Classification of Character of Rest Points of the Lotka-Volterra Operator Acting in $S^4$

In the paper, we consider the problem of studying the trajectories of points under the action of the Lotka-Volterra operator. In short, the goal is to study the dynamics of trajectories of interior points by finding fixed points and studying the Jacobian spectrum at these points of the operator in question. It turned out that in a number of applied problems, there are Lotka-Volterra mappings of exactly this type, and the points of the simplex in this case are considered as the states of the system under study. In this case, the simplex-preserving mapping defines the discrete law of evolution of the given system. Starting from a certain starting point, we can consider the sequence that determines the evolution of this point. The work explicitly shows sets of limit points for positive and negative trajectories, which in turn describe in applied problems the beginning and the end of the evolutionary process, respectively.