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Sensitivity of cycling measures derived from ecological flow analysis
Ecological Modelling, 48 (1989) 45-64 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
45
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
ROBERT W. BOSSERMAN Department of Biology, University of Louisville, Louisville, KY 40292 (U. S.A.) (Accepted 3 April 1989)
ABSTRACT Bosserman, R.W., 1989. Sensitivity of cycling measures derived from ecological flow analysis. Ecol. Modelling, 48: 45-64. Ecologists have developed and applied flow analyses to examine indirect interactions in ecosystem models. These methods generate measures of indirect influence such as recycling efficiencies, cycled throughflows, and cycling indices. Results of such analyses provide insight into the nature of interactions in ecosystems. This paper describes a sensitivity tectmique for examining flow structures of ecosystem models. The sensitivity technique relies on a method for determining the 'condition' of inverted matrices. Resulting sensitivity values describe effects of perturbations and errors on indirect flows, recycling efficiencies, cycling indices, cycled throughflows and other flow measures. I have applied the methods to three models of Ca flow for the Hubbard Brook forest.
INTRODUCTION M a t t e r a n d energy flows interconnect ecological c o m p o n e n t s in ecosystems, a n d give rise to n e t w o r k p h e n o m e n a i m p o r t a n t to ecology. Direct a n d indirect influences p r o p a g a t e t h r o u g h ecosystems a n d affect n u t r i e n t cycling, coevolution, trophic relationships, diffuse competition, a n d d y n a m i c behavior. Causes a n d effects which are t r a n s m i t t e d t h r o u g h an ecosystem n e t w o r k can generate n o n o b v i o u s consequences which c a n n o t be predicted b y analysis of c o m p o n e n t parts. Ecologists i n t r o d u c e d flow analyses to q u a n t i f y ecosystem characteristics ( H a n n o n , 1973; Patten, et al., 1976; Finn, 1977, 1980) such as indirect flow structure, cycling indices, a n d recycling efficiencies. These measures represent system-level properties that d e p e n d on the p a t t e r n of interactions a m o n g c o m p o n e n t s . In this paper, I present a sensitivity analysis for assessing measures derived f r o m flow analyses. 0304-3800/89/$03.50
© 1989 Elsevier Science Publishers B.V.
46
R.W. BOSSERMAN
Sensitivity techniques quantify effects of changes in model variables. Changes result from several sources: environmental variation, misconception, measurement error, and experimental manipulation. In complex models, interactions among model parameters often disguise consequences of variable change. Researchers have applied sensitivity techniques to different aspects of ecological modeling (Astor et al., 1976; Waide and Webster, 1976), but only recently have they been applied to flow analyses. In economics authors have studied effects of perturbations on input-output systems (Sherman and Morrison, 1950; Christ, 1955; Sebald, 1974; Bullard and Sebald, 1977). In ecology I derived sensitivities for flow analyses of ecosystem models (Bosserman, 1981, 1983). Here, I have extended previously published methods to include cycling indices, cycled throughflow, recycling efficiencies and pathlength as defined by Finn (1977, 1980). METHODS
Relationships for flow analyses Several relationships form the basis of the flow analyses used in ecology. They apply to compartmental models of energy/matter storage and dynamics. Flow analysis begins with the formation of flow" matrix, F; f~j depict amounts of matter or energy flow from compartments j to i. Column z of F contains all of the inflows to compartments from the environment, while row y contains all outflows from compffrtments to the environment. Summing the rows of F generates throughflow vector e', and summing the columns of F generates throughflow vector e". Element e~' of e' represents the amount of flow into compartment i; %l ! of e " represents the amount of flow out of compartment j. In a steady state model inflows equal outflows and e' equals e "T, where T represents the transpose operation. For nonsteady-state models, addition of state-variable derivatives (:~+, ~_) allows a dynamic flow analysis (Patten et al., 1976; Finn, 1980). Flow analysis may also include storage as a vector x; xj equals the amount of material in compartment j. Ecological flow analysis (Patten et al., 1976) either traces flows from compartments to model outflows or traces from model inflows to compartments. I will designate matrices by ' for the first (forward-looking) case and will designate matrices by " for the second (backward-looking) case. When the mathematical derivations for the two cases are the same, I will not use ' or
rp
Normalized flow matrices, G' and G", result from the next step in flow analysis. To form G' from F, each f~j is divided by the corresponding row
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
47
sum, e~'; to form G " from F, each f~j is divided by the corresponding column sum, e)', i.e. n !
