The boundedness and persistence conjectures for weakly reversible mass-action systems

Theorem 1.1: for any finite weakly reversible network with positive constant rates and any positive $x^0$, the solution exists for all $t\ge0$ and some $\varepsilon\in(0,1)$, depending on the network, the rates and $x^0$, gives $\varepsilon\le x_i(t)\le\varepsilon^{-1}$ for all $t\ge0$ and all $i$. The proof builds a compact convex forward-invariant polytope around each initial point from a finite concave piecewise-affine function. A remark derives convergence for complex-balanced systems, which the paper treats as known (citing Craciun's 2026 revision), not as new. The bounds are not uniform over a compatibility class (the permanence entry), and time-dependent rates are not treated.

Theorem 1.1: for any finite weakly reversible network with positive constant rates and any positive $x^0$, the solution exists for all $t\ge0$ and some $\varepsilon\in(0,1)$, depending on the network, the rates and $x^0$, gives $\varepsilon\le x_i(t)\le\varepsilon^{-1}$ for all $t\ge0$ and all $i$. The proof builds a compact convex forward-invariant polytope around each initial point from a finite concave piecewise-affine function. A remark derives convergence for complex-balanced systems, which the paper treats as known (citing Craciun's 2026 revision), not as new. The bounds are not uniform over a compatibility class (the permanence entry), and time-dependent rates are not treated.