Analysis and numerical approximations of hydrodynamical models of biological movements

Abstract

The aim of this thesis is to investigate some hydrodynamical models arising in biology, from both the analytical and the numerical point of view. We are interested in the movement of populations of cells, which can be influenced by changes in the environment. The movement of bacteria under the effect of a chemical substance, i.e. the chemotaxis, has been a widely studied topic in Mathematics in the last decades, and numerous models have been proposed. The first part of this thesis is devoted to the analytical and numerical study of hyperbolic models of chemotaxis. In Chapter 1 we present the derivation of parabolic and hyperbolic model of chemotaxis. At the beginning we introduce the standard Patlak-Keller-Segel model and some variations. Then we present hyperbolic models of chemotaxis showing two different possible derivations: the kinetic derivation with the moment closure methods by Hillen and the phenomenological derivation of a vasculogenesis model based on continuum mechanics proposed by Gamba et al. In the following chapter we investigate the existence and the behavior for large times of global smooth solutions to theCauchy problemon Rn, for a semilinear hyperbolic-parabolic model of chemotaxis. Initially we introduce some properties of partially dissipative hyperbolic systems, and we present some of the results obtained by Bianchini et al. on the decomposition of the Green Kernel. Thanks to the sharp decay estimates of the Green kernel of the parabolic and hyperbolic equations, we are able to prove the results. Moreover we prove an analogous result for perturbation of constant (non-null) stationary state. Finally, by the same technique, we compare the large time behavior of the solution with the behavior of solution to the parabolic Keller-Segel model. In Chapter 3 we investigate the existence and the behavior for large times of global smooth solution to the Cauchy problemon R for a quasilinear hyperbolic-parabolic systemwhich descibes the vasculogenisis process. Firstly we introduce hyperbolic partially dissipative systems. The global existence of solutions is proved by energy estimate, while the study of the large times behavior is based on the decay estimates of the Green Kernel of the linearized operators. We show that these results hold also for perturbation of small constant (non null) state. Chapter 4 is devoted to the numerical approximation of the hyperbolic-parabolic models studied analytically in the previous chapters. At the beginning we give an introduction to finite difference schemes, defining fundamental concepts like consistency, convergence, stability and monotonicity. Then we present 3-point finite difference schemes for hyperbolic conservation laws and we briefly present also some classical schemes for parabolic equations. A section is dedicated to the relaxation method, scheme used in our simulations. The second and the third section of this chapter are devoted to several numerical simulations in the two dimensional case of the hyperbolic-parabolic models. With reference to the chemotaxis model we show results of pattern formation and also simulations in agreement with our analytical result. Regarding the vasculogenesis problem, we show simulations of development of vascular network with different initial data and also simulations in agreement with the analytical results. The second part of this thesis is devoted to the modeling and the numerical approximation of two biological processes: inflammation during ischemic stroke and the growth of phototrophic biofilm. As a matter of fact in Chapter 5, we propose a model to describe the inflammatory process which occurs during ischemic stroke. Firstly, an introduction to some basic concepts about the biological phenomenon is given. Then, a detailed derivation of the model and the numerical scheme used are presented. Finally, the studies of the model robustness and sensitivity are showed and some numerical results on the time and space evolution of the process are presented and discussed. In the final chapter, a system of nonlinear hyperbolic partial differential equations is derived from the mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme are described in detail and several simulations are presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing.