Analysis and classification of fixed points of operators on a simplex

This paper investigates the dynamical behavior of Lotka–Volterra type operators defined on the four and five dimensional simplexes, focusing on their fixed points and structural representation through directed graphs (tournaments). For several classes of such operators, we derive algebraic and combinatorial conditions under which the configuration of fixed points exhibits transitive, cyclic, or homogeneous structures. Using methods from algebraic graph theory, Lyapunov stability theory, and Young’s inequality, explicit criteria are established for the existence, uniqueness, and stability of interior and boundary fixed points. A detailed analysis is provided for the class of operators whose associated skew-symmetric matrices are in general position. The connection between the minors of these matrices and the orientation of arcs in the tournament is clarified, revealing how dynamical transitions correspond to changes in tournament type. Furthermore, we demonstrate that under certain parameter regimes, fixed points coincide with evolutionarily stable strategies (ESS) in replicator dynamics, thus bridging discrete population models and evolutionary game theory. The obtained results enrich the theory of quadratic stochastic and Lotka–Volterra operators, providing new insights into nonlinear mappings on simplexes, combinatorial dynamics, and applications to models of interacting populations.