The development of a self-sustained quantum electrodynamical (QED) cascade in a single strong laser pulse is studied analytically and numerically. A hydrodynamical approach is used to construct an analytical model of cascade evolution, which includes the key features of the cascade observed in 3D QED particle-in-cell (QED-PIC) simulations, such as the magnetic field dominance in the cascade plasma and laser energy absorption. The equations of the model are derived in closed form and solved numerically. Direct comparison between the solutions of the model equations and 3D QED-PIC simulations shows that our model is able to describe the complex nonlinear process of cascade development qualitatively well. Various regimes of the interaction based on the intensity of the laser pulse are revealed in both the solutions of the model equations and the results of the QED-PIC simulations.