The movement under the effect of a stimulus has been a widely studied topic in the last decades and numerous models have been proposed. It is possible to describe this phenomena at different scales. For example, by considering the population as a continuum medium, it is possible to obtain macroscopic models of partial differential equations. Among them we can distinguish parabolic and hyperbolic types of systems. They are often expected to have the same asymptotic behaviour, but the initial evolution of solutions can be very different. For example, parabolic models are not always precise enough and prevent us from observing intermediate organized structures. On the other hand, hyperbolic models are more accurate providing better description but simultaneously make the mathematical analysis a highly non trivial task. In this talk I will show numerical comparison of behaviour of some parabolic and hyperbolic models. At first, I will consider the chemotaxis models proposed to describe the de novo formation of blood capillaries. I will present numerical and mathematical analysis of some non constant steady states on the one dimensional bounded domain with homogeneous Neumann boundary conditions. Then I fill focus on models describing the motion of pedestrians sharing the same goal such as leaving a room as soon as possible. I will present numerical analysis of the ability of parabolic and hyperbolic models to reproduce phenomena such as clogging at the exit, effect of an obstacle and existence of stop and go waves.