I don’t have any proof really, but I know that the system neither can cross into negative values, nor approach infinity (due to the nature of the model and both of these alternatives are biologically weird). Also, the system should be quite well behaved and I do not expect any strange cases. There will be at least one root.
In principle there could be more than one root, however, after playing around with this model for a while I am quite certain that this is not the case (and for this specific application I am satisfied with finding a single solution).