Let
M
M
be either
S
2
×
S
2
S^2\times S^2
or the one point blow-up
C
P
2
#
C
P
¯
2
{\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2
of
C
P
2
{\mathbb {C}}P^2
. In both cases
M
M
carries a family of symplectic forms
ω
λ
\omega _{\lambda }
, where
λ
>
−
1
\lambda > -1
determines the cohomology class
[
ω
λ
]
[\omega _\lambda ]
. This paper calculates the rational (co)homology of the group
G
λ
G_\lambda
of symplectomorphisms of
(
M
,
ω
λ
)
(M,\omega _\lambda )
as well as the rational homotopy type of its classifying space
B
G
λ
BG_\lambda
. It turns out that each group
G
λ
G_\lambda
contains a finite collection
K
k
,
k
=
0
,
…
,
ℓ
=
ℓ
(
λ
)
K_k, k = 0,\dots ,\ell = \ell (\lambda )
, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as
λ
→
∞
\lambda \to \infty
. However, for each fixed
λ
\lambda
there is essentially one nonvanishing product that gives rise to a “jumping generator"
w
λ
w_\lambda
in
H
∗
(
G
λ
)
H^*(G_\lambda )
and to a single relation in the rational cohomology ring
H
∗
(
B
G
λ
)
H^*(BG_\lambda )
. An analog of this generator
w
λ
w_\lambda
was also seen by Kronheimer in his study of families of symplectic forms on
4
4
-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of
ω
λ
\omega _\lambda
-compatible almost complex structures on
M
M
.