Abstract
An ordinary differential equation model for two competing populations with genetic variation in one population is presented. The degree of frequency dependence needed to produce various configurations of stable equilibria is discussed. For example, if the fitnesses are frequency independent then there may exist stable polymorphism although the genetically varying population becomes extinct in each fixation plane. Stable polymorphism where the genetically invariant population becomes extinct in each fixation plane requires frequency dependence in the fitness of the genetically invariant population.
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Crow, J. F., Kimura, M.: An introduction to population genetics theory. New York, Harper and Row 1970
Ginzburg, L. R.: Theory of natural selection and population growth. Menlo Park, Benjamin/Cummings 1983
Hadeler, K. P., Glas, D.: Quasimonotone systems and convergence to equilibrium in a population genetic model. J. Math. Anal. Appl. 95, 297–303 (1983)
Hirsch, M. W.: Systems of differential equations which are competitive or cooperative. I: Limit sets. SIAM J. Math. Anal. 13, 167–179 (1982)
Hirsch, M. W.: Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere. SIAM J. Math. Anal. 16, 423–439 (1985)
Hirsch, M. W., Smale, S.: Differential equations, dynamical systems, and linear algebra. New York, Academic Press 1974
Levin, S. A.: Some approaches to the modelling of coevolutionary interactions. Coevolution, Univ. Chicago Press, 21–65 (1984)
Levin, S. A., Udovic, J. D.: A mathematical model of coevolving populations. Amer. Natur. 111, 657–675 (1977)
May, R. M.: Stability and complexity in model ecosystems. Princeton Univ. Press 1973
Pimentel, D.: Population regulation and genetic feedback. Science 159, 1432–1437 (1968)
Rescigno, A., Richardson, I. W.: The struggle for life: I. Two species. Bull. Math. Biophysics 29, 377–388 (1967)
Selgrade, J. F.: Asymptotic behavior of solutions to single loop positive feedback systems. J. Diff. Eq. 38, 80–103 (1980)
Selgrade, J. F., Namkoong, G.: Stable periodic solutions for two species, density dependent coevolution. J. Math. Biol. 22, 69–80 (1985)
Waltman, P.: Competition models in population biology. CBMS — NSF 45: SIAM 1983
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Selgrade, J.F., Namkoong, G. Examples of the effect of genetic variation on competing species. J. Math. Biology 24, 193–206 (1986). https://doi.org/10.1007/BF00275998
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DOI: https://doi.org/10.1007/BF00275998