  Population dynamics provides a numerical tool allowing for the study of rare
events by means of simulating a large number of copies of the system,
supplemented with a selection rule that favours the rare trajectories of
interest. The cloning algorithm allows the estimation of a large deviation
function (LDF) of additive observables in Markov processes. However, such
algorithms are plagued by finite simulation time $t$ and finite population size
$N_c$ effects that can render their use delicate. First, using a non-constant
population approach, we analyze the small-$N_c$ effects in the initial
transient regime. These effects play an important role in the numerical
determination of LDF. We show how to overcome these effects by introducing a
time delay in the evolution of populations, additional to the discarding of the
initial regime of the population growth where these discreteness effects are
strong. Then, the study of the finite-$t$ and finite-$N_c$ scalings in the LDF
evaluation is done using two different versions of the algorithm, in discrete
and continuous-time. We show that these scalings behave as $1/N_c$ and $1/t$ in
the large-$N_c$ and large-$t$ asymptotics respectively. Moreover, we show that
one can make use of this convergence speed in order to extract the asymptotic
behavior in the infinite-$t$ and infinite-$N_c$ limits resulting in a better
LDF estimation. These scalings are later generalized and evidence of a
breakdown for large-size systems is presented.
