Simple and complex dynamics in the model of evolution of two populations coupled by migration with non-overlapping generations

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Abstract

Purpose is to study the mechanisms leading to genetic divergence (stable genetic differences between two adjacent populations). We considered the following classical model situation. Populations are panmictic with Mendelian rules of inheritance. The action of natural selection (differences in fitness) on each of population is the same and is determined by the genotypes of only one diallel locus. We assume that adjacent generations do not overlap and genetic transformations can be described by a discrete time model. This model describes the change in the concentration of one of the alleles in each population and the ratio (weight) of first population to the total size. Methods. We used the analogue of saddle charts to construct parametric portraits showing the domains of qualitatively different dynamic modes. The study is supplemented with phase portraits, basins of attraction and bifurcation diagrams. Results. We found that the model dynamic regimes qualitatively coincide with the regimes of a similar model with continuous time, but only for a weak migration. With a strong coupling, fluctuations of the phase variables are possible. We showed that the genetic divergence is possible only with reduced fitness of heterozygotes and is the result of a series of bifurcations: pitchfork bifurcation, period doubling, or saddle-node bifurcation. After these qualitative changes, the dynamics become bi- or quadstable. In the first case, the solutions corresponding to the genetic divergence are unstable and are just a part of the transient process to monomorphic state. In the second case, the divergence is stable and appears as 2-cycle for a strong migration coupling. Conclusion. In neighboring populations, movement towards an asymptotic genetic structure (monomorphism, polymorphism or divergence) can be strictly monotonous or in the form of damped unstable or undamped stable fluctuations with a period of 2 for biologically significant parameters. For insignificant parameters, we found a complex dynamics (chaos) that consist of divergent fluctuations around fixed points and quasi-random transitions between them.

