Schematic illustration of the partial differential equations (PDEs) for: A) an optically thick coupled proton-neutron system with proton injection (“inject”, from the acceleration zone), and B) an isotope in the nuclear cascade, possibly with injection (from the acceleration zone). Each dashed box corresponds to one species (or one PDE), to which the relevant cooling, escape, and injection processes are “attached”. We note that all disintegration, photo-meson, or decay channels destroying a species (p, n, or ^A Z) show up as escape terms in the PDE of that species.