n
I
g,j = L : / e i = f i J E f,j
and
tp
cp
gij = fij/ej = f i J E fij)
i=1
j=l
The following equations relate normalized flow matrices to outflows and inflows (Patten et al., 1976; Finn, 1977, 1980): e' = e ' G ' + y + :~+
(1)
e " = G " e " + ~ - :~_
(2)
where e', y, e", and z are as defined above, :~ is a vector of positive time derivatives of storages, and :f_ is a vector of negative time derivatives of storages. After rearranging the above equations: e'=(y+
:~+)(I - G ' ) -1
e " = (I -- G " ) - a ( z
- k_)
(3) (4)
H a n n o n (1973) called matrix ( I - G ' ) - a a "structure matrix", while Patten et al. (1976) called (I - G ' ) - 1 and (I - G ") - a ,, transitive closure matrices". (I - G , )~j -1 shows how m u c h flow from c o m p a r t m e n t j to i is required to generate a unit of flow to model outfows. (I - G " ' ~,,j- 1 shows how m u c h flow from c o m p a r t m e n t j to i is generated by a unit of flow from model inflows. In the discussion below, N ' and N " designate the transitive closure matrices ( I - G ' ) -1 and ( I - G " ) -1, respectively. Finn (1976, 1977, 1980) derived cycling indices, recycling efficiencies, cycled throughflow and other measures from N ' and N " (described below).
Condition of matrix The 'condition' of a matrix B refers to the a m o u n t of change that occurs in B -1 when element bij is varied. Condition can be expressed as 0 B - a / 0 b i j (derivations which are shown in the Appendix):
~)B-1/~)bij = - [ n - l ( O B / 0 b i j ) B -1]
(5)
Matrix OB/Obij contains all zeros except for the /jth element which is 1.0; therefore, sensitivity matrix, 0B-1/Sb~i, results from multiplying the ith column of B - 1 by the j t h row of B -1. (0B-1/0b~j)kl shows how m u c h (B-1)kl changes when b~j is changed. If bij of B is increased by an a m o u n t e to form B*, then B * - I equals B -1 +e(OB-1/3bij). Table 1 demonstrates the sensitivity analysis for a hypothetical matrix. Tables l a and l b show matrices B and B -1 and Tables l c and l d show matrices B* and B *-a. Matrix B* (Table lc) differs from B in that b13 has been increased by 0.0001 (i.e., b~'3 - b13 = 0.0001). Postmulti-
1 2 3 4
-0.2263 -0.2263 -0.0453 0.0000
1.0000 -1.0000 0.0000 0.0000
(c) B * 1 2 3 4
(e) ~ B - 1 / ~ b ] 3
1.0000 -1.0000 0.0000 0.0000
(a) B 1 2 3 4
1
-0.2263 -0.2263 -0.0453 0.0000
0.0000 1.0000 -0.2000 0.0000
0.0000 1.0000 -0.2000 0.0000
2
-1.1317 - 1.1317 -0.2263 0.0000
- 0.2999 0.0000 1.0000 0.0000
-0.30000 0.0000 1.0000 0.0000
3 (b) B -1 1 2 3 4
-1.0638 -1.0638 -0.2128 0.0000
(~ B - 1 - B * - I 1 2 3 4
(d) B* - l -0.7000 1 0.0000 , 2 -0.8000 3 1.0000 4
-0.70000 0.0000 -0.8000 1.0000
4
2.263X 10 - s 2.263X10 -5 4.53 X 10 - 6 0.00
1.0638 1.0638 0.2128 0.0000
1.0638 1.0638 0.2128 0.0000
1
2.263X10 - s 2 . 2 6 3 x 10 -3 4.53 x 1 0 -6 0.00
0.0638 1.0638 0.2128 0.0000
0.0638 1.0638 0.2128 0.0000
2
0.9999 0.9999 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000
4
1 . 1 3 1 7 x 1 0 4 1.0638 x 1 0 - 4 1.1317 x 10 - 4 1.0638 X 10 - 4 2.263 x 10 - s 2.128 x l 0 s 0.00 0.00
0.3190 0.3190 1.0638 0.0000
0.3191 0.3191 1.0638 0.0000
3
Demonstration of sensitivity analysis with arbitrary, invertible matrix in which p a r a m e t e r b13 is increased by 0.0001
TABLE 1
oo
SENSITIVITY OF CYCLING
MEASURES
DERIVED
FROM
ECOLOGICAL
FLOW ANALYSIS
49
plying column 1 of B-1 by row 3 of B-1 produces 0B-1/Obl3 (Table le). The difference between B -1 and B * - 1 (Table If) equals - 0 . 0 0 0 1 × c)B- 1/()b13 .
