FIG. 1: Schematic representation of the population dynamics considered in this paper. The arrows between the different types of individuals represent the possible population flows between them. The arrows pointing to the types themselves represent proliferation. The other arrows stand for death of each type of individuals. The thickness of the arrows stand for the realtive magnitude of each of the flows or processes. Introduction. Drug resistance is a very serious problem not least for chemotheaputic treatment of cancer and it is very important to understand, as far as possible, ways to countermeasure drug resistance in a number of biomedical contexts . In order to unravel the possible mechanisms responsible for this phenomena Iwasa and coworkers developed a prominent theoretical model of the evolutionary dynamics of escape[1, 2]. They base their approach on the assumption that n point mutations in some crucial parts of the genome are necessary for escape. They further assumed that the different mutants can be described by binary strings (with entries +1 or − 1) of length n . There are 2 n − 1 such mutants. It is assumed that treatment reduces the proliferation ratios of sensitive mutants: R < 1 whereas resistant mutants are such that R > 1. The corresponding evolutionary dynamics is modelled in terms of Galton-Watson multitype branching process (GWMBP) [3] where at each generation each individual of each type has a given (in general, mutant-dependent) probability of mutating and producing offspring belonging to a different type. The problem is to calculate the probability that a resistant mutant is reached within a population of size N . The model proposed by Iwasa et al. has been analysed in more detail in [4, 5]. Whilst this approach is both biologically and mathematically sound, we believe the efficiency with which evolutionary escape allows organisms to avoid extinction, when attacked by lethal 2