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

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).