otka-Volterra Dynamics on S 3 with a Partially Oriented Interaction Graph
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We study a five-parameter Lotka–Volterra quadratic stochastic operator on the three-
dimensional simplex S 3 in the degenerate case ce = ad. The corresponding skew-symmetric matrix has
exactly one zero interaction coefficient, and its partially oriented interaction graph contains two strong
three-vertex subtournaments. On the fixed labelled vertex set {1,2,3,4}, the isomorphism class of this
graph contains exactly twelve labelled realizations. The degeneracy reduces the four coordinate growth
forms to two independent linear forms. We determine the complete fixed-point set, prove that the kernel
intersects the simplex in a segment of fixed points, construct the six non-empty signature polytopes and
their adjacency cycle, and describe the admissible forward and backward crossing directions. We also prove
convergence of global negative trajectories contained in the relative interior to unique points of the fixed
segment and compute the Jacobian spectrum along that segment, including explicit eigenvalues and eigen-
vectors at its endpoints.
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