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We consider a continuous-time branching random walk on a multidimensionallattice in a random branching medium. It is theoretically known that, in suchbranching random walks, large rare fluctuations of the medium may lead toanomalous properties of a particle field, e.g., such as intermittency. However,the time intervals on which this intermittency phenomenon can be observed arevery difficult to estimate in practice. In this paper, branching mediacontaining only a finite and non-finite number of branching sources areconsidered. The evolution of the mean number of particles with a random pointperturbation and one initial ancestor particle at a lattice point is describedby an appropriate Cauchy problem for the evolutionary operator. We review somethe previous results about the long-time behavior of the medium-averagedmoments $\langle m_{n}^{p}\rangle$, $p\geq 1$, $n\geq 1$ for the particlepopulation at every lattice point as well as the total one over the lattice andpresent an algorithm for the simulation of branching random walks under variousassumptions about the medium, including the medium randomness. The effectsarising in random non-homogeneous and homogeneous media are then compared andillustrated by simulations based on the potential with Weibull-type upper tail.A wide range of models under different assumptions on a branching medium, aconfiguration of branching sources, and a lattice dimension were consideredduring the comparison. The simulation results indicate that intermittency canbe observed in random media even over finite time intervals.

Symmetric Branching Random Walks in Random Media: Comparing Theoretical and Numerical Results