In the last few years there has been a renewed interest in infinite systems of differential equations, similar to the classical birth-and-death system of population dynamics, due to their rôle in modelling the evolution of drug resistance in cancer cells. In [J. Banasiak, M. Lachowicz, Topological chaos for birth-and-death models with proliferation, Math. Models Methods Appl. Sci. 12 (6) (2002) 755] such systems were shown to generate a chaotic dynamics under, however, very restrictive assumptions on the growth of coefficients. In this paper, using recently developed concept of subspace chaos [J. Banasiak, M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for topological chaos with applications, Discrete Contin. Dyn. Syst. - A 12 (5) (2005) 959], we show that for a linear growth of the coefficients, which are more acceptable from biological point of view, the dynamics of these systems is chaotic in some subspaces of the original state space.