  In this paper, we prove global-in-time existence of strong solutions to a
class of fractional parabolic reaction-diffusion systems posed in a bounded
domain of $\mathbb{R}^N$. The nonlinear reactive terms are assumed to satisfy
natural structure conditions which provide non-negativity of the solutions and
uniform control of the total mass. The diffusion operators are of type
$u_i\mapsto d_i(-\Delta)^s u_i$ where $0<s<1$. Global existence of strong
solutions is proved under the assumption that the nonlinearities are at most of
polynomial growth. Our results extend previous results obtained when the
diffusion operators are of type $u_i\mapsto -d_i\Delta u_i$. On the other hand,
we use numerical simulations to examine the global existence of solutions to
systems with exponentially growing right-hand sides, which remains so far an
open theoretical question even in the case $s=1$.
