The Phd thesis is devoted to the numerical and mathematical analysis of systems of partial differential equations arising in the modeling of cells movement. The model of nutrient-dependent tumour growth is built and the asymptotic stability of constant steady states for small perturbations is proved. Then the parabolic and hyperbolic models of chemotaxis are approximated using finite differences and finite volume methods. In particular, a consistent scheme, which is well-balanced on steady states with constant velocity, preserves the non negativity of the density and treats the vacuum, is constructed. Having efficient and accurate numerical methods the behaviour of the solutions is analyzed. At first the pure diffusion problem, for which the waiting time phenomenon and regularity under the physical boundary condition are of main concern. Then the attention is focused on the study of existence and stability of non constant stationary solutions and long time behaviour of the hyperbolic model of chemotaxis on a bounded domain.
