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

• Published in: Mathematics (Basel) Vol. 11; no. 19; p. 4134
• Main Authors: Frisman, Efim, Kulakov, Matvey
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.10.2023
• Subjects: age structure; Analysis; Bifurcations; bistability; Differential equations; Divergence; Dynamic stability; dynamics; Equilibrium; Evolution; genetic divergence; Genotype & phenotype; Mathematical models; migration; Numerical analysis; Polymorphism; Population; Population genetics; Populations; Qualitative analysis; Stability analysis; Russia
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11194134

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).