# Toxin-mediated competition in weakly motile bacteria

## TL;DR

This paper models toxin-mediated competition between bacteria, showing how toxin strength and motility influence colony interactions, inhibition zones, and invasion dynamics through analytical and numerical methods.

## Contribution

It extends classic competition models to include toxin effects, deriving analytical solutions for non-motile bacteria and analyzing wave behavior in weakly motile bacteria.

## Key findings

- Inhibition zones form when toxin strength exceeds a threshold.
- Wave speed depends on toxin strength and competition, with producer colonies potentially outcompeting susceptibles.
- Higher bacterial motility requires stronger toxins for effective inhibition.

## Abstract

Many bacterial species produce toxins that inhibit their competitors. We model this phenomenon by extending classic two-species Lotka-Volterra competition in one spatial dimension to incorporate toxin production by one species. Considering solutions comprising two adjacent single-species colonies, we show how the toxin inhibits the susceptible species near the interface between the two colonies. Moreover, a sufficiently effective toxin inhibits the susceptible species to such a degree that an `inhibition zone' is formed separating the two colonies. In the special case of truly non-motile bacteria, i.e. with zero bacterial diffusivity, we derive analytical expressions describing the bacterial distributions and size of the inhibition zone. In the more general case of weakly motile bacteria, i.e. small bacterial diffusivity, these two-colony solutions become travelling waves. We employ numerical methods to show that the wavespeed is dependent upon both interspecific competition and toxin strength; precisely which colony expands at the expense of the other depends upon the choice of parameter values. In particular, a sufficiently effective toxin allows the producer to expand at the expense of the susceptible, with a wavespeed magnitude that is bounded above as the toxin strength increases. This asymptotic wavespeed is independent of interspecific competition and due to the formation of the inhibition zone; when the colonies are thus separated, there is no longer direct competition between the two species and the producer can invade effectively unimpeded by its competitor. We note that the minimum toxin strength required to produce an inhibition zone increases rapidly with increasing bacterial diffusivity, suggesting that even moderately motile bacteria must produce very strong toxins if they are to benefit in this way.