The probability and duration of immigration in microbial communities
Immigration is a fundamental feature of microbial communities. We propose a method to determine the probability (Pi) that a number of immigrants (i) can attain an abundance N using the ratio of the probabilities of death q and division or "birth" p and the gamblers ruin equation: Pi= (1-(q/p)i/(1-(q/p)N). We estimate the probability of successful bioaugmentation or transplantation, the fate of a mutation infection 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 immigration parameter used in neutral models, m is 1/(1-q/p). We calculated the long-term average value of m and q/p in a wastewater treatment plant. The value of m varies by >5 orders of magnitude, with a curious bimodal distribution. However, all the values of q/p are very close to, but greater than, 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 a q/p [≥]1 will go extinct with probability 1. The link between q/p and infectious dose is known and we demonstrate that, in principle, the gamblers ruin equation can estimate the infectious dose in naturally occurring infections, using Vibrio cholerae "carriers" to illustrate the point. We use the ratio q/p to estimate the time (measured in events or solar time) for a given change in abundance to happen. When q/p=1, extinction in even a small microbial population will take thousands of years.