Linear Homogeneous Inequalities and Trajectory Routes of the Degenerate Lotka–Volterra Operators

The work is dedicated to the study and finding of trajectory routes of degenerate Lotka–Volterra mappings. It is known that Lotka–Volterra mappings are automorphisms, which allows for the construction of both positive and negative orbits through the construction of a Lyapunov function and the analysis of the Jacobian matrix spectrum is accomplished in this work. The paper introduces a new definition of trajectory routes, the concept of positive and negative basins for stationary points, as well as velocities for mappings of this kind. Additionally, the proposed article presents a new approach to studying and finding routes by partitioning the simplex into parts according to the constructed signature. The analytical results obtained in this work are applicable in the fields of epidemiology, ecology, and economics.