Concentration phenomenon of single phytoplankton species with changing-sign advection term

Abstract

In this paper, a nonlocal reaction-diffusion equation modeling the growth of phytoplankton species with changing-sign advection in a vertical water column is investigated, where the species depends solely on light for its metabolism. We mainly study the concentration phenomenon of the phytoplankton with large advection amplitude and small diffusion rate. Firstly, we study the threshold-type dynamics of the population by critical death rate d⁎. Secondly, we examine the concentration phenomenon with large advection amplitude and small diffusion in two cases: (i) the advection function h(x) changes sign only once from positive to negative in water column [0,1]. We find that the phytoplankton will concentrate at certain critical point with large advection amplitude and small diffusion; (ii) the advection function h(x)<0 in [0,κ) with ∫0xh(s)a(s)ds<0 for all x∈(0,1], the phytoplankton will concentrate at the surface of water column with large advection amplitude and small diffusion. We also investigate the limiting distribution of phytoplankton as diffusion rate D→+∞: the phytoplankton tends to even distribution in water column.