On conservativity and shattering for an equation of phytoplankton dynamics

A model of phytoplankton dynamics introduced by Arino describes the evolution of aggregates of phytoplankton by a kinetic-type equation composed of terms describing the growth of the aggregates and their splitting, where the latter is modelled by a singular integral operator of the same form as in the classical fragmentation theory. In this paper we shall show that despite the presence of the growth term, the model displays the typical properties of the fragmentation models, in particular, if the fragmentation rate is unbounded as the size of aggregates tends to zero, then there occurs an unaccounted for loss of the phytoplankton though formally nothing is taken out of the system.