  The existence of true scale-invariance in slowly driven models of
self-organized criticality without a conservation law, as forest-fires or
earthquake automata, is scrutinized in this paper. By using three different
levels of description - (i) a simple mean field, (ii) a more detailed
mean-field description in terms of a (self-organized) branching processes, and
(iii) a full stochastic representation in terms of a Langevin equation-, it is
shown on general grounds that non-conserving dynamics does not lead to bona
fide criticality. Contrarily to conserving systems, a parameter, which we term
"re-charging" rate (e.g. the tree-growth rate in forest-fire models), needs to
be fine-tuned in non-conserving systems to obtain criticality. In the infinite
size limit, such a fine-tuning of the loading rate is easy to achieve, as it
emerges by imposing a second separation of time-scales but, for any finite
size, a precise tuning is required to achieve criticality and a coherent
finite-size scaling picture. Using the approaches above, we shed light on the
common mechanisms by which "apparent criticality" is observed in non-conserving
systems, and explain in detail (both qualitatively and quantitatively) the
difference with respect to true criticality obtained in conserving systems. We
propose to call this self-organized quasi-criticality (SOqC). Some of the
reported results are already known and some of them are new. We hope the
unified framework presented here helps to elucidate the confusing and
contradictory literature in this field. In a second accompanying paper, we
shall discuss the implications of the general results obtained here for models
of neural avalanches in Neuroscience for which self-organized scale-invariance
in the absence of conservation has been claimed.