Sensitivity matrices for flow analyses 'Condition' analyses apply to the relationships between the normalized matrices G and the transitive closure matrices N = (I - G)-1. In the following discussion, ' and " analyses parallel one another. Sensitivity matrices S(i, j ) = 0 N / a ( N - 1 ) i j = 0 N / 0 ( I - G)i j can be determined. Elements of ( I - G) take nonpositive values except along the main diagonal. A change in any element of G will cause an opposite change in ( I - G), so S(i, j ) = 0 N / 0 ( I - G)~j = -ON/Ogij. Changes in the negative elements of ( I - G ' ) and (I - G ' ) cancel the negative sign that occurs in equation (5). Elements of G ' or G " indicate the fraction of flow to come from or go to each part of the model. Each row or column in G ' or G " must sum 1.0, i.e. t/
Y'. gi'j= 1.0 j=l or n it
E gij = 1.0
i=1
The fact that the row sums or column sums are constrained to equal 1.0 ! means that a single element cannot be changed by itself. If one element g~j I! or &j is changed then another element (or several elements) in the same row (for G ' ) or column (for G " ) must be changed to maintain the constraint. The most reasonable elements to change are those that represent inflow z or ! ,,¢ outflow y of the model (i.e., either &z or gyj). Sensitivity can therefore be calculated as S'(i, j ) - S'(i, z) or S ' ( i , j ) - S ' ( y , j ) . S ' ( i , z) contains all zeros, except for the column that corresponds to inflow z, and S " ( y , j ) contains all zeros except for the row that corresponds to outflow y. Other combinations of elements in the row or column can also be changed, as long as the row sum in G ' or the column sum in G " equals 1.0. For very small changes in elements, this constraint m a y not be important. M a t r i c e s SPL' a n d SPL" can be formed, where: n
s.L;j= E s'(i, J)k, k,l=l
and n
SPLi'2 = Y'~ S " ( i, j ) kt k,l=l
50
R.W. BOSSERMAN
Elements of SPL' and Sl'L" show the total change in N ' and N " due to ? changes in gij and g~'j.
Relationships between sensitivity analysis and power series The relationship between the power series of G and transitive closure matrix N demonstrates the mechanism of the sensitivity analysis. Patten et al. (1976) used the equality: oo
E G i= ( l - G) -1 i=1
to trace flow through paths of increasingly longer lengths. G 2 shows the proportion of material that has traveled along two-length paths, G 3 the proportion of material that has traveled along three-length paths, and in general, G" the proportion along n-length paths. The product series can also be used to trace the effects of parameter modification along the multilength paths in a model. If g~ij and gi~ are increased by e and - e , respectively, to form (], then ~ i = 1 G ' = ( I - G ) -a. The difference between ~o¢ i = l G~ and Eoo (], traces the effect of modification through the model network. G - (~ i=1 shows the difference between one-length paths, G 2 - (~2 the difference between modified and unmodified two-length paths, etc. The network structure of the model governs the propagation of the modified flows. The series E~i = 1 (G ~ - (~i) converges to the sensitivity matrix S(i, j ) - S(i, z).
Sensitivity of flow pathlengths The sums of the rows and the columns of the transitive closure matrices N ' and N " represent the average pathlengths that material or energy have traveled to the outflow or from the inflow (Finn, 1979, 1980). YPL can be defined as a vector of pathlengths from compartments to outflows where elements Y P L k = ~nj=l nkj' is the average pathlength from compartment k to the model outflows; Zl'L is a vector of pathlengths from the inflow to compartments, where element zPL k = ~"~=1 nik "' is the average pathlength from the model inflow to compartment k. The total sum of all elements in the transitive closure matrices is the average pathlength from inflow to outflow, YPL, or to outflow from inflow, ZPL. Each row sum of S'(i, j) represents the change that occurs in the average pathlength from compartment k to the model outflow when gi'j changes; i.e. OYeLI,/Ogi'j = ~t"=l S'(i, J)kt. Likewise, each column sum of S"(i, j ) represents the amount of change that occurs in the average pathlength from the It model inflow to compartment k when g~j changes; i.e. O Z P L k / O g i j It = Y~7<=l Sit( i, J)kl" SYPLk can be defined for compartment k to be the matrix
SENSITIVITY
OF CYCLING
MEASURES
DERIVED
FROM
ECOLOGICAL
FLOW
ANALYSIS
51
of pathlength sensitivities where each element SYPLij is 0 YPLk/Ogi'j. SZPL k can be defined to be the matrix of pathlength sensitivities where each tf element SZPL~j is 3 ZPLk/3g~j.
