On a nonlocal reaction–diffusion–advection equation modelling phytoplankton dynamics

Abstract

We investigate a reaction–diffusion–advection equation that models the dynamics of a single phytoplankton species in a eutrophic vertical water column. First, we extend the results of Du and Hsu (2010 SIAM J. Math. Anal.42 1305–33) to show that even with variable diffusion and sinking rates, the global dynamics of the model is completely determined by its unique steady-state solution. This implies that the bistable behaviour observed through numerical simulation in Ryabov et al (2010 J. Theor. Biol. 263 120–33) for the phytoplankton dynamics can only occur when one assumes limitation of nutrients in the model. Second, we examine the asymptotic profiles of the positive steady-state solution for small diffusion, large diffusion and deep water column, respectively. Our results reveal that for small diffusion, the phytoplankton population concentrates at the bottom of the water column, while for large diffusion, the population tends to distribute evenly in the water column, and when all the other factors are the same, in a water column with positive background turbidity, the total biomass is bigger in the large diffusion case than in the small diffusion case, and in a water column with zero (or negligible) background turbidity, the total biomass tends to the same limit in both cases; when the water column depth goes to infinity, the population distribution approaches that obtained in Ishii and Takagi (1982 J. Math. Biol. 16 1–24) with infinite water depth, and it reaches a unique maximum at a certain finite water level. We also give a complete answer to a question left open in Hsu and Lou (2010 SIAM J. Appl. Math. 70 245–54) regarding the behaviour of the critical death rate for deep water column, which plays a key role in determining whether a critical water depth exists.