Finite conservative production destruction systems

Fix an integer $m\ge 1$ and a domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^m$; its admissible nonnegative states are ${\mathrm{\Omega }}_{+}:=\mathrm{\Omega }\cap [0,\mathrm{\infty }{)}^m$. For every ordered pair $i,j\in \{1,\dots ,m\}$ let $p_{ij},d_{ij}:\mathrm{\Omega }\to \mathbb{R}$ be production and destruction transfer rates that are nonnegative on ${\mathrm{\Omega }}_{+}$. A finite conservative production destruction system (finite conservative PDS) is the autonomous ODE

$y_i^{\mathrm{\prime }}=F_i(y):={\sum }_{j=1}^m(p_{ij}(y)-d_{ij}(y)),i=1,\dots ,m,$

with internal-transfer identities

$p_{ij}(y)=d_{ji}(y),p_{ii}(y)=d_{ii}(y)=0.$

The index $j$ is the donor/source of $p_{ij}$ and the index $i$ is its recipient/destination; $d_{ji}$ is the same transfer viewed as destruction from $j$. The total mass is $M(y):={\sum }_{i=1}^m{y}_i$. The ODE is considered at states in ${\mathrm{\Omega }}_{+}$. The definition describes a closed system: external sources and sinks must be specified separately and are not included in the conservation claims on this page. No existence, positivity, or regularity assertion about solutions is part of this definition.