Sensitivity of recycling efficiencies Finn defined the recycling efficiency of compartment k, REk, as: p
?
t?
pt
REk= [nkk-- 1]/nkk = [nkk-- 1]/nkk = 1 -- (1/nkk)
(6)
Recycling efficiencies reflect the degree to which material which has passed through a compartment once will return and pass through again. The sensitivity of REk with respect to a change in an element of G' or G " (derived in Appendix) is:
3 RF,k/Og,j= --(Onkk/Og, j)/n2kk =
-
S(i, j ) / n 2 k
(7)
The sensitivity of the recycling index for compartment k when the sensitivity due to a change in the inflow is removed is:
[s(i,
s(i,
(8)
Sensitivity of cycled throughflow The amount of flow that passes through a compartment, Tk, can be partitioned into a cycled part, T~, and a noncycled part (T k - T~k); Tc~ can be determined by the relationship:
T~, = REk(Tk)
(9)
Sensitivity of the cycled throughflow for compartment k with respect to changes in gij can be calculated as follows:
OTck/Ogij
-
OTck/Ogiz = Tk(OREk/Ogij ) = Tk
S(i
,
"
z
(lO)
Tk and Tck for each compartment can be added to get the total system
throughflow (TST) and total system cycled throughflow (rsrc) through the entire model; i.e. TST = E'k=l Tk, and TSTo = Y~7,=aT~. Sensitivities of the total cycled throughflow, TST¢, equal the sums of sensitivities of compartmental cycled throughflows: n
O rSTc/Og,j = Y'~ OTc~/Og,y k=l
(11)
52
R.W. B O S S E R M A N
Sensitivity of cycling index Finn (1976, 1980) defined a cycling index as follows: n
CI= rSrc/rSr=
n
ELk/ E rk k=l
(12)
k=l
r s r will be assumed to be constant in these derivations because a change in gij will be balanced by a change in g/z or giy. The sensitivity of ci with respect to changes in g,j (derivation in Appendix) is:
OCI/OgiJ=O(TSTc/TST)/OgiJ= [ ~ ~
(13)
Therefore, after substituting from equation (10) the sensitivity of the cycling index cI with respect to the element gij is:
14, APPLICATION TO THE FOREST CALCIUM MODELS The flow analysis sensitivities strongly depend on both the model structure and the values of flows among compartments. Different researchers often will generate different models of the same system. Sensitivity analyses can provide perspectives into the differences between various models and can generate questions about the differences between models. In this paper, the sensitivity analysis has been applied to three models of Ca flow through the Hubbard Brook experimental forest; each model was developed by a different group of authors. Finn (1976, 1980) did the original flow analyses for this model. Figure 1 shows the diagrams for these model.
Waide et al. 's model of Hubbard Brook temperate forest Waide et al. (1974) constructed a model of Ca flow through a temperate hardwood forest at the Hubbard Brook experimental forest. The model contains vegetation (X1), litter (X2), and available nutrients (X3), soil and rock minerals (X4). A single cycle contains vegetation, litter and available nutrients, while no cycle contains mineral soil. Flow matrix F (Table 2a), which consists of flows of Ca (kg ha -a year -1) to compartment i from compartment j, shows that inflows and outflows are balanced. Tables 2b and 2c show transitive closure matrices N ' and N " . The fifth row and column depict inflows from the environment, while the sixth row and
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
o.
Waide
b. Jordan
53
Model
Model
c. Finn Model
Fig. 1. Three compartmental models of Ca flow through Hubbard Brook experimental forest: (a) model developed by Waide et al. (1974), (b) model developed by Jordan et al. (1972), (c) model developed by Finn (1980). All models were calibrated by Finn (1976, 1980).
column depict outflows from the model. N ' and N " also contain average pathlengths from compartments to outflows and to compartments from inflows, respectively (row sums and column sums of N ' and N " ) .
Jordan mineral model for temperate and tropical forests Jordan et al. (1972) developed a general mineral model for temperate and tropical forests (Fig. lb), which Finn (1976) calibrated with Ca data from Hubbard Brook. This model contains canopy (X1), litter (X2), soil (X3), and Wood (X4). Cycles interconnect all four of the compartments. Flow matrix F (Table 3a) shows that the model is at steady state where inflows balance outflows. Tables 3b and 3c) also contain transitive closure matrices N ' and N PP.
