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978-3-8439-0323-3, Reihe Mathematik
Manuel Villegas Caballero
Random Delay Differential Equations: Application to Biofilm Modeling
111 Seiten, Dissertation Technische Universität München (2012), Softcover, A5
The RDDEs and their application to different systems, specially to biofilm modeling, have been the focus of this work.
A population model for the effect of antibiotics in a bacteria population within a biofilm has been presented. The model considers the QS mechanism via the inclusion of a delay term in the equations. Linearization of the system around the equilibrium points for stability analysis yields an ODE and a DDE system. The potential difficulties appearing in the analysis of the dynamics of the system due to the delay are addressed through the introduction of the Lambert W function. The special structure of the linearized DDE system allows the use of this function to calculate the eigenvalues. Thus, a simple criterion for the stability is obtained, based on the computation of the eigenvalue with the largest real part. This is used for a qualitative study of the system. With this criterion the dynamical behavior of the system for certain choices of the value of the parameters is determined.
Nevertheless, there are certain situations where a qualitative description is not sufficient, and solutions must be computed in order to assess the evolution of the population. Here, the inclusion of uncertainties as an inherent feature of biological systems is considered. The random solution is computed by means of the non-intrusive PC scheme and with the RADAR software for the solution of the system in each iteration. It is seen that the uncertainties imply a total different diagnosis. Finally the PC results were validated with a MC simulation, where the convergence in the expected value and the variance was shown.