Context. Stellar companions of accreting neutron stars in ultra compact X-ray binaries (UCXBs) are hydrogen-deficient. Their helium or C/O accretion discs are strongly X-ray irradiated. Both the chemical composition and irradiation determine the disc stability with respect to thermal and viscous perturbations. At shorter periods, UCXBs are persistent, whereas longer-period systems are mostly transient. Aims. To understand this behaviour one has to derive the stability criteria for X-ray irradiated hydrogen-poor accretion discs. Methods. We use a modified and updated version of the Dubus et al. code describing time-dependent irradiated accretion discs around compact objects. Results. We obtained the relevant stability criteria and compared the results to observed properties of UCXBs Conclusions. Although the general trend in the stability behaviour of UCXBs is consistent with the prediction of the disc instability model, in a few cases the inconsistency of theoretical predictions with the system observed properties is weak enough to be attributed to observational and/or theoretical uncertainties. Two systems might require the presence of some amount of hydrogen in the donor star.