Skip to main content
SpringerLink
Log in

Effects of natural selection by fertility on the evolution of the dynamic modes of population number: bistability and multistability

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A discrete-time model of a structured population dynamics considering both density-dependent regulation and natural selection is studied. In the model, the dynamics of two population groups corresponding to different development stages is described, and birth rate limitation is considered. Fertility is assumed to change during microevolution. The stability loss of nontrivial fixed points was shown to realise according to the Neimark–Sacker scenario and the Feigenbaum one. Bifurcations, dynamic modes and possible shifting for the proposed model are explored. Bistability and multistability are also revealed. The phase space structure of bistability and multistability areas in which a variation in population sizes or population genotype compositions can lead to shift in the dynamic modes is examined using attraction basins. The multistability of genetic structure dynamics is also investigated. The presence of multistability fundamentally increases the quantity and variety of possible evolutionary scenarios and makes them dependent on both parameter values and initial conditions. In particular, shifting dynamic mode in a population can occur because of an increase in an individual’s reproductive potential during natural evolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Darwin, C.: On the Origin of Species by MEANS of Natural Selection. Murray, London (1859)

    Google Scholar 

  2. Fisher, R.A.: The Genetical Theory of Natural Selection. Clarendon Press, Oxford (1930) [Variorum edition, Bennett J.H. (ed), 1999, Oxford University Press, Oxford]. https://doi.org/10.5962/bhl.title.27468

  3. Wright, S.: The genetical theory of natural selection: a review. J. Hered. 21, 340–356 (1930)

    Google Scholar 

  4. Haldane, J.B.S.: The Causes of Evolution. Longman Green, London (1932)

    Google Scholar 

  5. Chetverikov, S.S.: About some moments of evolutionary process from the point of view of contemporary genetics. Zhurnal Eksp. Biol. 2, 3–54 (1926). (in Russ.)

    Google Scholar 

  6. Volterra, V.: Variations and fluctuations of the number of individuals in animal species living together. ICES J. Mar. Sci. 3(1), 3–51 (1928)

    MathSciNet  Google Scholar 

  7. Gauze, G.F.: Analysis of struggle for survival in mixed populations. Zool. Zhurnal 14(2), 243–270 (1935)

    Google Scholar 

  8. Verhulst, P.F.: Deuxième mémoire sur la loi d’accroissement de la population. Mém. Acad. R. Sci. Lett. Beaux-arts Belgique 20, 1–32 (1847)

    Google Scholar 

  9. Chitty, D.: Mortality among voles (Microtus agrestis) at Lake Vyrnwy, Montgomeryshire in 1936–9. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci. 236(638), 505–552 (1952). https://doi.org/10.1098/rstb.1952.0009

    Article  Google Scholar 

  10. Chitty, D.: Population processes in the vole and their relevance to general theory. Can. J. Zool. 38, 99–113 (1960). https://doi.org/10.1139/z60-011

    Article  Google Scholar 

  11. Pimentel, D.: Population regulation and genetic feedback. Science 159, 1432–1437 (1968). https://doi.org/10.1126/science.159.3822.1432

    Article  Google Scholar 

  12. Birch, L.C.: Selection in Drosophila pseudoobscura in relation to crowding. Evolution 1955, 389–399 (1955)

    Google Scholar 

  13. Charlesworth, B.: Selection in density-regulated populations. Ecology 52(3), 469–474 (1971)

    Google Scholar 

  14. Gottlieb, L.D.: Genetic stability in a peripheral isolate of Stephanomeria exigua ssp. coronaria that fluctuates in population size. Genetics 76(3), 551–556 (1974)

    Google Scholar 

  15. Gaines, M.S., Leroy, R., McClenaghan Jr., L.R., Rose, R.K.: Temporal patterns of allozymic variation in fluctuating populations of Microtus ochrogaster. Evolution 1978, 723–739 (1978)

    Google Scholar 

  16. Frisman, E.Ya.: Primary Genetic Divergence (Theoretical Analysis and Modeling). Dal’nevost. Nauch. Tsentr Akad. Nauk SSSR, Vladivostok (1986) (in Russ.)

  17. Boonstra, R., Boag, P.T.: A test of the Chitty hypothesis: inheritance of life-history traits in meadow voles Microtus pennsylvanicus. Evolution 41, 929–947 (1987). https://doi.org/10.2307/2409183

    Article  Google Scholar 

  18. Carroll, S.P., Hendry, A.P., Reznick, D.N., Fox, C.W.: Evolution on ecological time-scales. Func. Ecol. 21(3), 387–393 (2007)

    Google Scholar 

  19. Endler, J.A.: Natural selection on color patterns in Poecilia reticulata. Evolution 34(1), 76–91 (1980)

    Google Scholar 

  20. Reznick, D.N., Bryga, H.: Life-history evolution in guppies (Poecilia reticulata): 1. Phenotypic and genetic changes in an introduction experiment. Evolution 41(6), 1370–1385 (1987)

    Google Scholar 

  21. Reznick, D.A., Bryga, H., Endler, J.A.: Experimentally induced life-history evolution in a natural population. Nature 346(6282), 357–359 (1990)

