  This study is devoted to the long-term behavior of nucleation, growth and
fragmentation equations, modeling the spontaneous formation and kinetics of
large polymers in a spatially homogeneous and closed environment. Such models
are, for instance, commonly used in the biophysical community in order to model
in vitro experiments of fibrillation. We investigate the interplay between four
processes: nucleation, polymeriza-tion, depolymerization and fragmentation. We
first revisit the well-known Lifshitz-Slyozov model, which takes into account
only polymerization and depolymerization, and we show that, when nucleation is
included, the system goes to a trivial equilibrium: all polymers fragmentize,
going back to very small polymers. Taking into account only polymerization and
fragmentation, modeled by the classical growth-fragmentation equation, also
leads the system to the same trivial equilibrium, whether or not nucleation is
considered. However, also taking into account a depolymer-ization reaction term
may surprisingly stabilize the system, since a steady size-distribution of
polymers may then emerge, as soon as polymeriza-tion dominates depolymerization
for large sizes whereas depolymerization dominates polymerization for smaller
ones-a case which fits the classical assumptions for the Lifshitz-Slyozov
equations, but complemented with fragmentation so that " Ostwald ripening "
does not happen.
