Single Phytoplankton Species Growth with Light and Advection in a Water Column

We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coefficient, water column depth and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. Under a general reproductive rate which is an increasing function of light intensity, we establish the existence of a critical death rate; i.e., the phytoplankton survives if and only if its death rate is less than the critical death rate. The critical death rate is a strictly monotone decreasing function of sinking or buoyant coefficient and water column depth, and it is also a strictly monotone decreasing function of turbulent diffusion rate for buoyant species. In contrast to critical death rate, critical sinking or buoyant velocity, critical water column depth and critical turbulent diffusion rate may or may not exist. For instance, it is shown that if the death rate is suitably small with respect to the water column depth, the phytoplankton can persist for any sinking or buoyant velocity; i.e., there is no critical sinking or buoyant velocity under such situation. We further show that critical water column depth, critical sinking or buoyant velocity and critical turbulent diffusion rate for buoyant species can exist for some intermediate range of phytoplankton death rates and, whenever they exist, are always unique. In strong contrast, we show that there may exist two critical turbulent diffusion rates for sinking species. The phytoplankton forms a thin layer at the surface of the water column for sufficiently large buoyant rate, and it forms a thin layer at the bottom of the water column for sufficiently large sinking rate. Precise characterizations of these thin layers are also given.