54
R.W. BOSSERMAN
TABLE 2 Flow and transitive closure matrices for Waide Ca model (a) Flow matrix F (kg Ca m-1 year-1) X1 X2 X3 X4 Inflow Outflow
X1
X2
X3
0.00 50.03 0.00 0.00
0.00 0.00 50.21 0.00
47.69 0.00 0.00 0.00
0.00 0.00
0.00 0.08
0.00 11.32
X4
Inflow
Outflow
0.00 0.00 8.80 0.00
2.34 0.26 0.00 9.00
0.00 0.00 0.00 0.00
0.00 0.20
0.00 0.00
0.00 0.00
(b) Transitive closure matrix N ' S1
X1 X2 X3 X4
X2
5.18 5.15 4.38 0.00
4.20 5.18 4.41 0.00
S3
4.94 4.91 5.18 0.00
X4
0.74 0.73 0.77 1.00
Row
sum
16.16 17.07 15.76 2.00
(c) Transitive closure matrix N " Xl
X1 X2 X3 X4 Column sum
X2
S3
X4
5.18 5.18 5.17 0.00
4.18 5.18 5.17 0.00
4.19 4.19 5.18 0.00
4.09 4.09 5.06 1.00
16.53
15.53
14.56
15.24
Finn's model of Hubbard Brook forest F i n n (1980) d e v e l o p e d a t h i r d m o d e l o f C a f l o w t h r o u g h a H u b b a r d B r o o k forest w i t h d a t a f r o m L i k e n s et al. (1977). T h i s m o d e l (Fig. l c ) consists of a b o v e g r o u n d b i o m a s s (X1), forest f l o o r ( X 2 ) , m i n e r a l soil (X3) , b e l o w g r o u n d b i o m a s s ()(4), a n d a v a i l a b l e n u t r i e n t p o o l ()(5); it differs f r o m W a i d e ' s a n d J o r d a n ' s m o d e l s b e c a u s e it is n o t in s t e a d y state; t h e r e f o r e , d e r i v a t i v e s of s t a t e v a r i a b l e s are n o n z e r o . Cycles c o n n e c t a b o v e g r o u n d b i o m a s s , b e l o w g r o u n d b i o m a s s , forest floor, a n d a v a i l a b l e n u t r i e n t s , w h i l e n o cycles include m i n e r a l soil. F l o w m a t r i x F ( T a b l e 4a) i n c l u d e s t i m e d e r i v a t i v e s of s t a t e v a r i a b l e s (:~+ = 0 if d x k / d t <_O, .it+ = d x ~ / d t if d x k / d t > 0; :~_ = 0 if d x k / d t >__O, At_ = d x ~ / d t if d x k / d t < 0) ( F i n n , 1980). F l o w a n d sensitivity a n a l y s e s p r o c e e d as in the s t e a d y s t a t e case, e x c e p t t h a t r o w a n d c o l u m n s u m s u s e d to f o r m G ' a n d G " i n c l u d e :~+ a n d : ~ .
