In Sebastian Aniţa, Viorel Arnăutu, Vincenzo Capasso. An Introduction to Optimal Control Problems in Life Sciences and Economics (Birkhäuser. 2011), this equation appears as a special case of the Sharpe–Lotka–McKendrick’s equation ; in the latter there is inflow, and the math is based on directional derivative. The McKendrick’s equation appears extensively in the context of cell biology as a good approach to model the eukaryotic cell cycle.