I think I would agree with your (final, including the fixed typo) answer, lock, stock, and barrel. If you had to include the growth factor of the population, you'd have already done that by plugging in the solution of the P differential equation into the F differential equation. The expression for P includes the growth and death factors.
So that's my opinion, though I could be missing something. I've pondered your problem over a few days! It's not a trivial interpretation problem.