Convergence of the age structure of general schemes of population processes

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 (Proc. Steklov Inst. Math. 282 (2013) 90-105), the Law of Large Numbers as K→∞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.