  In this paper, we consider systems of semilinear elliptic equations
\displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v, \displaystyle
-\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, in the whole of Hyperbolic space
$\mathbb{H}^{N}$. We establish decay estimates and symmetry properties of
positive solutions. Unlike the corresponding problem in Euclidean space
$\mathbb{R}^N$, we prove that there exists a nonnegative nontrivial solution of
problem.
