  We consider general linear non-degenerate weakly-coupled cooperative elliptic
systems and study certain monotonicity properties of the generalized principal
eigenvalue in $\mathbb{R}^d$ with respect to the potential. It is shown that
monotonicity on the right is equivalent to the recurrence property of the
twisted operator which is, in turn, equivalent to the minimal growth property
at infinity of the principal eigenfunctions. The strict monotonicity property
of the principal eigenvalue is shown to be equivalent with the exponential
stability of the twisted operators. An equivalence between the monotonicity
property on the right and the stochastic representation of the principal
eigenfunction is also established.
