A protocell model consisting of mutually catalyzing molecules is studied, in order to investigate how chemical compositions are transferred recursively through cell divisions under replication errors. Depending on the numbers of molecules and species, and the path rate, three phases are found: fast switching state without recursive production, recursive production, and itinerancy between the above two states. The number distributions of the molecules in the recursive states are shown to be log-normal except for those species that form a core hypercycle, and are explained with the help of a heuristic argument.