This paper focuses on the stationary portion of file download in an unstructured peer-to-peer network, which typically follows for many hours after a flash crowd initiation. The model includes the case that peers can have some pieces at the time of arrival. The contribution of the paper is to identify how much help is needed from the seeds, either fixed seeds or peer seeds (which are peers remaining in the system after obtaining a complete collection) to stabilize the system. The dominant cause for instability is the missing piece syndrome, whereby one piece becomes very rare in the network. It is shown that stability can be achieved with only a small amount of help from peer seeds--even with very little help from a fixed seed, peers need dwell as peer seeds on average only long enough to upload one additional piece. The region of stability is insensitive to the piece selection policy. Network coding can substantially increase the region of stability in case a portion of the new peers arrive with randomly coded pieces.
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