In practice, low Earth orbit (LEO) and medium Earth orbit (MEO) satellite networks consist of multiple orbits which are populated with many satellites. A widely used spatial architecture for LEO or MEO satellites is the Walker constellation, where the longitudes of orbits are evenly spaced and the satellites are equally spaced along the orbits. In this paper, we develop a stochastic geometry model for the Walker constellations. This proposed model enables an analysis based on dynamical system theory, which allows one to address essential structural properties such as periodicity and ergodicity. It also enables a stochastic geometry analysis under which we derive the performance of downlink communications of a typical user at a given latitude, as a function of the key constellation parameters.