Source-factored transfer rates for zero compartments

Let $(p_{ij},d_{ij})$ be the rates of a finite conservative PDS Finite conservative production destruction systems. A source-factored transfer representation on an open set $U\subseteq \mathrm{\Omega }\subseteq {\mathbb{R}}^m$ containing a compact set $K\subseteq [0,\mathrm{\infty }{)}^m$ consists of functions $a_{ij}:U\to \mathbb{R}$, $i\mathrm{\ne }j$, such that each $a_{ij}$ is $C^2$ on the open subset $U\subseteq {\mathbb{R}}^m$ in the multivariable sense of $C^k$ maps and multi-index derivative notation in Euclidean space, is nonnegative on $U\cap [0,\mathrm{\infty }{)}^m$, and, for every $y\in U\cap [0,\mathrm{\infty }{)}^m$,

$p_{ij}(y)=y_j{a}_{ij}(y),d_{ij}(y)=y_i{a}_{ji}(y),a_{ii}:=0.$

Here $a_{ij}$ is the transfer rate from donor $j$ to recipient $i$ per unit of donor quantity. Nonnegativity is required on admissible nonnegative states; the $C^2$ extension to the open neighborhood is used for local analysis and need not be nonnegative at inadmissible states with negative coordinates. The factorization is an additional hypothesis, not a consequence of nonnegative rates and the identity $p_{ij}=d_{ji}$. On strictly positive states the ratio $p_{ij}(y)\mathrm{/}{y}_j$ equals $a_{ij}(y)$; at a zero donor coordinate the factorized coefficient remains defined and avoids the indeterminate expression $0\mathrm{/}0$.