Reaction vector fields and quasi-positivity

Let $m\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open, and let $F:U\to {\mathbb{R}}^m$ be locally Lipschitz. Write $F_i$ for its $i$th component. The autonomous reaction equation is $z^{\mathrm{\prime }}(t)=F(z(t))$. The field is quasi-positive on $U\cap [0,\mathrm{\infty }{)}^m$ if, for every $i$ and every $z\in U\cap [0,\mathrm{\infty }{)}^m$ with $z_i=0$, one has $F_i(z)\ge 0$. This condition concerns only the coordinate faces of the nonnegative orthant.

For a rectangle $R={\prod }_{i=1}^m[a_i,b_i]\subset U$ with $a_i<b_i$, the field is inward-pointing on $R$ if $F_i(z)\ge 0$ whenever $z\in R$ and $z_i=a_i$, and $F_i(z)\le 0$ whenever $z\in R$ and $z_i=b_i$. If $a_i=0$ for every $i$, quasi-positivity supplies the lower-face half of this condition, but the upper-face inequalities remain separate hypotheses. Neither boundary condition by itself asserts that a numerical update preserves $R$.

For smoothness-dependent results below, $F\in C^4(U,{\mathbb{R}}^m)$ will be stated explicitly. No smoothness beyond local Lipschitz continuity is built into this definition.