Free boundary approach to modelling multispecies biofilms

Abstract

The work presents the free boundary approach to modelling the formation and growth of multispecies biofilms with special attention to recent new problems. The mathematical free boundary value problem is formalized by a system of nonlinear hyperbolic partial differential equations that governs the multispecies biofilm growth and a system of semi-linear parabolic partial differential equations that governs the substrate diffusion within the biofilm. The biofilm thickness is the free boundary of the problem. Its evolution over time is governed by a differential equation that involves the growth velocity of microbial mass, attachment and detachment. A generalized model is introduced to include the possibility that the planktonic bacterial cells could penetrate an already constituted biofilm and colonize the regions where favorable conditions for their growth are found. The new process is governed by parabolic equations mutually connected with all the others. Moreover, the biological process of the initial biofilm formation is modelled. The proposed model does not require any initial conditions as the biofilm thickness is zero and the species composition is modelled according to the environmental conditions. The related mathematical problem is interesting since the free boundary is a space-like line and its initial value is zero. This last problem is also discussed numerically. Since the initial biofilm thickness is zero, the free boundary domain cannot be transformed to a rectangular domain and most numerical methods cannot be applied. It is shown that the method of characteristics, successively used for the qualitative analysis, still works for the numerical solutions.