We present a new mathematical model for the development of biofilm that extends
the nonlinear density-dependent dependent continuum model introduced by Eberl et al.,
2001. It is a coupled nonlinear density-dependent diffusion-reaction model for biomass
spreading, describing the interaction of nutrient availability and biomass production.
The model by Eberl has a degenerate and singular diffusion coefficient. The model
considered in this thesis relaxes the singularity but imposes an inequality constraint on the biomass amount.
We first consider a simplified zero-dimensional nonlinear ODE system. To understand
the basic behavior of the ODE system we use linearization and examine
the phase plane. We also consider several Finite Difference schemes to approximate
the solutions. These include Forward Euler, Sequential method and Newton's method.
Next we modify the PDE model vby Eberl et al by introducing a new density-dependent diffusion coefficient and introduce an inequality constraint. Applying Newton's method, we discretize the system in space and time. The use of Neumann boundary conditions allows us to study the morphology of biofilm as well as the dynamics of the total amounts. Additionally, we impose a condition on the time step and run several some numerical experiments with varying initial conditions to show robustness of the new model.