Bifurcations in a two-dimensional population evolution model (Q1395117)

The author considers the following population model for two species \[ \begin{aligned} dx/dt &= mxy/(x+y)+ \psi x -p_{12}| \psi_1| x+ p_{21}|\psi_2| y,\\ dy/dt &=-mxy/(x+y)+ \psi_2 y +p_{12}|\psi_1| x- p_{21}|\psi_2| y.\end{aligned} \] Here, \(\psi_i= a_i- b_{i1}x- b_{i2}y\), \(i= 1,2\), and \(p_{ij}\geq 0\) are small mutation probabilities. The present paper is a continuation of the former study by \textit{I. K. Volkov} and \textit{A. P. Krishchenko} [Differ. Equations 32, No. 11, 1452--1461 (1996); translation from Differ. Uravn. 32, No. 11, 1457--1465 (1996; Zbl 0899.92028)]. Concerning the seven parameters \(m\), \(a_i\), \(b_{ij}\) which, by rescaling \(x\), \(y\), \(t\) could easily be reduced to five, the following assumptions hold: \(b_{11}= b_{21}> 0\), \(a_1> 0\), \(b_{12}> 0\), \(a_2< 0\), \(b_{22}< 0\). Let \(\sigma_i= \text{sign\,}\psi\) and \(\mu= m/(1- \sigma_1 p_{12}- \sigma_2 p_{21})+ a_1- a_2\). Then, for any set of parameters, the number, stability, and bifurcation of equilibria in the quadrant \(x\geq 0\), \(y\geq 0\) are determined as a function of the real parameter \(\mu\). The exhaustive results show that, for any admissible set of parameters, there are at most two internal equilibria (\(x>0\), \(y> 0\)). Since the given system is not smooth along the straight lines \(\psi_i= 0\), the investigation of the stability of those equilibria which happen to belong to at least one of these straight lines, requires some care in the process of linearization. Dynamic bifurcations such as Hopf bifurcation are not discussed in this paper.