On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients

Abstract

We investigate the steady state solutions of a phytoplankton-nutrient system proposed by Huisman et al. in [14] that models the dynamics of a single phytoplankton species whose growth is limited by light and nutrients in a vertical water column. We first study the existence and nonexistence problem of the model and prove there is at least one positive solution to the system when the parameters involved are in a suitable range. We then analyze the limiting profiles of the positive solutions as the specific phytoplankton loss rate approaches zero and as the diffusion coefficient of the system tends to zero, respectively. In the small diffusion case, we show that the phytoplankton species all die out regardless of how large the nutrient supply is, and the nutrients distribution approaches a linear function determined by the parameters of the system. This phenomenon is in sharp contrast to that of the model studied by Du and Hsu [3, 4], where the phytoplankton species can concentrate at the bottom, at the surface or at a specific depth of the water column decided by the amplitude of the nutrient supply. We also study the asymptotic profile of the positive solution when the diffusion coefficient is very large. Our results reveal that in such a case the phytoplankton and the nutrients distribute evenly in the water column. Our concentration results also reveal that passive diffusion and active movement (sinking or floating) should be in proportion to the oscillation phenomena showed in [14, 24] to occur.