We provide explicit examples of positive and power-bounded operators on $c_0$ and $\ell^\infty$ which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature.
Finally, we prove that if $T$ is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of $T$.