    Google Scholar 

  22. Stearns, S.C.: The genetic basis of differences in life-history traits among six populations of mosquitofish (Gambusia affinis) that shared ancestors in 1905. Evolution 37, 618–627 (1983)

    Google Scholar 

  23. Williams, C.K., Moore, R.J.: Phenotypic adaptation and natural selection in the wild rabbit, Oryctolagus cuniculus, Australia. J. Anim. Ecol. 58, 495–507 (1989)

    Google Scholar 

  24. Sinervo, B., Svensson, E., Comendant, T.: Density cycles and an offspring quantity and quality game driven by natural selection. Nature 406, 985–988 (2000). https://doi.org/10.1038/35023149

    Article  Google Scholar 

  25. Yoshida, T., Jones, L.E., Ellner, S.P., Fussmann, G.F., Hairston, N.G.: Rapid evolution drives ecological dynamics in a predator–prey system. Nature 424, 303–306 (2003). https://doi.org/10.1038/nature01767

    Article  Google Scholar 

  26. Pelletier, F., Garant, D., Hendry, A.P.: Eco-evolutionary dynamics. Philos. Trans. R. Soc. B 364, 1483–1489 (2009). https://doi.org/10.1098/rstb.2009.0027

    Article  Google Scholar 

  27. Mallet, J.: The struggle for existence. How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution and speciation. Evol. Ecol. Res. 14, 627–665 (2012)

    Google Scholar 

  28. Bertram, J., Masel, J.: Density-dependent selection and the limits of relative fitness. Theor. Popul. Biol. 129, 81–92 (2019)

    MATH  Google Scholar 

  29. Waxman, D., Gavrilets, S.: 20 questions on adaptive dynamics. J. Evol. Biol. 18, 1139–1154 (2005). https://doi.org/10.1111/j.1420-9101.2005.00948.x

    Article  Google Scholar 

  30. Fussmann, G.F., Loreau, M., Abrams, P.A.: Eco-evolutionary dynamics of communities and ecosystems. Funct. Ecol. 21(3), 465–477 (2007)

    Google Scholar 

  31. Govaert, L., Fronhofer, E.A., Lion, S., Eizaguirre, C., Bonte, D., Egas, M., Hendry, A.P., Martins, A.D.B., Melian, C.J., Raeymaekers, J.A.M., et al.: Eco-evolutionary feedbacks—theoretical models and perspectives. Funct. Ecol. 33, 13–30 (2019)

    Google Scholar 

  32. Stearns, S.C.: The Evolution of Life Histories. Oxford University Press, Oxford (1992)

    Google Scholar 

  33. Ellner, S.: Environmental fluctuations and the maintenance of genetic diversity in age or stage-structured populations. Bull. Math. Biol. 58(1), 103–127 (1996)

    MATH  Google Scholar 

  34. Barton, N., Briggs, D., Eisen, J., Goldstein, D., Patel, N.: Evolution. Cold Spring Harbor Laboratory Press, New York (2007)

    Google Scholar 

  35. Yamamichi, M., Ellner, S.P.: Antagonistic coevolution between quantitative and Mendelian traits. Proc. R. Soc. B Biol. Sci. 283, 20152926 (2016)

    Google Scholar 

  36. Yamamichi, M., Hoso, M.: Roles of maternal effects in maintaining genetic variation: maternal storage effect. Evolution 71(2), 449–457 (2017)

    Google Scholar 

  37. Shapiro, A.P.: To the question about cycles in the recurrent series. Upravlenie i Informaciya 3, 96–118 (1972). (in Russ.)

    Google Scholar 

  38. May, R.M.: Biological populations obeying difference equations: stable points, stable cycles, and chaos. J. Theor. Biol. 51(2), 511–524 (1975)

    Google Scholar 

  39. Syta, A., Litak, G.: Stochastic description of the deterministic Ricker’s population model. Chaos Soliton Fractal 37, 262–268 (2008)

    MathSciNet  MATH  Google Scholar 

  40. Sacker, R.J., von Bremen, H.F.: A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters. Appl. Math. Comput. 217, 1213–1219 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Leslie, P.H.: On the use of matrices in certain population mathematics. Biometrika 33(3), 183–212 (1945)

    MathSciNet  MATH  Google Scholar 

  42. Leslie, P.H.: Some further notes on the use of matrices in population mathematics. Biometrika 35(3/4), 213–245 (1948)

    MathSciNet  MATH  Google Scholar 

  43. Lefkovitch, L.P.: The study of population growth in organisms grouped by stages. Biometrics 21, 1–18 (1965)

    Google Scholar 

  44. Caswell, H.: Matrix Population Models: Construction, Analysis, and Interpretation, 2nd edn. Sinauer Associates, Sunderland (2001)

    Google Scholar 

  45. Frisman, E.Y.A., Kulakov, M.P., Revutskaya, O.L., Zhdanova, O.L., Neverova, G.P.: The key approaches and review of current researches on dynamics of structured and interacting populations. Comput. Res. Model 11(1), 119–151 (2019). https://doi.org/10.20537/2076-7633-2019-11-1-119-151

    Article  Google Scholar 

  46. Dajor, R.: Precis d’Ecologie. Paris (1972)

  47. Svirezhev, I., Logofet, D.O.: The Stability of Biological Communities. Nauka, Moscow (1978). (in Russ.)

    Google Scholar 

  48. Frisman, E.Y., Skaletskaya, E.I.: Strange attractors in elementary models for dynamics of the quantity of biological populations. Obozrenie Prikladnoy i Promyshlennoy Matematiki 1(6), 988–1008 (1994). (in Russ.)

