Steady-state solution of a two-species biofilm problem

Through a thorough investigation of the boundary conditions for a general two-species biofilm model, a simple and fast method for solving the steady-state case is developed and presented. The methods used may be extended to biofilm models in which more than two species are considered. Four different sets of boundary conditions are possible for the two-species biofilm model. Each set is shown to be asymptotically stable. A biofilm model describing the competition between autotrophic and heterotrophic bacteria and a biofilm model considering only Nitrosomonas and Nitrobacter are used for illustration. A parameter L(crit), critical film thickness for bacterial coexistence, is introduced from which criteria on the bulk concentrations for coexistence are derived. From these criteria it is seen that the thinner the biofilm, the more restrictive the conditions are for steady-state coexistence. For thin biofilms there may, in many cases, be no point in considering more than one species in the biofilm model. Furthermore, the gradients of the bacterial concentrations are in many cases negligible in thin biofilms, and the biofilm may then be assumed to be homogeneous. The criteria on the bulk concentrations together with the four sets of boundary conditions provide the necessary information for a direct solution of the steady-state two-species biofilm model by means of an ordinary differential and algebraic equation solver.