Analysis of the dynamics of quadratic mappings of a simplex with skew-symmetric matrices that are not in general position
Analysis of the dynamics of quadratic mappings of a simplex with ske
DOI:
https://doi.org/10.56143/ujmcs.v1i1.4Keywords:
fixed point; homogeneous tournament; quadratic Lotka -- Volterra mapping; simplex.Abstract
The Lotka -- Volterra systems arise in questions of biology, population genetics, epidemiology, ecology, economics as well as in some branches of theoretical physics, in particular, in solid state physics. Some important questions of ecology (for example, biogens cycles) can be studied using Lotka -- Volterra mappings operating in a four-dimensional simplex with homogeneous tournaments. In this regard, the work is devoted to the construction and study of cards of fixed points of Lotka -- Volterra mappings operating in a four-dimensional simplex in the case of homogeneous tournaments (for arbitrary coefficients of a skew-symmetric matrix). The card of fixed points gives us a more detailed understanding of the asymptotic behavior of the trajectories of discrete dynamical Lotka -- Volterra systems.
In the paper, we show that even if the tournaments corresponding to the Lotka -- Volterra mappings are homogeneous, among them it is possible to distinguish a class of mappings with skew-symmetric matrices that are not matrices in a general position. It is not possible to generalize this kind of mappings; each of them represents a map of fixed points of a different type. This is clearly noted in the work. It is also shown that even in the case when the tournament corresponding to the Lotka -- Volterra mapping is homogeneous, the set of fixed points is infinite and the card of fixed points consists of a convex hull of fixed points belonging to strong faces.
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