The periodic swarming of bacteria is one of the simplest examples for pattern formation produced by the self-organized collective behavior of a large number of organisms. In the spectacular colonies of Proteus mirabilis (the most common species exhibiting this type of growth), a series of concentric rings are developed as the bacteria multiply and swarm following a scenario that periodically repeats itself. We have developed a theoretical description for this process in order to obtain a deeper insight into some of the typical processes governing the phenomena in systems of many interacting living units. Our approach is based on simple assumptions directly related to the latest experimental observations on colony formation under various conditions. The corresponding one-dimensional model consists of two coupled differential equations investigated here both by numerical integrations and by analyzing the various expressions obtained from these equations using a few natural assumptions about the parameters of the model. We determine the phase diagram corresponding to systems exhibiting periodic swarming, and discuss in detail how the various stages of the colony development can be interpreted in our framework. We point out that all of our theoretical results are in excellent agreement with the complete set of available observations. Thus the present study represents one of the few examples where self-organized biological pattern formation is understood within a relatively simple theoretical approach, leading to results and predictions fully compatible with experiments.