Spreading of N diffusing species with death and birth features

The number of distinct sites visited by a random walker after t steps is of great interest, as it provides a direct measure of the territory covered by a diffusing particle. We review the analytical solution to the problem of calculating S_N(t), the mean number of distinct sites visited by N random walkers on a d-dimensional lattices, for d = 1, 2, 3 in the limit of large N. There are three distinct time regimes for S_N(t). A remarkable transition, for dimension ≥ 2, in the geometry of the set of visited sites is found. This set initially grows as a disk with a relatively smooth surface until it reaches a certain size, after which the surface becomes increasingly rough. We also review the results for a model for migration and spreading of populations and diseases. The model is based on N diffusing species, where each species has a probability α_- of dying (or recovery from a disease) and a probability α₊ to give birth (or to infect another species). It is found analytically that when α₊ ≈ α_- ≠ 0, after a crossover time t_x ∼ N/2α_-, the territory covered by the population is localized around its center of mass while the center of mass diffuses regularly. When α₊ > α_-, the localization breaks down after a second crossover time and the species diffuse and spread around their center of mass. These results may explain the phenomena of migration and spreading of diseases and population appearing in nature.