55
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
TABLE 3 Flow and transitive closure matrices for Jordan Ca model (a) Flow matrix F (kg Ca m -2 year -1) X1 X2 X3 X4 Inflow Outflow
)(1
X2
X3
Xa
Inflow
Outflow
0.00 49.90 13.00 0.00
0.00 0.00 49.90 0.00
0.00 0.00 0.00 59.30
59.30 0.00 0.00 0.00
3.60 0.00 39.50 0.00
0.00 0.00 0.00 0.00
0.00 0.00
0.00 0.00
0.00 43.10
0.00 0.00
0.00 0.00
0.00 0.00
(b) Transitive closure matrix N' X1
X1 X2 X3 X4
X2
2.38 2.38 1.46 1.46
1.09 2.09 1.16 1.16
X3
2.24 2.24 2.38 2.38
X4
2.24 1.38 2.38
Row
sum
7.95 8.95 6.48 7.48
(c) Transitive closure matrix N "
s~
x2
x3
x4
X1 X2 X3 X4
2.38 1.89 2.38 1.38
1.38 2.09 2.38 1.38
1.38 1.09 2.38 1.38
2.38 1.89 2.38 2.38
Column sum
8.03
7.23
6.23
9.03
RESULTS
T r a n s i t i v e closure matrices for W a i d e ' s m o d e l (Tables 2b a n d 2c) s h o w that vegetation (X1), litter (X2), a n d available n u t r i e n t s (X3) o c c u r in C a cycles while soil (X4) does n o t (n n = / ' / 2 2 = / / 3 3 = 5.18, /144 = 1.00). F o r J o r d a n ' s m o d e l (Tables 3b a n d 3c), nij's v a r y b e t w e e n 2.09 a n d 2.38, indicating that all c o m p a r t m e n t s cycle Ca. Cycling in F i n n ' s m o d e l lies b e t w e e n the o t h e r two, as s h o w n b y n , ( T a b l e s 4b a n d 4c) w h i c h v a r y b e t w e e n 1.00 a n d 3.26. T a b l e 5 c o n t a i n s REk, T~k, TST, TSTc, a n d cI for e a c h model. G r e a t e s t REk'S Occur in W a i d e ' s m o d e l ( T a b l e 5a), lowest REp's in J o r d a n ' s m o d e l a n d i n t e r m e d i a t e REk'S in F i n n ' s m o d e l J o r d a n ' s m o d e l differs f r o m the o t h e r two m o d e l s b e c a u s e m i n e r a l soil (X3) lies inside C a cycles a n d t h e r e f o r e has a n o n z e r o recycling efficiency, T~'s show p a t t e r n s similar to REk'S; m i n e r a l soil in the J o r d a n m o d e l has high cycled t h r o u g h f l o w , while m i n e r a l soil in
56
R.W. BOSSERMAN
TABLE 4 Flow a n d transitive closure matrices for the F i n n Ca model (a) Flow matrix F (kg Ca m -2 year - 1 )
X1 X2 X3 X4 X5 Input Output - .~_
X1
X2
X3
X4
X5
Input
Output
~+
0.00 40.70 0.00 0.00 6.70
0.00 0.00 0.00 0.00 42.40
0.00 0.00 0.00 0.00 21.10
52.80 3.20 0.00 0.00 3.50
0.00 0.00 0.00 62.20 0.00
0.00 0.00 0.00 0.00 2.20
0.00 0.10 0.20 0.00 13.70
0.00 0.00 21.30 0.00 0.00
0.00 0.00 5.40
0.00 0.10 1.40
0.00 0.20 0.00
0.00 0.00 2.70
2.20 13.70 0.00
0.00 0.00 0.00
0.00 0.00 0.00
(b) Transitive closure matrix N ' X1
X1 X2 X3 X4 X5
2.98 2.90 0.00 1.98 1.98
X2
1.82 2.82 0.00 1.82 1.82
X3
0.91 0.91 1.00 0.91 0.91
S 4
3.26 3.26 0.00 3.26 2.26
X5 3.26 3.26 0.00 3.26 3.26
Row
sum
11.23 12.15 1.00 10.23 9.23
(c) Transitive closure matrix N "
X1 X2 X3 X4 X5 C o l u m n sum
X1
g 2
g 3
g 4
X5
2.98 2.41 0.00 2.33 2.84 10.56
2.19 2.82 0.00 2.58 3.15 10.74
2.24 1.87 1.00 2.64 3.23 10.98
2.77 2.41 0.00 2.33 2.84 10.35
2.27 1.88 0.00 2.67 3.26 10.08
the other two models has no cycled throughflow. Jordan's model has highest TST and Waide's model lowest; Finn's model has highest TSTc and Waide's model the lowest. The cycling indices for the three models show that Waide's model has the highest cycling index (ci = 0.76). Total sensitivities which result from modifying each element gijt or gijt t for the three models are shown in Table 6; SPL'ij a n d SPL~. represent the sums of all elements in the sensitivity matrices S'(i, j) and S"(i, j), respectively. If element gi~ is changed by e, then the amount of change in all elements of N ' equals e SPL'ij; SPLPij equals the sensitivity of pathlength from model inflow tP to model outflow to changes in gi'j, while SPLij equals the sensitivity of the tt pathlength from model inflow to model outflow due to changes in g~j. Total sensitivity matrices SPL' and SPL" for Waide's model (Tables 6a, 6b) show that changes in direct flows among vegetation (X1) , litter (X2) and
