9. Simulation of Stochastic Processes with Generation and Transport of Particles

In modeling of a cell population evolution, the key characteristics are the existence of several sources where cells can proliferate their copies or die, and migration of cells over an environment. One of the study aims is to obtain the threshold value of a parameter which separates different types of the cell proliferation process at the sources. Continuous-time branching random walks on multidimensional lattices with a few sources of branching can be used for modeling of a cell population dynamics. For example, active growth of the cancer cellular population in the frame of branching random walk models may be explained by the excess of the threshold value. Branching random walks is an appropriate tool to describe such processes in terms of generation and transport of particles. The effect of phase transitions on the asymptotic behavior of a particle population in the frame of branching random walks was studied analytically in detail by many authors. Simulation of branching random walks is applied for numerical estimation of a threshold value of the parameter on limited time intervals. Obtained results are used to define strategies that may delay a cell population progression to some extent. The work may be treated as the first step to the simulation of branching random walks. We assume that the process started by the initial particle which walks on the lattice until it reaches one of the sources where its behavior changes, and new copies may appear. All particles behave independently of each other and of their history. We present an approach to simulation of the mean number of particles over the lattice and in every point of the lattice. Simulation of the process is based on a well-known algorithm of queue data structures and the Monte Carlo method.