Phytoplankton depth profiles and their transitions near the critical sinking velocity

We consider a simple phytoplankton model introduced by Shigesada and Okubo which incorporates the sinking and self-shading effect of the phytoplankton. The amount of light the phytoplankton receives is assumed to be controlled by the density of the phytoplankton population above the given depth. We show the existence of non-homogeneous solutions for any water depth and study their profiles and stability. Depending on the sinking rate of the phytoplankton, light intensity and water depth, the plankton can concentrate either near the surface, at the bottom of the water column, or both, resulting in a "double-peak" profile. As the buoyancy passes a certain critical threshold, a sudden change in the phytoplankton profile occurs. We quantify this transition using asymptotic techniques. In all cases we show that the profile is locally stable. This generalizes the results of Shigesada and Okubo where infinite depth was considered.