Investigation of the Asymptotics of Quadratic Lotka-Volterra Mappings and Their Connections with Elements of Graph Theory

Abstract

The relevance of studying discrete Lotka-Volterra mappings lies in their applicability to modeling epidemiological and environmental problems. In this regard, this work focuses on the asymptotic behavior of these mappings and their connection with graph theory. The paper demonstrates that, when these mappings are in a general position, they can be associated with complete oriented graphs. However, in degenerate cases, it may be associated with partially oriented graphs. Furthermore, the paper proves that, if the skew-symmetric matrix corresponding to a Lotka-Volterra mapping is not in general position, then the set of fixed points becomes infinite. In addition, sets of fixed points are constructed for systems that are discrete analogues of continuous compartmental models, SIR; SIRD; and the characteristics of these fixed points are studied by analyzing the spectrum of the Jacobian matrix and constructing a phase portrait of the trajectories of the interior points.