SENSITIVITYOF CYCLINGMEASURESDERIVEDFROMECOLOGICALFLOWANALYSIS
57
TABLE 5 Recycling efficiencies, cycled throughflow, cycling efficiencies, and total system throughflow for forest Ca models (a) Recycling efficiencies for forest Ca models Waide model
Jordan model
Finn model
X3
0.81 0.81 0.81
0.58 0.52 0.58
X4
0.00
0.58
0.66 0.65 0.00 0.69 0.69
X1 Xz
X5 (b) Cycled throughflow, T~k (kg Ca
m -2
Waide model X1 40.4 X2 40.6 X3 47.6 X4 0.0 X5 (c) Total system throughflow,
TST" (kg
year -1) Jordan model
Finn model
36.4 26.0 59.3 34.3
31.5 25.4 0.0 41.3 60.6
Ca m
-2
year -1)
Waide model
Jordan model
Finn model
168.3
274.5
256.1
(d) Total system cycled throughflow, TST (kg Ca m
-2
year -1)
Waide model
Jordan model
Finn model
128.6
156.1
158.8
Waide model
Jordan model
Finn model
0.76
0.57
0.62
(e) Cycling index, ci
available n u t r i e n t s (X3) cause largest changes in N ' a n d N " . F l o w s to a n d f r o m m i n e r a l soil (X4) cause smaller changes in N ' a n d N " b e c a u s e the cycle that c o n n e c t s X 1, )(2, X 3 does n o t include X 4. T h e t o t a l sensitivity matrices SPL' a n d SPL" for J o r d a n ' s m o d e l ( T a b l e s 6c a n d 6d) c o n t a i n elements that are a b o u t 4 - 5 times smaller t h a n for W a i d e ' s m o d e l . P r o p o r tions of flow into soil f r o m litter (g32), to soil f r o m c a n o p y (g;1) a n d to w o o d f r o m c a n o p y (g41) m o s t c h a n g e N ' . P r o p o r t i o n s of o u t f l o w f r o m c a n o p y to soil (g43), f r o m soil to litter (g23), a n d f r o m w o o d to c a n o p y elements (g4'1) m o s t c h a n g e N " . Flows g41 a n d g23 h a v e zero values in J o r d a n ' s m o d e l ; however, the sensitivity analysis shows the a m o u n t o f c h a n g e t h a t w o u l d o c c u r if t h e y were m a d e n o n z e r o . SPL' a n d SPL" for F i n n ' s m o d e l (Tables 6e a n d 6f) c o n t a i n e l e m e n t s w h i c h are i n t e r m e d i a t e to those o f the o t h e r two models. Sensitivity m a t r i x SPL'
221.5 207.6 226.3 48.7
234.9 220.2 240.0 51.7
x2
111.1 93.6 52.3 136.1
120.2 101.3 56.6 147.2
0.0 0.0 0.0 0.0
91.5 77.1 43.0 112.0
x,
56.5 40.5 68.0 60.6
x,
x5
x~ 289.4 27O.8 252.7 265.4
X1 X2 X3 X4
60.1 54.1 46.6 67.6
x~ 55.7 50.2 43.2 62.6
x~
101.3 85.4 47.7 124.0
X1 X2 X3 X5
144.6 147.1 150.5 151.8
128.4 130.6 133.6 134.8
x~
x~
X1 X2 X3 X4
x2
x~
48.8 35.0 58.7 52.4
x~
273.9 256.3 239.1 251.1
x~
(d) SPL" for J o r d a n model
X1 X2 X3 X4
(b) SPL" for Waide model
(f) SPL" for Finn model
68.6 49.2 82.6 73.6
x2
14.7 13.7 15.0 3.2
x~
x,
60.9 43.7 73.3 65.4
x~
216.8 203.2 221.5 47.7
x3
(e) SPL' for Finn model
X1 X2 X3 X4
(c) SPL' for Jordan model
X1 X2 X3 X4
x~
(a) SPL' for Waide model
Total sensitivities for three forest Ca models
TABLE 6
x~
0.0 0.0 0.0 0.0
x3
76.2 68.6 59.1 85.6
x~
319.6 299.0 279.0 293.0
184.5 187.6 192.0 193.6
x4
52.1 46.9 40.4 58.6
x4
15.5 14.5 13.5 14.2
x4
x~ 160.8 163.5 167.2 153.5
59
SENSITIVITY OF C Y C L I N G MEASURES D E R I V E D F R O M ECOLOGICAL FLOW ANALYSIS
TABLE 7
Sensitivity values for Waide Ca model (a) Sensitivities of recycling efficiencies
i,j
1,3 t
ORE 1//~gij t ORE 2 / Ogij
0RE3/ 0gijt i
ORE 4 / Ogij
i,j
2,1
3,2
3,4
0.85
0.81
0.95
0.00
0.85
0.81
0.95
0.00
0.85
0.81
0.95
0.00
0.00
0.00
0.00
0.00
1,3
2,1
3,2
3,4
ORE1/Ogijtt 0REz / 0gijtt ORE3/Ogijt!
