  A systematic analysis of low-lying quadrupole and octupole collective states
is presented, based on the microscopic energy density functional framework. By
mapping the deformation constrained self-consistent axially symmetric
mean-field energy surfaces onto the equivalent Hamiltonian of the $sdf$
interacting boson model (IBM), that is, onto the energy expectation value in
the boson condensate state, the Hamiltonian parameters are determined. The
study is based on the global relativistic energy density functional DD-PC1. The
resulting IBM Hamiltonian is used to calculate excitation spectra and
transition rates for the positive- and negative-parity collective states in
four isotopic chains characteristic for two regions of octupole deformation and
collectivity: Th, Ra, Sm and Ba. Consistent with the empirical trend, the
microscopic calculation based on the systematics of $\beta_{2}$-$\beta_{3}$
energy maps, the resulting low-lying negative-parity bands and transition rates
show evidence of a shape transition between stable octupole deformation and
octupole vibrations characteristic for $\beta_{3}$-soft potentials.
