Mathematical Modeling of Biofilms

A continuum approach to mathematical modeling of multispecies biofilm formation and growth is presented. The equations governing the biological process are derived from mass balance principles in the general 3D situation. The 1D equations are inferred as special cases. A system of n nonlinear hyperbolic partial differential equations (PDEs) for the n bacterial species forming the biofilm, m semilinear parabolic (or elliptic) PDEs for the m substrates present in the biofilm, and an ordinary differential equation (ODE) for the motion of the biofilm boundary are obtained. In addition, a further system of parabolic PDEs is added in some situations, such as the invasion of new bacterial species and colonization into an already constituted biofilm. All equations mentioned are mutually connected and must be solved simultaneously in a domain that is a further unknown within the mathematical problem. Transforming the problem to characteristic coordinates yields positive answers about model consistency (uniqueness, existence, positiveness of solutions). The invasion problem is illustrated with some simulations based on the method of characteristics. The biofilm-reactor model is also discussed.