  When it comes to the nonlinear heat equation $u_t - \Delta u = u^p$, the
stability of positive radial steady states in the supercritical case was
established in the classical paper by Gui, Ni and Wang. We extend this result
to systems of reaction-diffusion equations by studying the positive radial
steady states of the parabolic H\'enon-Lane-Emden system $$\left\{
\begin{aligned} u_t - \Delta u &= |x|^k v^p &\mbox{ in } \mathbb R^n \times
(0,\infty),\\ v_t - \Delta v &= |x|^l u^q &\mbox{ in } \mathbb R^n \times
(0,\infty), \end{aligned} \right.$$ where $k,l\geq 0$, $p,q\geq 1$ and $pq>1$.
Assume that $(p,q)$ lies either on or above the Joseph-Lundgren critical curve
which arose in the work of Chen, Dupaigne and Ghergu. Then all positive radial
steady states have the same asymptotic behavior at infinity, and they are all
stable solutions of the parabolic H\'enon-Lane-Emden system in $\mathbb R^n$.
