  We study a Bose-Fermi mixture within the framework of the mean-field theory,
including three possible regimes for the fermionic species: fully polarized,
BCS, and unitarity. Starting from the 3D description and using the variational
approximation (VA), we derive 1D and 2D systems of equations, under the
corresponding confining potentials. This method produces a pair of nonlinear
Schr\"{o}dinger (NLS) equations coupled to algebraic equations for the
transverse widths of the confined state. The equations incorporate interactions
between atoms of the same species and between the species, assuming that the
latter can be manipulated by means of the Feshbach resonance (FR). As an
application, we explore spatial density correlations in the ground state (GS)
between the species, concluding that they strongly depend on the sign and
strength of the inter-species interaction. Also studied are the dynamics of the
mixture in a vicinity of the GS and the corresponding spatiotemporal
inter-species correlation. The correlations are strongly affected by the
fermionic component, featuring the greatest variation in the unitary regime.
Results produced by the VA are verified by comparison with full numerical
solutions.
