We present a stochastic reaction-diffusion-taxis model to describe the picophytoplankton dynamics along a water column. The model, which is valid for poorly mixed waters, typical of the Mediterranean Sea, considers intraspecific competition of picophytoplankton for light and nutrients. Random fluctuations of environmental variables are taken into account by adding a source of multiplicative noise to the diffusion equation for the picophytoplankton biomass concentration, whose distribution along the water column shows a maximum at a certain depth. After converting our results into chlorophyll a concentrations, we compare theoretical distributions, obtained for different noise intensities, with the experimental chlorophyll a distribution sampled in a site of the Strait of Sicily. Specifically, we find that position and height of the chlorophyll a peak concentration obtained from the model are in a very good agreement with field observations. Finally, we consider the effects of seasonal variations on phytoplankton dynamics by adding an oscillating term in the equation for the light intensity.