  We consider a continuous-time branching random walk on a multidimensional
lattice in a random branching medium. It is theoretically known that, in such
branching random walks, large rare fluctuations of the medium may lead to
anomalous properties of a particle field, e.g., such as intermittency. However,
the time intervals on which this intermittency phenomenon can be observed are
very difficult to estimate in practice. In this paper, branching media
containing only a finite and non-finite number of branching sources are
considered. The evolution of the mean number of particles with a random point
perturbation and one initial ancestor particle at a lattice point is described
by an appropriate Cauchy problem for the evolutionary operator. We review some
the previous results about the long-time behavior of the medium-averaged
moments $\langle m_{n}^{p}\rangle$, $p\geq 1$, $n\geq 1$ for the particle
population at every lattice point as well as the total one over the lattice and
present an algorithm for the simulation of branching random walks under various
assumptions about the medium, including the medium randomness. The effects
arising in random non-homogeneous and homogeneous media are then compared and
illustrated by simulations based on the potential with Weibull-type upper tail.
A wide range of models under different assumptions on a branching medium, a
configuration of branching sources, and a lattice dimension were considered
during the comparison. The simulation results indicate that intermittency can
be observed in random media even over finite time intervals.
