The permanence conjecture for weakly reversible mass-action systems with fixed rates

Theorem 1.1: for a finite weakly reversible network with fixed positive rates and any positive stoichiometric class $P$, there is a nonempty compact convex forward-invariant $K_P\subset P$ that every solution from $P$ enters in finite time; hence one $\varepsilon_P$ bounds every trajectory in $P$ between $\varepsilon_P$ and $\varepsilon_P^{-1}$ after its entry time, also on unbounded classes. The proof adds strict increase off the trapping set and a passage between scales to the companion's construction. Uniformity holds within one class and one choice of rates; entry times may depend on the initial point, and time-varying rates are not treated.

Theorem 1.1: for a finite weakly reversible network with fixed positive rates and any positive stoichiometric class $P$, there is a nonempty compact convex forward-invariant $K_P\subset P$ that every solution from $P$ enters in finite time; hence one $\varepsilon_P$ bounds every trajectory in $P$ between $\varepsilon_P$ and $\varepsilon_P^{-1}$ after its entry time, also on unbounded classes. The proof adds strict increase off the trapping set and a passage between scales to the companion's construction. Uniformity holds within one class and one choice of rates; entry times may depend on the initial point, and time-varying rates are not treated.