [16]
    Slatkin, M.: Gene flow and selection in a cline. Genetics 75, 733–756 (1973).Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • W. H. Fleming
    • 1
  1. 1.Lefschetz Center for Dynamical Systems Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations