Population 1 G$_{1}$ ( ⃗ x ) = 1 − e − µN + e − $^{µN}$r$_{12}$ x 2 + e − $^{µN}$r$_{13}$ x 2 3 + e − $^{µN}$(1 − r$_{12}$ − r$_{13}$ ) x 2 1 Population 2 G$_{2}$ ( ⃗ x ) = α + (1 − α ) x$_{2}$ Population 3 G$_{3}$ ( ⃗ x ) = ϵ + r$_{31}$e − $^{µN}$x$_{1}$ + (1 − ϵ − r$_{31}$e − $^{µN}$) x$_{3}$ TABLE II: Model B. Generating functions of the probabilities of the number of offspring for individuals of type 1, i.e. well adapated to harsh conditions, G$_{1}$ ( ⃗ x ), type 2, i.e. those for which the environment is lethal, G$_{2}$ ( ⃗ x ), and type 3, i.e. quiescent individuals, G$_{3}$ ( ⃗ x ). The parameter µ is the carrying capacity, which accounts for the limitation in resources, α is a measure of the remanent resilience of inviduals of type 2 to the environment, and ϵ is the death rate (probability of death per individual per generation) of the quiescent cells. The quantity r$_{1}$$_{j}$ with j = 1 , 2 can be interpreted as the muation rate of individuals of type 1 into individuals of type j . The quantity r$_{31}$ is the probability per quiescent individual per generation of a type 3 individual to revert to ype 1. TABLE I: Model A . Generating functions of the probabilities of the number of offspring for individuals belonging to our population previous to the introduction of a hostile agent. In this situation, both type 1 and type 2 indviduals are well adapated and can proliferate subject to the restrictions imposed by the carrying capacity. Type 3 correspond to quiescent individuals. The new parameters introduced here are r$_{21}$ , i.e. the mutation probability of type 2 into type 1, the probability of a type 2 individual to enter quiescence, r$_{23}$ , and the probability of a quiescent individual to revert to type 2. The rest of the parameters have the same physical meaning as in Table II Population 1 G$_{1}$ ( ⃗ x ) = 1 − e − µN + e − $^{µN}$r$_{12}$ x 2 + e − $^{µN}$r$_{13}$ x 2 3 + e − $^{µN}$(1 − r$_{12}$ − r$_{13}$ ) x 2 1 Population 2 G$_{2}$ ( ⃗ x ) = 1 − e − µN + e − $^{µN}$r$_{21}$ x 2 + e − $^{µN}$r$_{23}$ x 2 3 + e − $^{µN}$(1 − r$_{21}$ − r$_{13}$ ) x 2 1 Population 3 G$_{3}$ ( ⃗ x ) = ϵ + r$_{31}$e − $^{µN}$x$_{1}$ + r$_{32}$e − $^{µN}$x$_{2}$ + (1 − ϵ − ( r$_{31}$ + r$_{32}$ ) e − $^{µN}$) x$_{3}$ [1] Y. Iwasa, F. Michor, and M. A. Nowak, Proc. R. Soc. B 270 , 2573 (2003). [2] Y. Iwasa, F. Mchor, and M. A. Nowak, J. theor. Biol. 226 , 205 (2004). [3] M. Kimmel and D. E. Axelrod, Branching processes in Biology (Springer-Verlag, New York, 16