An Analytical Model of Microbial Interactions

Abstract

Interactions between bacteria which produce antibiotics and those which produce degrading enzymes to counteract the antibiotics are common in nature. They have been proposed as a mechanism enabling robust species coexistence and the stability of communities with many species. Current approaches to this rely on complex simulations, without effective mathematical treatment or insight. As a possible approach to better understand this system, I describe these bacterial interactions as a Wright-Fisher evolutionary model on a randomly colored graph. By taking an infinite-population limit, I derive a replicator equation, which is studied at varying degrees of simplicity. I show that the interaction can be captured by a three-strategy matrix game which exhibits stability. Three-strategy matrix games are then studied more generally, and I prove a theorem connecting coexistence of all species in the replicator equation to the ability of smaller sub-communities to be invaded.