  We consider a family of general branching processes with reproduction
parameters depending on the age of the individual as well as the population age
structure and a parameter $K$, which may represent the carrying capacity. These
processes are Markovian in the age structure. In a previous paper the Law of
Large Numbers as $K\to\infty$ was derived. Here we prove the Central Limit
Theorem, namely the weak convergence of the fluctuation processes in an
appropriate Skorokhod space. We also show that the limit is driven by a
stochastic partial differential equation.
