The asymptotic behavior of nonextinction probabilityof a branching process with pairwise interaction of particles

The model of continuous time branching processes is considered in which each pair of particles of the population produces offspring independently of all other particles. In the critical case we “explicitly” solve under certain regularity conditions on the infinitesimal generating function of the process, the forward Kolmogorov equation for the Laplace transform of the generating function of the process and show that the nonextinction probability decreases exponentially as $\tau\to\infty$. For the particular case when the number of offspring of any pair of particles does not exceed three we “estimate” from below the probability of nonextinction.