  The nature of very low luminosity objects with the internal luminosity L_obj
<= 0.1 Lsun is investigated by means of numerical modeling coupling the core
collapse simulations with the stellar evolution calculations. The gravitational
collapse of a large sample of model cores in the mass range 0.1-2.0 Msun is
investigated. Numerical simulations were started at the pre-stellar phase and
terminated at the end of the embedded phase when 90% of the initial core mass
had been accreted onto the forming protostar plus disk system. The disk
formation and evolution was studied using numerical hydrodynamics simulations,
while the formation and evolution of the central star was calculated using a
stellar evolution code. Three scenarios for mass accretion from the disk onto
the star were considered: hybrid accretion in which a fraction of accreted
energy absorbed by the protostar depends on the accretion rate, hot accretion
wherein a fraction of accreted energy is constant, and cold accretion wherein
all accretion energy is radiated away. Our conclusions on the nature of VeLLOs
depend crucially on the character of protostellar accretion. In the hybrid
accretion scenario, most VeLLOs (90.6%) are expected to be the first
hydrostatic cores (FHSCs) and only a small fraction (9.4%) are true protostars.
In the hot accretion scenario, all VeLLOs are FHSCs due to overly high
photospheric luminosity of protostars. In the cold accretion scenario, on the
contrary, the majority of VeLLOs belong to the Class I phase of stellar
evolution. The reason is that the stellar photospheric luminosity, which sets
the floor for the total internal luminosity of a young star, is lower in cold
accretion, thus enabling more VeLLOs in the protostellar stage. VeLLOs are
relatively rare objects occupying 7%-11% of the total duration of the embedded
phase and their masses do not exceed 0.3 Msun. (abridged).
