Probability, Parameters, and Duration of Immigration and Extinction in Microbial Communities

We propose a suite of simple equations to estimate the probability and duration of two important processes in microbial ecology: immigration and extinction. Our work is based on the gambler's ruin equation, which determines the probability that a number of immigrants (i) can attain an abundance N given the ratio of the probabilities of death q and division (or birth) p. We estimate the probability of an organism attaining a value of N in the context of bioaugmentation, transplantation, infection, mutation, and extinction. For example, an inoculum of 108 bacteria with a q/p of 1.00000001 has a 10-43 chance of attaining an abundance of 1010. The ratio of deaths to births controls the immigration parameter used in neutral models (m), and infectious dose in pathogens. We use Vibrio cholerae infections to demonstrate that the gambler's ruin equation can be used to estimate the infectious dose in naturally occurring infections. We calculated the long-term average value of m and q/p in a wastewater treatment plant. All values of q/p were ≥1. We expect the long-term average value of q/p to be ~1 in all stable microbial communities. In the absence of migration, bacterial populations with q/p ≥1 will go extinct with probability 1. We use the ratio q/p and simple recurrence relationships to estimate the time for a given change in abundance to occur. When q/p=1, extinction in even a small microbial population will take thousands of years. Our simple mechanistic models could play a powerful role in theory and practice.