#### Date of Award

Spring 1-1-2012

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Applied Mathematics

#### First Advisor

David M. Bortz

#### Second Advisor

Mark Ablowitz

#### Third Advisor

Keith Julien

#### Fourth Advisor

Stephen McCormick

#### Fifth Advisor

Michael Solomon

#### Abstract

In this dissertation, we investigate two important problems in mathematical biology that are best modeled using partial differential equations. We first consider the question of how surface-adherent bacterial biofilm communities respond in flowing systems, simulating the interaction and separation process using the immersed boundary method. We use the incompressible viscous Navier-Stokes (N-S) equations to describe the motion of the flowing fluid. In these simulations we can assign different density and viscosity values to the biofilm than that of the surrounding fluid. The simulation also includes breakable springs connecting the particles in the biofilm. This allows the inclusion of erosion and detachment into the simulation. We discretize the fluid equations using finite differences and use a multigrid method to solve the resulting equations at each time step. The use of multigrid is necessary because of the dramatically different densities and viscosities between the biofilm and the surrounding fluid. We investigate and simulate the model in both two and three dimensions.

We also consider the spread of favorable genes in a population as described by the time varying coefficient Fisher's equation. We construct analytical solutions by using the Painlevé property for partial differential equations as defined by Weiss in 1983. We use this technique to find solutions to Fisher's equation with time-dependent coefficients for both diffusion and nonlinear terms. Finally, we compute specific solutions to illustrate their behaviors.

#### Recommended Citation

Hammond, Jason Frank, "Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation" (2012). *Applied Mathematics Graduate Theses & Dissertations*. 31.

https://scholar.colorado.edu/appm_gradetds/31

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