Diffusion and Multiplication in Random Media

TL;DR

This paper studies how populations of particles grow and spread in random environments with nutrients, revealing super-exponential growth under certain conditions and highlighting differences between average and typical growth behaviors.

Contribution

It provides a detailed analysis of population dynamics in quenched random media, emphasizing the impact of nutrient distribution and spatial correlations on growth rates.

Findings

Population exhibits super-exponential growth with unbounded nutrient distributions.

Significant difference between average and typical growth rates.

Spatial correlations in nutrients critically influence growth behavior.

Abstract

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. We focus on the situation where the distribution of nutrients is a stationary quenched random variable, and show that the population 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.