  This continues work started in part I on a general branching-within-branching
model for host-parasite co-evolution. Here we focus on asymptotic results for
relevant processes in the case when parasites survive. In particular, limit
theorems for the processes of contaminated cells and of parasites are
established by using martingale theory and the technique of size-biasing. The
results for both processes are of Kesten-Stigum type by including equivalent
integrability conditions for the martingale limits to be positive with positive
probability. The case when these conditions fail is also studied. For the
process of contaminated cells, we show that a proper Heyde-Seneta norming
exists such that the limit is nondegenerate.
