A linear model for the dynamics of fish larvae

We consider a linear model for the growth and the dispersion of fish larvae of certain species. Dispersion is modeled as entailed by the combination of transport and vertical diffusion. We generalize the work of Boushaba, Arino and Boussouar [5,6] in the sense that horizontal velocities are uniform throughout the water column; but we deal with vertical component velocity and vertical diffusion depending on the space variables and on time, which was not the case in [5,6]. This new vision leads us to non-autonomous problems, the aim of this work is to show the existence, uniqueness, and positivity of solutions.