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Transition from Bi- to Quadro-Stability in Models of Population Dynamics and Evolution

Author

Listed:
  • Efim Frisman

    (Institute of Complex Analysis of Regional Problems FEB RAS, Sholom-Aleikhem St. 4, 679016 Birobidzhan, Russia
    These authors contributed equally to this work.)

  • Matvey Kulakov

    (Institute of Complex Analysis of Regional Problems FEB RAS, Sholom-Aleikhem St. 4, 679016 Birobidzhan, Russia
    These authors contributed equally to this work.)

Abstract

The article is devoted to a review of bistability and quadro-stability phenomena found in a certain class of mathematical models of population numbers and allele frequency dynamics. The purpose is to generalize the results of studying the transition from bi- to quadro-stability in such models. This transition explains the causes and mechanisms for the appearance and maintenance of significant differences in numbers and allele frequencies (genetic divergence) in neighboring sites within a homogeneous habitat or between adjacent generations. Using qualitative methods of differential equations and numerical analysis, we consider bifurcations that lead to bi- and quadro-stability in models of the following biological objects: a system of two coupled populations subject to natural selection; a system of two connected limited populations described by the Bazykin or Ricker model; a population with two age stages and density-dependent regulation. The bistability in these models is caused by the nonlinear growth of a local homogeneous population or the phase bistability of the 2-cycle in populations structured by space or age. We show that there is a series of similar bifurcations of equilibrium states or fixed or periodic points that precede quadro-stability (pitchfork, period-doubling, or saddle-node bifurcation).

Suggested Citation

  • Efim Frisman & Matvey Kulakov, 2023. "Transition from Bi- to Quadro-Stability in Models of Population Dynamics and Evolution," Mathematics, MDPI, vol. 11(19), pages 1-28, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4134-:d:1251519
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    References listed on IDEAS

    as
    1. Frisman, E.Y. & Neverova, G.P. & Revutskaya, O.L., 2011. "Complex dynamics of the population with a simple age structure," Ecological Modelling, Elsevier, vol. 222(12), pages 1943-1950.
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