  For a competition-diffusion blow-up system involving the fractional Laplacian
of the form \begin{equation*}\label{syst1}
-(-\Delta)^su=uv^2,\quad-(-\Delta)^sv=vu^2,\quad u,v>0 \ \mathrm{in} \
\mathbb{R}^N, \end{equation*} whith $s\in(0,1)$, we prove that the maximal
asymptotic growth rate for its entire solutions is $2s$. Moreover, since we are
able to construct symmetric solutions to the problem, when $N=2$ with
prescribed growth arbitrarily close to the critical one, we can conclude that
the asymptotic bound found is optimal. Finally, we prove existence of genuinely
higher dimensional solutions, when $N\geq 3$.
