  We analyse numerically the effects of small population size in the initial
transient regime of a simple example population dynamics. These effects play an
important role for the numerical determination of large deviation functions of
additive observables for stochastic processes. A method commonly used in order
to determine such functions is the so-called cloning algorithm which in its
non-constant population version essentially reduces to the determination of the
growth rate of a population, averaged over many realizations of the dynamics.
However, the averaging of populations is highly dependent not only on the
number of realizations of the population dynamics, and on the initial
population size but also on the cut-off time (or population) considered to stop
their numerical evolution. This may result in an over-influence of discreteness
effects at initial times, caused by small population size. We overcome these
effects by introducing a (realization-dependent) time delay in the evolution of
populations, additional to the discarding of the initial transient regime of
the population growth where these discreteness effects are strong. We show that
the improvement in the estimation of the large deviation function comes
precisely from these two main contributions.
