In this article we extend our previous analysis of a cold beam tube vacuum in a superconducting proton collider to include ion desorption in addition to thermal desorption and synchrotron radiation induced photodesorption. The ion desorption terms introduce the possibility of vacuum instability. This is similar to the classical room temperature case but is now modified by the inclusion of ion desorption coefficients for cryosorbed (physisorbed) molecules which can greatly exceed the coefficients for tightly bound molecules. The sojourn time concept for physisorbed H-2 is generalized to include photodesorption and ion desorption as well as the usually considered thermal desorption. The ion desorption rate is density dependent and divergent so at the onset of instability the sojourn time goes to zero. Experimental data are used to evaluate the H-2 sojourn time for the conditions of the Large Hadron Collider (LHC) and the situation is found to be stable. The sojourn time is dominated by photodesorption for surface density s(H-2) less than a monolayer and by thermal desorption for s(H-2) greater than a monolayer. For a few percent of a monolayer, characteristic of a beam screen, the photodesorption rate exceeds the ion desorption rate by more than two orders of magnitude. The photodesorption rate corresponds to a sojourn time of approximately 100 s. The article then turns to the evaluation of stability margins and the inclusion of gases heavier than H-2 (CO CO2, and CH4), where ion desorption introduces coupling between molecular species. Stability conditions are worked out for a simple cold beam tube, a cold beam tube pumped from the ends, and a cold beam tube with a coaxial perforated beam screen. In each case a simple inequality for stability of a single component is replaced by a determinant that must be greater than zero for a gas mixture. A connection with the general theory of feedback stability is made and it is shown that the gains of the diagonal uncoupled feedback loops are first order in the ion desorption coefficients whereas the gains of the off-diagonal coupled feedback loops are second order and higher. For this reason it turns out that in practical cases stability is dominated by the uncoupled diagonal elements and the inverse of the largest first-order closed loop Pain is a useful estimate of the margin of stability. In contrast to the case of a simple cold beam tube, the stability condition for a beam screen does not contain the desorption coefficient for physisorbed molecules, even when the screen temperature is low enough that there is a finite surface density of them on the screen surface. Consequently there does not appear to be any particular advantage to operating the beam screen at a high enough temperature to avoid physisorption. Numerical estimates of ion desorption stability are given for a number of cases relevant to the LHC and all of the ones likely to be encountered were found to be stable. The most important case, a 1% transparency beam screen at similar to 4.2 K, was found to have a stability safety margin of approximately 17 determined by ion desorption of CO. Ion desorption of H-2 is about a factor of 75 less stringent than CO. For these estimates the beam tube surface was assumed to be chemically cleaned but otherwise untreated, for example, by a vacuum bakeout or by glow discharge cleaning.