in which the dynamics of evolutionary escape was put forward, as hypoxic cells are known to play a major role in the resistance to chemo- and radio-therapy in tumours [11]. To study the influence of such a sub-population on the global dynamics of the total population, we model it as a dormant or quiescent population with neglegible proliferation rate and small death rate. Previous works have analysed some of the effects of quiescent cells in tumour growth (see for example [12, 13]). All these previous studies hint to the role of quiescence cells in the dynamics of tumour growth and its role in resistance to tumour growth but the issue of the evolutionary dynamics involved is not directly addressed. We now turn to the study of the survival probability. We disentangle the interplay between drug susceptibility, type (2), and dormancy, type (3), by analysing three different versions of the population dynamics, all are depicted in the diagram 1 and formulated as the following three Models. Model A – This is the ”normal” situation, where no drug is present. Type (1) and type (2) are essentially equivalent, except that when type (1) reproduces it may undergo a ”mutation” and end up as type (3). When type (2) reproduce it may mutate and become type (1). Model B – This is the situation in the presence of the drug. The death rate of type (2) is now significantly bigger than the death rates of type (1) and type (3). Moreover, the drug makes type (2) unable to reproduce and therefore the flow from type (2) to type (1) is absent. We will below find that the dormant stage increases the survival probability of the population in the presence of the drug. To emphasis that the increased ability of the population to escape extinction is in fact caused by the presence of a dormant stage, we also consider an extreme version of the dynamics in which type (1) is unable to die. Namely, Model C – This version of the the dynamics is equivalent to model B except that type (1) now is assumed not to be able to die directly, but has to flow through either type (2) or type (3) to do so. We then demonstrate that even in this extreme situation does the availability 4