Dynamics of Degenerate Quadratic Lotka-Volterra Mappings Acting in a Four-Dimensional Simplex and Bigraphs

Abstract

The work is devoted to the problem of studying the trajectories of points under the action of the Lotka-Volterra operator acting in a four-dimensional simplex. We show the connection between oriented bigraphs and Lotka-Volterra mappings, and with the skew-symmetric matrices corresponding to them. The connection between such type operators and the corresponding skew-symmetric matrices is considered for the first time. In the paper, we introduce a definition of bigraphs as oriented graphs whose vertices are separated into two groups. Based on this, bigraphs, the vertices of which are separated as follows: one vertex in assigned to one group, and the remaining four are assigned to another group, are divided into three subclasses, taking into account the isomorphity of the bigraphs included in these groups. One representative from each isomorphic group was examined for the presence of fixed points of operators corresponding to them, and the characters of these points are established.