of the generating functions given in Table III. Its rationale is as follows. Let us assume that a population of cells is divided in two types: those in which the pathways regulating cell division are activated (type 1) and those in which the active pathways are those regulating apoptosis, i.e. cell death, (type 2). Type 1 cells can either proliferate, or stay as they are, or suffering activation of the apoptotic pathways, thus becoming type (2) cells. The flux of population between types (1) and (2) is controlled by the parameter r$_{12}$ . Type 2 cells are those marked for cell death, but the model provides for their staying within the population for a while (not dying immediately), but they cannot proliferate. Type (3) cells are, as in the previous model, quiescent cells. In this context the introduction of a hostile agent (drug, removal of oxygen, etc.) corresponds simply to increasing the value of r$_{12}$ . In the absence of quiescence, standard arguments reveal that extinction with probability 1 occurs when r$_{12}$ ≥ 1 / 2. The question is once again whether quiescence can rescue the population from extinction under such conditions. Fig. 7 shows that this is indeed the case, provided that the flux out of the quiescent state back into type (1) does not exceed a critical value. This results are completely analogous to those obtained for the previous model. Invasion dynamics. We know discuss under which conditions a small proportion of in- dividuals, y , which can undergo quiescence, modelled by Model A and B depending on whether a drug is present or not, respectively, and with r$_{13}$ = 0, can take over a population of fully growing individuals, i.e. a population modelled by Model A and B with r$_{13}$ = 0. In particular, we will analyse the corresponding invasion probability, P$_{F}$ ( y ), by direct simula- tion of the population dynamics and by using the analytical approximation provided by the so-called evolutionary formalism [16–19] which allows a more thorough exploration of the behaviour of the invasion probability as a function of the model parameters. The problem we address here relates to whether a small population of mutants with can take over an incumbent population. Demetrius and coworkers [17, 18] have developed a formalism based on the application of ideas from ergodic theory to evolutionary problems [16] and the diffusion approximation [20]. This formalism allows us to estimate the fixation probability of a mutant population in the presence of an incumbent species. Here we gener- alise this formalism to apply it to the problem of whether a genetic inactivation generating new phenotypes can invade the incumbent population. The starting point of this formalism is the following fundamental equation: 10