  We consider the speed of propagation of a {continuous-time continuous-space}
branching random walk with the additional restriction that the birth rate at
any spatial point cannot exceed $1$. The dispersion kernel is taken to have
density that decays polynomially as $|x|^{- 2\alpha}$, $x \to \infty$. We show
that if $\alpha > 2$, then the system spreads at a linear speed, {while for
$\alpha \in (\frac 12 ,2]$ the spread is faster than linear}. We also consider
the mesoscopic equation corresponding to the microscopic stochastic system. We
show that in contrast to the microscopic process, the solution to the
mesoscopic equation spreads exponentially fast for every $\alpha > \frac 12$.
