Cultivated in vitro, toxin-producing bacteria usually outcompete toxin-sensitive bacteria. However, in many natural ecosystems (e.g., animal guts, soil, lakes), the toxin-sensitives co-exist with the toxin-producers, and sometimes are even in the majority. We posit that a possible explanation for this puzzle might be the random (re)occurrence of extreme events (such as flushing/dilution in the gut), which interrupt the natural course of bacterial competition.

Using several control-theoretic models, we show that the toxin-producers cannot always win even if they develop quorum sensing and have evolved to take advantage of the known frequency of such extreme events. Mathematically, this involves solving a range of Hamilton-Jacobi-type equations using semi-Lagrangian numerical methods. Joint work with Andrea Giometto and Alex Vladimirsky.