Past exam of the mathematical physiology course of the University of Oxford / 2023 / 1 / d / Solution

For each state, add its incoming reaction rates and subtract its outgoing reaction rates. This gives the master equationSumming gives ${\sum }_{i,j}{\dot{S}}_{ij}=0$, verifying the conservation law for total channel density. Nonnegative initial densities stay nonnegative because every loss term vanishes when its own density reaches zero.