Bifurcations in the symmetric viability model with weak selection

Abstract

In this paper we discuss bifurcations of equilibria in the symmetric viability model with weak selection. Overcoming the difficulties from irrational fractions of the 4 parameters in determining the signs of determinant and trace, we find that the system has at most 5 interior equilibria, 4 edge equilibria and 4 corner equilibria and give a complete qualitative analysis for all equilibria in both hyperbolic cases and non-hyperbolic cases, including the case of fully null degeneracy. On this basis we display all bifurcations of equilibria such as pitchfork bifurcation, transcritical bifurcation and bifurcations at corner equilibria of fully null degeneracy, which is of codimension 2 but none of known results of codimension 2 can be applied to. We further discuss the unfolding system of fully null degeneracy, displaying all its bifurcations and giving bifurcation curves analytically. Our results provide an intuitive basis for us to predict the genetic composition of a population and provide the basic mathematical theory for the protection and loss of alleles.