Discrete Lotka–Volterra Dynamical Systems on the S5 Simplex Associated with Bigraphs

Abstract

Purpose. This paper investigates discrete Lotka–Volterra dynamical systems on the simplex S5 associated with bigraphs, with emphasis on the degenerate case in which some interaction coefficients vanish, potentially yielding an infinite set of fixed points. Methods. For several classes of such systems, we analyze the asymptotic behavior of trajectories and the geometric structure of the associated invariant sets by employing Lyapunov-type functions, Jacobian analysis, and graph-induced polyhedral decompositions. Results. We classify fixed points, describe invariant sets, and characterize trajectory routes and attraction basins. For the simplest bigraph configuration (I = {1}, II = {2, 3, 4, 5, 6}), all trajectories are shown to converge to specific invariant polyhedral regions, and the asymptotic behavior is described via Lyapunov functions. For more complex bigraph structures (I = {1, 2}, II = {3, 4, 5, 6}), we identify three qualitatively distinct sub-cases depending crucially on the configuration of characteristic point sets. In one sub-case, positive trajectories do not converge and approach the boundary ∂S5 infinitely often, while negative trajectories converge to well-defined invariant sets. In the remaining two sub-cases, both positive and negative trajectories converge, with basin dimensions explicitly determined by the Jacobian spectrum. Conclusion. These results provide a systematic geometric and dynamical description of discrete Lotka–Volterra systems on S5, showing how bigraph structure and coefficient degeneracy influence fixed-point sets, invariant regions, and trajectory behavior. The developed framework may be applicable to discrete population models and related problems in mathematical biology.