A vector  bundle morphism  of  a vector  bundle with  strongly ordered
Banach spaces as fibers is studied.  It is assumed that the fiber maps
of  this morphism are compact  and strongly positive. The existence of
two      complementary (dimension-one and codimension-one)  continuous
subbundles invariant under the morphism is established.  Each fiber of
the first bundle is  spanned by a positive  vector (that is, a nonzero
vector lying in the order cone), while  the fibers of the other bundle
do  not contain a  positive vector.  Moreover,   the ratio between the
norms of the  components  (given by the splitting  of  the bundle)  of
iterated  images   of  any  vector in  the     bundle approaches  zero
exponentially (if the positive component is in the denominator).  This
is  an  extension of the  Krein-Rutman  theorem which  deals  with one
compact strongly positive map only. The existence of invariant bundles
with the  above    properties  appears to  be  very    useful   in the
investigation of  asymptotic   behavior of  trajectories   of strongly
monotone discrete-time  dynamical systems, as demonstrated by
Pol\'a\v cik and Tere\v s\v c\'ak (1991) and Hess and Pol\'a\v cik (1993).
The present paper also  contains  some new  results on typical  asymptotic
behavior in scalar periodic parabolic equations.