Modified Patankar Euler for conservative transfer systems

Let $m\ge 1$, let $(p_{ij},d_{ij})$ be the rates of a finite conservative PDS Finite conservative production destruction systems with state domain $\mathrm{\Omega }$, and take an input state $y^n\in {\mathbb{R}}^m$, a step size $h$, and declared denominators ${\sigma }_i$. If $y^n\mathrm{\notin }{\mathrm{\Omega }}_{+}$, $h\le 0$, or any ${\sigma }_i\le 0$, return an inadmissible-input status before evaluating any rate. Otherwise the modified Patankar Euler step returns the unique solution $y^{n+1}$ of

$y_i^{n+1}=y_i^n+h{\sum }_{j\mathrm{\ne }i}(p_{ij}(y^n)\frac{y_j^{n+1}}{{\sigma }_j}-d_{ij}(y^n)\frac{y_i^{n+1}}{{\sigma }_i}),i=1,\dots ,m\mathrm{.}$

Equivalently, form $M\in {\mathbb{R}}^{m\times m}$ by

$M_{ii}=1+h{\sum }_{j\mathrm{\ne }i}\frac{d_{ij}(y^n)}{{\sigma }_i},M_{ij}=-h\frac{p_{ij}(y^n)}{{\sigma }_j}(i\mathrm{\ne }j),$

and solve $M{y}^{n+1}=y^n$. The rates in the solve are evaluated only at the admissible state $y^n\in {\mathrm{\Omega }}_{+}$. For every input passing the branch, return the unique exact solution of the stated linear system. The standard first-order choice is ${\sigma }_i=y_i^n$, which is admissible only when every $y_i^n>0$. A separately defined rate-factor form extends a first-order step to zero donor coordinates under an additional source-factorization hypothesis.