Siphon Calculus and Lyapunov Functions for Generalized Lotka-Volterra Systems: A Reaction Networks Perspective

Generalized Lotka-Volterra (GLV) systems, with roots in ecology, constitute one of the most studied classes of positive ODEs. Recently, a reaction-network perspective for a generalization useful in mathematical epidemiology, called block GLV systems, was offered by Adenane, Avram and Halanay. These authors developed a "siphon calculus" in which the boundary stability of block GLV equations is studied through (i) invariant faces associated with minimal siphons, (ii) transversal Jacobians, (iii) invasibility expressed via R-invasion functions, (iv) relay graphs, (v) exclusion partitions and (vi) Lyapunov functions, without leaving the original state space. Another reaction-network perspective for GLV systems was offered by Rojas La Luz, Yu and Craciun, who developed a global stability theory for positive equilibria by introducing associated polyexponential systems obtained through the logarithmic change of variables xi=eξi. In these logarithmic coordinates, compatibility classes become affine subspaces and simple quadratic Lyapunov functions establish global convergence of complex-balanced systems. The purpose of the present paper is to combine and compare these two perspectives. Our block GLV results here start with a general Perron-Volterra relay theorem (Theorem 7) and its explicit verification for the rank-one multi-strain class. The theorem constructs a face-adapted Lyapunov function in which resident blocks enter through Perron-weighted entropy terms and missing blocks through positive left Perron functionals, and reduces global convergence to resident and transversal closing conditions together with compactness. For the rank-one block model, the canonical Perron normalization gives the closing terms explicitly on an arbitrary resident support I: for every missing block k∉I, the corresponding coefficient has the sign of Rk(EI)-1. Hence, the relay-sink conditions Rk(EI)<1,k∉I, verify the transversal closing hypothesis and yield convergence to the resident invariant set selected by the resident closing condition (Corollary 20); when that set consists of a single equilibrium EI, the convergence is global to EI. For nonlinear scalar GLV systems, Theorem 6 gives an exact characterization of global Volterra admissibility through the Jacobian averaged along the segment joining the positive equilibrium x* to each point x: a∈W(x*)⇔(x-x*)TAJ¯(x;x*)+J¯(x;x*)TA(x-x*)≤0foreveryx∈R>0n. It also identifies the averaged-Jacobian matrix inequality as a sufficient global certificate, shows that uniform diagonal stability of the pointwise Jacobian family is a stronger sufficient condition, and shows, via a nonlinear counterexample, that even strict diagonal stability of Df(x*) at the equilibrium does not imply global Volterra admissibility. Then, we give a complex-balanced GLV example for which no positive Volterra weight yields a Lyapunov function on the whole positive orthant (Theorem 11). Thus, complex balance does not imply global Volterra admissibility. Interestingly, Volterra decrease is recovered on the compatibility manifold in this example, which leads to the open question whether such compatibility-restricted Volterra functions exist more generally for complex-balanced GLV systems (Problem 2). Finally, we show that all four logical combinations of complex balance and global Volterra admissibility occur among GLV systems, three of them already among affine systems (Theorem 12).