The authors provide a detailed description of the long time dynamics of the semigroup associated with constant coefficient infinite birth-and-death systems with proliferation. They identify a range of parameters for which the semigroup is both stable and topologically chaotic. The results extend earlier stability results by A. Bobrowski
and M. Kimmel
[J. Biol. Syst. 7, No. 1, 33–43 (1999), doi:10.1142/S0218339099000048
]. Moreover, for a range of parameters, they give an explicit description of subspaces which cannot generate chaotic orbits.