math.DS — Dynamical Systems

Endotactic permanence and quantitative recovery with vanishing reverse rates

Contributed by Michael Mislan

For a finite ordinary endotactic reaction network, measurable reaction rates in a fixed positive compact box admit a compact convex invariant absorbing set in every positive stoichiometric class. Global positive Carathéodory solutions are unique, and entry times are uniform over compact initial sets and all admitted rate histories. The argument combines the finite affine construction of OpenAI with an endotactic witness estimate that retains the reaction projection lengths. We then quantify what permanence permits as reverse rates shrink. For autonomous weakly reversible networks with $k_e=\bar k_e\varepsilon^{a_e}$, eventual reciprocal concentration bounds depend on the exponent spread $\Delta=a_+-a_-$ through $\varepsilon^{\Delta/(2H)}$, where $H>0$ is a network-dependent geometric constant chosen before the individual rates. In an explicit three-species system with small positive rate parameters $p$ and $h$, prepared concentrations have scales $ph^3$, $ph^2$, $ph$, and the worst-case prepared recovery time lies between $(400/1313)(p^2h^7)^{-1}$ and $(31/10)(p^2h^7)^{-1}$. A weighted inventory gives necessary restoration and depletion times from arbitrary positive seeds. Further results treat common clocks, reverse-exponent families and a conventional stochastic lower comparison. Two registered Lean developments support the permanence theorem, continuous dependence and selected deterministic quantitative results. Their precise coverage is separated from conventional extensions and numerical illustrations. Permanence at fixed positive rates supplies neither a uniform guarantee at zero reverse rate nor a global sharp-corridor recovery deadline.