Diffusion and multiplication in random media

We investigate the evolution of a population of non-interacting particles which undergo diffusion and multiplication. Diffusion is assumed to be homogeneous, while multiplication proceeds with different rates reflecting the distribution of nutrients. The distribution of nutrients is considered as a stationary quenched random variable with zero average, so the population size would remain constant if there were no fluctuations in the distribution of nutrients. We show that fluctuations drastically affect the behavior, e.g. the population size exhibits a super-exponential growth whenever the nutrient distribution is unbounded. We elucidate a huge difference between the average and typical asymptotic growths and emphasize the role played by the spatial correlations in the nutrient distribution.