The present paper is a continuation of [A. N. Startsev, Theory Probab. Appl., 46 (2002), pp. 431-447] in which limit theorems are established for the number of particles changing their types to the terminal moment of the process given that the initial numbers of particles of both types tend to infinity. Here this problem is solved under the condition that the initial number of particles having "energy" is fixed. This assumption leads to models more actual for applications, in particular, in epidemiology. A part of the obtained results (Theorems 1 and 2) has been announced in [M. Mirzaev and A. N. Startsev, Proceedings of the International Conference "Advances in Statistical Inferential Methods" (Almaty, 2003), NITS "Fylym," Almaty, 2003, pp. 81-85].
