  Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences
of a fibration p:E\to B. We make a general study of this monoid, especially in
rational homotopy theory. When E and B are simply connected CW complexes with E
finite, we identify the rational Samelson Lie algebra of the identity component
of Aut(p) as the homology of a certain DG Lie algebra of derivations arising
from the Koszul-Sullivan model of p. We obtain related identifications for the
rational homotopy groups of fibrewise mapping spaces and for the
rationalization of a natural nilpotent subgroup of Aut(p).
