  We study the evolution with redshift, from z=5 to z=0, of the Lyman-alpha
forest in a CDM model using numerical simulations including collisionless
particles only. The baryonic component is assumed to follow the dark matter
distribution. We distinguish between two populations of particles: Population
P{s} traces the filamentary structures of the dark matter, evolves slowly with
redshift and, for N(HI)>10^14 cm^-2, dominates the number density of lines at
z<3; most of population P{u} is located in underdense regions and for the same
column densities, disappears rapidly at high redshift. We generate synthetic
spectra from the simulation and show that the redshift evolution of the
Lyman-alpha forest (decrement, N(HI) distribution) is well reproduced over the
whole redshift range for Omega_b*h^2=0.0125 and J_-21=0.1 at z=3 where J_-21 is
the UV background flux intensity in units of 10^-21 erg cm^-2 s^-1 Hz^-1 sr^-1.
The total number of lines with N(HI)>10^12 cm^-2 remains approximately constant
from z=4 to z=1. At z=0, the number density of lines per unit redshift with log
N(HI)>12, 13, 14 is of the order of 400, 100, and 20 respectively. Therefore,
at low redshift, if most of the strong (w_r>0.3 \AA) lines are expected to be
associated with galaxies, the bulk of the Lyman-alpha forest however should
have lower equivalent width and should not be tightly correlated with galaxies.
