Finite-dimensional coagulation-fragmentation dynamics

We examine a finite dimensional truncation of the discrete coagulation-fragmentationequations that is designed to allow mass to escape from the system into clusters largerthan those in the truncated problem. The aim is to be able to model gelation with afinite system. The main result is a centre manifold calculation that gives the asymptoticbehaviour of the truncated model as time t->infinity. Detailed numerical results show thattruncated system solutions are often very close to this centre manifold, and the range ofvalidity of the truncated system as a model of the full infinite problem is explored forsystems with and without gelation. The latter cases are mass conserving, and we providean estimate using quantities from the centre manifold calculations of the time period thetruncated system can be used for before loss of mass is apparent. We also include someobservations on how numerical approximation can be made more reliable and efficient.