  The concept of an attractor or constrictor was used by several mathematicians
to characterize the asymptotic behavior of operators. In this paper we show
that a positive LR-net on KB-spaces is mean ergodic if the LR-net has a weakly
compact attractor. Moreover if the weakly compact attractor is an order
interval, then a Markovian LR-net converges strongly to the finite dimensional
fixed space. As a consequence we investigate also stability of LR-nets of
positive operators and existence of lower bound functions on KB-spaces.