About the authors

Matvej Pavlovich Kulakov

Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

ul. Sholom-Aleikhema, 4, Birobidzhan, 679016, Russia

Efim Yakovlevich Frisman

Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

ul. Sholom-Aleikhema, 4, Birobidzhan, 679016, Russia

References

  1. Haldane J. B. S. A mathematical theory of natural and artificial selection. Part II. The influence of partial self-fertilisation, inbreeding, assortative mating, and selective fertilisation on the composition of Mendelian populations, and on natural selection // Biological Reviews. 1924. Vol. 1, no. 3. P. 158-163. doi: 10.1111/j.1469-185X.1924.tb00546.x.
  2. Fisher R. A. The Genetical Theory of Natural Selection. Oxford: Clarendon Press, 1930. 272 p. doi: 10.5962/bhl.title.27468.
  3. Wright S. Evolution in Mendelian populations // Genetics. 1931. Vol. 16, no. 2. P. 97-159. doi: 10.1093/genetics/16.2.97.
  4. Фрисман Е. Я., Шапиро А. П. Избранные математические модели дивергентной эволюции популяций. М.: Наука, 1977. 152 с.
  5. Свирежев Ю. М., Пасеков В. П. Основы математической генетики. М.: Наука, 1982. 512 с.
  6. Фрисман Е. Я., Первичная генетическая дивергенция (Теоретический анализ и моделирование). Владивосток: ДВНЦ АН СССР, 1986. 160 с.
  7. Burger R. A survey of migration-selection models in population genetics // Discrete & Continuous Dynamical Systems - B. 2014. Vol. 19, no. 4. P. 883-959. doi: 10.3934/dcdsb.2014.19.883.
  8. Carroll S. P., Hendry A. P., Reznick D. N., Fox C. W. Evolution on ecological time-scales // Functional Ecology. 2007. Vol. 21, no. 3. P. 387-393. doi: 10.1111/j.1365-2435.2007.01289.x.
  9. Pelletier F., Garant D., Hendry A. P. Eco-evolutionary dynamics // Phil. Trans. R. Soc. B. 2009. Vol. 364, no. 1523. P. 1483-1489. doi: 10.1098/rstb.2009.0027.
  10. Yeaman S., Otto S. P. Establishment and maintenance of adaptive genetic divergence under migration, selection, and drift // Evolution. 2011. Vol. 65, no. 7. P. 2123-2129. doi: 10.1111/j.1558-5646.2011.01277.x.
  11. Bertram J., Masel J. Different mechanisms drive the maintenance of polymorphism at locisubject to strong versus weak fluctuating selection // Evolution. 2019. Vol. 73, no. 5. P. 883-896. doi: 10.1111/evo.13719.
  12. Neverova G. P., Zhdanova O. L., Frisman E. Y. Effects of natural selection by fertility on the evolution of the dynamic modes of population number: bistability and multistability // Nonlinear Dyn. 2020. Vol. 101, no. 1. P. 687-709. doi: 10.1007/s11071-020-05745-w.
  13. Zhdanova O. L., Frisman E. Y. Genetic polymorphism under cyclical selection in long-lived species: The complex effect of age structure and maternal selection // Journal of Theoretical Biology. 2021. Vol. 512. P. 110564. doi: 10.1016/j.jtbi.2020.110564.
  14. Telschow A., Hammerstein P., Werren J. H. The effect of Wolbachia on genetic divergence between populations: Models with two-way migration // The American Naturalist. 2002. Vol. 160, no. S4. P. S54-S66. doi: 10.1086/342153.
  15. Fussmann G. F., Loreau M., Abrams P. A. Eco-evolutionary dynamics of communities and ecosystems // Functional Ecology. 2007. Vol. 21, no. 3. P. 465-477. doi: 10.1111/j.1365-2435.2007.01275.x.
  16. Tellier A., Brown J. K. M. Stability of genetic polymorphism in host-parasite interactions // Proc. R. Soc. B. 2007. Vol. 274, no. 1611. P. 809-817. doi: 10.1098/rspb.2006.0281.
  17. Nagylaki T., Lou Y. The dynamics of migration-selection models // In: Friedman A. (ed) Tutorials in Mathematical Biosciences IV. Vol. 1922 of Lecture Notes in Mathematics. Berlin, Heidelberg: Springer, 2008. P. 117-170. doi: 10.1007/978-3-540-74331-6_4.
  18. Akerman A., Burger R. The consequences of gene flow for local adaptation and differentiation: a two-locus two-deme model // J. Math. Biol. 2014. Vol. 68, no. 5. P. 1135-1198. doi: 10.1007/s00285-013-0660-z.
  19. Пасеков В. П. К анализу слабого двулокусного отбора по жизнеспособности и квазиравновесия по сцеплению // Доклады Академии наук. 2019. Т. 484, № 6. С. 781-785. doi: 10.31857/S0869-56524846781-785.
  20. Фрисман Е. Я., Кулаков М. П. О генетической дивергенции в системе двух смежных популяций, обитающих на однородном ареале // Известия вузов. ПНД. 2021. Т. 29, № 5. С. 706-726. doi: 10.18500/0869-6632-2021-29-5-706-726.
  21. Фрисман Е. Я., Жданова О. Л., Кулаков М. П., Неверова Г. П., Ревуцкая О. Л. Математическое моделирование популяционной динамики на основе рекуррентных уравнений: результаты и перспективы. Ч. II // Известия РАН. Серия биологическая. 2021. № 3. С. 227-240. doi: 10.31857/S000233292103005X.
  22. Altrock P. M., Traulsen A., Reeves R. G., Reed F. A. Using underdominance to bi-stably transform local populations // Journal of Theoretical Biology. 2010. Vol. 267, no. 1. P. 62-75. doi: 10.1016/j.jtbi.2010.08.004.
  23. Laruson A. J., Reed F. A. Stability of underdominant genetic polymorphisms in population networks // Journal of Theoretical Biology. 2016. Vol. 390. P. 156-163. doi: 10.1016/j.jtbi.2015.11.023.
  24. Гонченко А. С., Гонченко С. В., Казаков А. О., Козлов А. Д. Математическая теория динамического хаоса и её приложения: Обзор. Часть 1. Псевдогиперболические аттракторы // Известия вузов. ПНД. 2017. Т. 25, № 2. С. 4-36. doi: 10.18500/0869-6632-2017-25-2-4-36.

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