On the extinction probability of a branching process with two kinds of interaction of particles

A continuous-time Markov branching process with a single type T of particles is considered in which any, pair of particles T + T produces offspring independently of all other particles. In addition, any, particle of type T also produces children. Representations for the extinction probabilities of this process are obtained under certain assumptions, oil the offspring distribution. To establish this result we apply the exponential generating function method for the solution of a stationary backward Kolmogorov system of differential equations (see [A. V. Kalinkin. Theory, Probab. Appl.. 27 (1982) pp. 201-205]).