    MATH  Google Scholar 

  49. Frisman, E.Y., Zhdanova, O.L.: Evolutionary transition to complex population dynamic patterns in a two-age population. Russ. J. Genet. 45(9), 1124–1133 (2009)

    Google Scholar 

  50. Frisman, E.Y., Neverova, G.P., Revutskaya, O.L.: Complex dynamics of the population with a simple age structure. Ecol. Model. 222(12), 1943–1950 (2011)

    Google Scholar 

  51. Neverova, G.P., Zhdanova, O.L., Frisman, E.Y.A.: The emergence of complex dynamics during the evolution of a structured limited population. Russ. J. Genet. (2020). https://doi.org/10.31857/s0016675820060065

    Article  Google Scholar 

  52. Kuznetsov, A.P., Sedova, J.V.: Bifurcations of three- and four-dimensional maps: universal properties. Izvestiya VUZ. Appl. Nonlinear Dyn. 20(5), 26–43 (2012). https://doi.org/10.18500/0869-6632-2012-20-5-26-43. (in Russ.)

    Article  MATH  Google Scholar 

  53. Kuznetsov, A.P., Savin, A.V., Sedova, Y.V., Tyuryukin, L.V.: Bifurcations of Maps. Publishing Center Science, Saratov (2012). (in Russ.)

    Google Scholar 

  54. Neverova, G.P., Abakumov, A.I., Yarovenko, I.P., Frisman, E.Y.: Mode change in the dynamics of exploited limited population with age structure. Nonlinear Dyn. 94, 827–844 (2018). https://doi.org/10.1007/s11071-018-4396-6

    Article  Google Scholar 

  55. Frisman, E.Y., Neverova, G.P., Kulakov, M.P.: Change of dynamic regimes in the population of species with short life cycles: results of an analytical and numerical study. Ecol. Complex. 27, 2–11 (2016). https://doi.org/10.1016/j.ecocom.2016.02.001

    Article  Google Scholar 

  56. Zhivotovsky, L.A., Gharret, A.J., McGregor, A.J., Glubokovsky, M.K., Feldman, M.W.: Gene differentiation in pacific salmon (Oncorhynchus sp.): facts and models with reference to pink salmon (O. gorbuscha). Can. J. Fish. Aquat. Sci. 51(1), 223–232 (1994)

    Google Scholar 

  57. Gordeeva, N.V., Salmenkova, E.A., Altukhov, Y.P., Makhrov, A.A., Pustovoit, S.P.: Genetic changes in pink salmon Oncorhynchus gorbuscha Walbaum during acclimatization in the White sea basin. Russ. J. Genet. 39(3), 322–332 (2003)

    Google Scholar 

  58. Golovanov, I.S., Marchenko, S.L., Pustovoit, S.P.: Genetic monitoring of northern sea of Okhotsk populations of pink salmon (Oncorhynchus gorbuscha). Cytol. Genet. 43(6), 379 (2009)

    Google Scholar 

  59. Beacham, T.D., McIntosh, B., MacConnachie, C., Spilsted, B., White, B.A.: Population structure of pink salmon (Oncorhynchus gorbuscha) in British Columbia and Washington, determined with microsatellites. Fish. Bull. 110(2), 242–256 (2012)

    Google Scholar 

  60. Sato, S., Urawa, S.: Genetic variation of Japanese pink salmon populations inferred from nucleotide sequence analysis of the mitochondrial DNA control region. Environ. Biol. Fish. 100(10), 1355–1372 (2017)

    Google Scholar 

  61. Carson, H.L.: Increased genetic variance after a population bottleneck. Trends Ecol. Evol. 5(7), 228–230 (1990)

    Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Russian Foundation for Basic Research (Nos. 15-18-04-00073a, 18-51-45004 IND_a).

Author information

Authors and Affiliations

  1. Institute of Automation and Control Processes Far Eastern Branch, Russian Academy of Sciences, 5 Radio St., Vladivostok, Russia, 690041

    G. P. Neverova & O. L. Zhdanova

  2. Institute for Complex Analysis of Regional Problems, Far Eastern Branch, Russian Academy of Sciences, 4 Sholom-Aleikhem St., Birobidzhan, Russia, 679016

    E. Ya. Frisman

Authors
  1. G. P. Neverova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. L. Zhdanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Ya. Frisman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. P. Neverova.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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 101, 687–709 (2020). https://doi.org/10.1007/s11071-020-05745-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05745-w

Keywords

Search

Navigation