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In this paper, we address two main topics: steady propagation fields for positive streamers in air and streamer deceleration in fields below the steady propagation field. We generate constant-velocity positive streamers in air with an axisymmetric fluid model, by initially adjusting the applied voltage based on the streamer velocity. After an initial transient, we observe steady propagation for velocities of $3\times10^4$ m/s to $1.2\times10^5$ m/s, during which streamer properties and the background field do not change. This propagation mode is not fully stable, in the sense that a small change in streamer properties or background field eventually leads to acceleration or deceleration. An important finding is that faster streamers are able to propagate in significantly lower background fields than slower ones, indicating that there is no unique stability field. We relate the streamer radius, velocity, maximal electric field and background electric field to a characteristic time scale for the loss of conductivity. This relation is qualitatively confirmed by studying streamers in N2-O2 mixtures with less oxygen than air. In such mixtures, steady streamers require lower background fields, due to a reduction in the attachment and recombination rates. We also study the deceleration of streamers, which is important to predict how far they can propagate in a low field. Stagnating streamers are simulated by applying a constant applied voltage. We show how the properties of these streamers relate to the steady cases, and present a phenomenological model with fitted coefficients that describes the evolution of the velocity and radius. Finally, we compare the lengths of the stagnated streamers with predictions based on the conventional stability field.