Models induced from critical birth–death process with random initial conditions

In this work, we study a linear birth–death process starting from random initial conditions. First, we consider these initial conditions as a random number of particles following different standard probabilistic distributions – Negative-Binomial and its closest Geometric, Poisson or Pólya–Aeppli distributions. It is proved analytically and numerically that in these cases the random number of particles alive at any positive time follows the same probability law like the initial condition, but with different parameters depending on time. The random initial conditions cannot change the critical parameter of branching mechanism, but they impact the extinction probability. Finally, the numerical model is extended to an application for studying branching processes with more complex initial conditions. This is demonstrated with a linear birth–death process initialised with Pólya urn sampling scheme. The obtained preliminary results for particle distribution show close relation to Pólya–Aeppli distribution.