Inward reaction rectangles

Let $F:U\to {\mathbb{R}}^m$ be locally Lipschitz, and let $R={\prod }_{i=1}^m[a_i,b_i]\subset U$ with finite $a_i<b_i$. The rectangle is reaction-inward if, for every $i$, $F_i(z)\ge 0$ on the lower face $z_i=a_i$ and $F_i(z)\le 0$ on the upper face $z_i=b_i$, with all other coordinates in their intervals. It is reaction-invariant if the solution map ${\mathrm{\Phi }}_F^t$ is defined for all $t\ge 0$ on $R$ and ${\mathrm{\Phi }}_F^t(R)\subseteq R$ for every $t\ge 0$.

For a rectangle with $a_i=0$ for all $i$, quasi-positivity from Reaction vector fields and quasi-positivity implies the lower-face condition. That condition tests only points in $R$ and does not imply quasi-positivity on all of $U\cap [0,\mathrm{\infty }{)}^m$. The upper-face condition is independent and must be checked separately. The inward-sign condition is a hypothesis, not by definition an invariance theorem.