Supplementary Information: Derivations for input and output interactions
Competition-interaction landscapes for the joint response
of forests to climate change
JAMES S. CLARK*† , DAVID M. BELL* , MATTHEW C. KWIT* and KAI ZHU*
Here we summarize background for input and output interactions, based on the
approach detailed Clark et al. (2010) of the main text. For simplicity, consider two
responses in the response vector yij,t, diameter growth dij,t and fecundity fij,t. Let r
represent one of the responses yr Î ln dij,t ,ln fij,t to one of the input variables
{
}
xq Î{1,… ,Q} . A sensitivity coefficient calculated from models lacking input and output
interactions would look like this,
s(rq) =
dyir ¶yir ¶xiq A(qr )
=
= D
dxiq¢ ¶xiq ¶xiq¢
xq
S1
The j and t subscripts are omitted to reduce clutter. This is the proportionate (log)
D
response of growth or fecundity to an input xq, standardized by the range of variation xq
to allow comparison of effects across different input variables,
(
( ))
xij,t
¢ ,q = xij,t,q - min xq
( ( )
( ))
xqD , where xqD = max xq - min xq .
When there are interactions among inputs and outputs (within individuals), we have a
wide range of responses,
æ ¶y
¶y ¶y ö ¶xiq
si(qq ¢r ¢ ®r ) = ç ir + r ir ¢ ÷
è ¶xiq ¶yr ¢ ¶xiq ø ¶xiq¢
=
S2
1
é Bi(qr ) + grr ¢ Bi(qr ¢ ) ùû
xqD ë
where
grr ¢ =
¶yr
= S rr ¢ S -1
r¢
¶yr ¢
is the (output) dependence of a response yr on another response variable yr', and
1
Bi(qr ) = Aqr +
åA
q ¢ ÎI q
x
(qq ¢ )r i, q ¢
S3
is the sum of the main effect of xq on r, Aqr, and other inputs xq' that interact with xq.
Note that grr' is not the full dependence of yr on yr', but rather is limited to only the
dependence not related to inputs, which are taken up by matrix A of eqn 2 in the main
text. The prime notation on r' indicates the response that is not r, and q' refers to variables
that interact with q. Input interactions (between input variables xq and xq') enter through
Bi(qr). Output interactions (between response variables r and r') enter through covariance
rr', the effects of which also depend on input interactions Bi(qr'). The sensitivity subscript
now includes not only i, but also r' and q', the interactions. In other words, interactions
shift the response scale from species to individual. The terms in eqn S3 match those of
eqn 3 of the text.
The survival submodel allows for the negative effects of decreasing growth and
increasing size on survival probability,
( )
p ( Si,t ) = Bernoulli z ij,t
(
z ij,t = 1 - md,ij,t + m D,ij,t - md,ij,t m D,ij,t
)
(
)
Each individual in each year has a pair of risks md,ij,t , m D,ij,t associated with growth rate
(monotonically decreasing) and diameter (monotonically increasing), with survival
probability from year t-1 to year t. Sensitivity to input xq depends on tree size and growth
rate
sz ,q D =
=
dz ij,t
dxq,ij,t
¢
(
=
dz ij,t ¶md,ij,t
d md,ij,t ¶xq,ij,t
¢
)
(
= 1 - m D,ij,t
¶md,ij,t ¶ ln dij,t ¶xq,ij,t
) ¶ln d
ij,t
¶xij,t ,q ¶xq,ij,t
¢ ,q
si(qq ¢r ¢ ®r ) 1 - m D,ij,t ¶md,ij,t
x
D
q
¶ln dij,t
S4
Note that the first factor on the right-hand side is eqn S2.
2