  A numerical and analytical study of the role of exponentially truncated
L\'evy flights in the superdiffusive propagation of fronts in
reaction-diffusion systems is presented. The study is based on a variation of
the Fisher-Kolmogorov equation where the diffusion operator is replaced by a
$\lambda$-truncated fractional derivative of order $\alpha$ where $1/\lambda$
is the characteristic truncation length scale. For $\lambda=0$ there is no
truncation and fronts exhibit exponential acceleration and algebraic decaying
tails. It is shown that for $\lambda \neq 0$ this phenomenology prevails in the
intermediate asymptotic regime $(\chi t)^{1/\alpha} \ll x \ll 1/\lambda$ where
$\chi$ is the diffusion constant. Outside the intermediate asymptotic regime,
i.e. for $x > 1/\lambda$, the tail of the front exhibits the tempered decay
$\phi \sim e^{-\lambda x}/x^{(1+\alpha)} $, the acceleration is transient, and
the front velocity, $v_L$, approaches the terminal speed $v_* = (\gamma -
\lambda^\alpha \chi)/\lambda$ as $t\to \infty$, where it is assumed that
$\gamma > \lambda^\alpha \chi$ with $\gamma$ denoting the growth rate of the
reaction kinetics. However, the convergence of this process is algebraic, $v_L
\sim v_* - \alpha /(\lambda t)$, which is very slow compared to the exponential
convergence observed in the diffusive (Gaussian) case. An over-truncated regime
in which the characteristic truncation length scale is shorter than the length
scale of the decay of the initial condition, $1/\nu$, is also identified. In
this extreme regime, fronts exhibit exponential tails, $\phi \sim e^{-\nu x}$,
and move at the constant velocity, $v=(\gamma - \lambda^\alpha \chi)/\nu$.
