Appendix S1
Recursion for wt
We derive a recursive equation for the increment in age structure when the vital rate
matrix X t is perturbed to X t  Ct . Let the age-structure vector u t change to
ˆ t = u t   w t where  is small. Then û t iterates as
u
ˆt =
u
ˆ t 1
( X t  C t )u
*t
(1)
,
where
*t =< e, ( X t  C t )uˆ t 1 > .
(2)
Here e denotes a vector with all elements equal to 1 and < x, y >= i , j x(i ) y ( j ) for two
vectors x and y . Expanding equation (1) ,
X t u t 1   (C t u t 1  X t w t 1 )  O( 2 )
ut   w t =
,
t  t  O( 2 )
(3)
where
 t = < e, (Ct u t 1  X t w t 1 ) > .
Note that real-time elasticity Et =  t /t . Equating coefficients of  on both sides we
get
wt =
(Ct u t 1  X t w t 1 )
t

t
( X t u t 1 )
t2
Finally use the basic recursion t u t = X t u t 1 to write
1
w t = I  u t e'C t u t 1  X t w t 1 .
t
(5)
Consistency of Estimators
Here we show that our estimators are consistent, i.e., we show that (1/T )t =1( t /t )
T
converges to E , the long-run elasticity of  S (Tuljapurkar, 1990, p-89). First evaluate

T
ˆ T = uT   w T iterates as
( t /t ) by observing that the new age-structure u
t =1
following:
ˆT =
u
(4)
(XT  CT )(X1  C1 )u 0
*T 1*
 T

 G t u 0 T

t =1

= uT  
 (  t /t )u T   O( 2 )
 T 

t =1
 t =1 t



2
= u T  w T  O( ),
(6)
where
 T

 G t u 0 T

t =1

wT =
 (  t /t )u T ,
 T 

t =1
 t =1 t



(7)
with
G t = XT  X t 1Ct X t 1  X1 .
Note that | w T |= 0 (since | u *T |=| u T |= 1 ) so that from equation (7) we see that
T
T
 (  / ) =
t
| G t u 0 |
t =1
 Tt=1t
t
t =1
.
(8)
Now we show that our estimator for elasticities has the same limit as that given by
equation (11.2.7) in Tuljapurkar (1990) p-89: multiply both sides of equation (7) by v T 
so that we have
T
T
(1/T )(  t /t ) = (1/T )
t =1
v 'T
G
t
u0
t =1
v 'T u T   t
 (1/T )( v 'T w T )/( v'T u T ).
(9)
ˆ T and v'T uT are uniformly bounded positive numbers
both v'T u
ˆ T )/T = limT  ( v'T uT )/T = 0 so that from equation (6) we see that
limT  ( v'T u
limT  ( v'T wT )/T = 0 . Hence limT  (1/T )( v'T wT )/( v'T uT ) = 0 and from equation (9)
it follows that
Since
T
T
lim (1/T )(  t /t ) = lim (1/T )
T 
t =1
v 'T
G u
t
0
t =1
v 'T u T   t
T
v't Ct u t 1
= lim (1/T )
T 
t =1 t < v t , u t >
T 
which is the elasticity of the long-term growth rate  S as derived in Tuljapurkar (1990,
p-89). In the last step above we used iterative properties of the vectors u t and v t
(Tuljapurkar (1990, p-89).
(10)
Normality of Estimators
The environmental process given by the sequence of matrices {X t }t 1 is assumed to be
stationary and mixing with finite moments which guarantees that age-structure {u t }t 0 is
also stationary and mixing (Cohen 1977, Tuljapurkar 1990). Hence it follows that first
component of elasticity e R ,t is stationary and mixing. From equation (5) we see that
increment in age-structure {w t }t 0 forms a stationary mixing process being continuous
measurable functions of X t and u t proving that second component of elasticity eU ,t is
also stationary and mixing. Hence the central limit theorem holds for elasticities E t by
Corollary (5.1) of Hall and Heyde (1980).
Appendix S2: Additional figures
Figure 1
a
Population counts
150
(0-2)
(2-10)
> 10
100
50
0
1974
1978
1983
1988
1993
1998
b
Age structure
0.8
0.6
(0-2)
(2-10)
> 10
0.4
0.2
0
1974
1978
1983
1988
Years
1993
1998
Figure 2
a
0.8
(1,1)
(1,2)
(1,3)
0.6
0.4
Vital rates
0.2
0
1974
1978
1983
1988
1993
1998
b
1
0.8
(2,1)
(2,2)
(3,2)
(3,3)
0.6
0.4
0.2
0
1974
1978
1983
1988
Years
1993
1998
Figure Legends
Figure 1: a) Population counts of red deer ( Cervus elaphus) from 1974 to 2000; b)
Annual age-structure from 1974 to 2000, age-classes are (0  2) , (2  10) and (> 10)
Figure 2: a) Vital rates represented by matrix elements (1,1), (1,2) and (1,3) for years
1974 to 2000; b) Vital rates represented by matrix elements (2,1), (2,2), (3,2) and (3,3)
for years 1974 to 2000.