Trajectory Routes and Velocity-Signature Classification of Transitive and Mixed Lotka–Volterra Tournament Operators on S 3

We give a complete velocity-signature classification of two discrete Lotka–Volterra op-
erators generated by fully oriented tournaments on the three-dimensional simplex S 3 : the transitive
tournament T 1 and the mixed tournament T 2 , in which one vertex is dominated by all others while
the remaining three vertices form a directed 3-cycle. The two matrices coincide in their first row
and column, so any difference in the resulting dynamics is attributable solely to the internal order
of the three subordinate strategies. Using the identity x 0
k − x k
= x k (Ax) k , we convert the sixteen
formal sign patterns of Ax into systems of linear inequalities and classify all admissible ones for both
models, obtaining N(T 1 ) = 3 and N(T 2 ) = 7. For each operator we give the explicit convex-hull
domains (where tractable), the complete Jacobian spectrum at every vertex, a Lyapunov-type argu-
ment for global convergence, the fixed-point set, and the evolutionary-game reading. Although both
operators eliminate the same dominated strategy, T 1 converges to a single dominant vertex, whereas
the restriction of V 2 to the complementary invariant face coincides with the classical non-convergent
three-cycle Volterra operator.