A biofilm model that avoids a tragedy of the commons

Abstract

The study of competition between multiple species is of great significance in biology. Competitive behavior is often observed to occur in biofilms and understanding cooperation between multiple species in a single biofilm is the center of much research. The species that grow in biofilms are frequently studied in chemostats, which have a rich history in mathematical modeling. In this thesis, a review of a mathematical chemostat model is presented in which a tragedy of the commons occurs. The chemostat model is then developed into a biofilm model to see if a tragedy occurs in a biofilm under similar conditions. The biofilm and chemostat model consist of two species, a cooperator and a cheater. The cooperator produces an enzyme that combines with a substrate to produce a nutrient. The nutrient is then consumed by the cooperator and cheater. The cooperator is at a disadvantage since it must allocate some of its nutrient uptake towards enzyme production. A one dimensional biofilm model is developed with reaction advection equations governing the behavior of the species and reaction-diffusion equations governing the behavior of the substrate, nutrient ,and enzyme. A set of numerical methods is then outlined on how to solve the system of equations. It is found that a tragedy of the commons is avoided in the biofilm and both species can persist when numerical simulations are run for a finite amount of time. It is then argued that the cooperative behavior exhibited by the two species is a stable equilibrium by approximating the steady state solutions. Further evidence is provided for the existence of a stable equilibrium by perturbing the system and finding that the perturbed system tends back to the equilibrium. Finally, the eigenvalues of the discretized linear system are computed and the results suggest that either the equilibrium is stable or moves away from the equilibrium slowly.