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of the dormant stage enhance the population’s chance for avoiding extinction. 

We notice that the different types (1), (2) and (3) can be thought in epidemiological terms as susceptibles, infected and immune. The version of the dynamics defined as model B can be thought of as representing an age structured population. Type (3) is then juveniles, type (1) mature reproduction active individuals and type (2) are individuals in the post-reproductive stage. 

An economical and precise way to present the dynamics is in terms of the generator functions for the corresponding Galton-Watson multitype branching process, we include these generator functions in the tables in the appendix. Within the theory of multi-type branching process, the condition for eventual non-extinction is given in terms of the spectral radius, ρ , of the matrix A = ( a$_{ij}$ ), whose entries are the expected values of the number of offspring of type j produced of an indvidual of type i . If ρ > 1, there is a finite probability of N ( t → ∞ ) > 0. The quantities a$_{ij}$ are calculate in terms of the corresponding generating functions (Table II): A$_{ij}$ = ∂$_{j}$G$_{i}$ |$_{⃗}$ x = $_{1}$. From these standard arguments of the theory of multi- type branching process, which carry on in a straightforward manner when size-dependence is taken into account [14], we can establish from Tables I and II, Appendix A, that the condition for asymptotic survival, that is for P [ N ( t → ∞ ) > 0] > 0, is 

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i.e. r$_{12}$ < 1 / 2. Otherwise, the probability of eventual extinction is 1. Consider now the presence of a third type (type 3) within the population, namely, quiescent individuals. These individuals are not allowed to proliferate but are resilient to the environment and can survive under hostile conditions, hence we assume that their death rate ϵ << 1. In addition, quiescent cells are assumed to be able to revert back to type 1 at a given rate which depends on the availabity of resources. The issue we intend to analyse is whether the introduction of a quiescent sub-population helps to escape from the whole population being extinct. More precisely, the question we aim to address is: Assume r$_{12}$ ≥ 1 / 2, is it possible for a population whose dynamics is described in the by Model B and Table II to elude eventual extinction? Specifically this scenario has been considered within the context of modelling of tumour growth, in particular in the response of cancer cells to hypoxia (low levels of oygen) [12]. Cancer cells appear to become quiescent in response to hypoxia, which is thought to give them an advatage in their competition with their normal counterparts as 

