Population Models with Size and Spatial Structure - Well-posedness & Applications

This thesis is concerned with the investigation of size-and-spatially structured population mod- els and related applications. The first part contains the theoretic fundamentals for this thesis, upon which we rely for subsequent properties and applications. This involves proving well- posedness of the model in terms of deriving generalized solutions via the method of character- istics and utilizing the explicit form of solutions to prove that they admit the structure of a strongly continuous semigroup, called the size-diffusion semigroup. Next, favorable properties of the parabolic structure in the spatial variable are used to derive regularizing effects of the size-diffusion semigroup, and its generator is characterized. The theoretic part is rounded off with several consequences of the explicit form of the size-diffusion semigroup including the proof that the generator has compact resolvent, a perturbation result based on the theory of Miyadera and Voigt, and an ansatz for investigating quasilinear problems. The second part of this thesis contains two applications for size-and-spatially structured popu- lation equations. The first one models growth and dispersion of tumor cells inside healthy tissue and consists of several coupled equations. The individual growth of the proliferating tumor cells in combination with their spatial movement being composed of size-dependent diffusion and haptotactic behavior present the main issues in the investigation of local existence and positivity of mild solutions. The chapter is closed off with global existence of mild solutions under stronger assumptions on the spatial diffusion of tumor cells. The second application consists of a perturbed model for cell population evolution with division. The inclusion of a divisional process as a linear perturbation yields a particular asymptotic behavior for the resulting semigroup called asynchronous exponential growth. This entails that the long-term behavior of the solutions can be described by exponential growth or decay in time while stabilizing to a finite-dimensional space with respect to the size-variable. A crucial prerequisite for this result is eventual compactness of the first generation of the perturbed semigroup, which is then combined with spectral properties of the perturbed generator, as well as a result by Krein and Rutman on the point spectrum of compact and strongly positive operators.