Degenerate cases in Lotka–Volterra systems acting in the simplex $S^3$

In this work, we examine the dynamical behavior of discrete Lotka–Volterra operators associated with a given signature. The primary objectives are as follows: determine the polytopes corresponding to the signature structure of the operator; establish the order of their arrangement based on the trajectory operator; identify invariant polytopes under the action of the operator; compute the $\alpha$-limit and $\omega$-limit sets of the trajectories; determine the transition routes governed by the transformation $T_{\sigma_1}$. The transition of points between polytopes is analyzed for understanding the asymptotic behavior of the system.