1.00 1.00 1.00
0.81 0.81 0.81
0.81 0.81 0.81
0.00 0.00 0.00
ORE4 /, 4(1gij,,
0.00
0.00
0.00
0.00
(b) Sensitivities of cycled throughflows
i,j
1,3
2,1
3,2
3,4
0 T~3/ 0 gu
42.35 42.57 49.95
40.58 40.79 47.86
47.44 47.69 55.96
0.00 0.00 0.00
OT~ / Ogut
0.00
0.00
0.00
0.00
l
aT~a/ 0gu ! OT~2/Ogij t
i,j it
OTcl//Ogij
¢!
Tc2//O gij t!
Tc3/ 0gu 0TeA/ Ogutt
1,3
2,1
3,2
3,4
49.95 50.21 58.92 0.00
40.37 40.58 47.61 0.00
40.43 40.64 47.69 0.00
0.00 0.00 0.00 0.00
2,1
3,2
3,4
(c) Sensitivities of cycling indices
i,j
1,3 t
OCI/Ogij i,j
0.80 1,3
tt
OCI//Ogij
0.95
0.77 2,1 0.76
0.90 3,2 0.76
0.00 3,4 0.00
shows that flows to and from mineral soil (X3) change less than those to and from other compartments. Elements of sPIY indicate that flows from mineral soil (X3) change N " less than from other compartments; flows to mineral soil (X3) are as influential as flows from the other four compartments. Recycling efficiency sensitivities for the models are shown in Tables 7a, 8a, and 9a. Only sensitivities for parameters which represent nonzero flows have been calculated. Recycling efficiencies of Waide's model change the greatest amount if proportion of inflow into nutrients from litter (g32) or if the proportion out outflow from nutrients into vegetation (g~3) are changed. In Jordan's model the greatest changes in recycling efficiencies occur if the
60
R.W. BOSSERMAN
TABLE 8
Sensitivity values for Jordan Ca model (a) Sensitivities of recycling efficiencies
i,j
1,4 t
0~1/~gij
!
Oe,Ez/~gij!
RE3/ 0 gij
RE4//~gijt i, j
0.61 0.63 0.61 0.61 1,4
tt
ORE1/~gij ~RE2/ogij "" "
Oe~3/Ogijpt tt
~RE4/~gij
2,1 0.46 0.52 0.46 0.46 2,1
3,1 0.94 0.56 0.94 0.94 3,1
3,2 0.94 1.07 0.94 0.94 3,2
4,3 0.58 0.59 0.58 0.58 4,3
0.58 0.59 0.58
0.58 0.66 0.58
0.58 0.34 0.58
0.46 0.52 0.46
1.00 1.02 1.00
0.58
0.58
0.58
0.46
1.00
(b) Sensitivities of cycled throughflows
i,j 3T~a/3gijt ~T~2/Ogi'J
1,4
2,1
3,1
3,2
0 Tc3/ 0 gij p Tc4/ 0 gij
38.64 31.38 62.90 36.43
28.90 26.04 47.04 27.24
59.30 27.89 96.54 55.91
59.30 53.44 96.54 55.91
i,j
1,4
2,1
3,1
3,2
4,3
36.43 29.58 59.30 34.34
36.43 32.83 59.30 34.34
36.43 17.13 59.30 34.34
28.90 26.04 47.04 27.24
62.90 51.08 102.40 59.30
1,4 0.31
2,1 0.10
3,1 0.78
3,2 0.64
4,3 0.21
1,4
2,1
3,1
3,2
4,3
t
tt
T~a/ 0 gij
~Tc2/agijt t ~T~3/Ogijpt 0 T~ / ~ gijt !
4,3 36.43 29.58 59.30 34.34
(c) Sensitivities of cycling indices
i,j cl/~gijt i,j c,/~g'~
0.51
0.59
0.54
0.47
1.00
proportion of inflow into soil from canopy (g31), of inflow into soil from litter (g32) or of outflow from soil into wood (g43) are changed. In Finn's model the greatest changes in recycling efficiencies occur if the proportion of inflow into available nutrients from above ground biomass (gEl), into available nutrients from forest floor (g~2), or into nutrients from below ground biomass (g~4) are changed. The proportion of outflow from nutrients to below ground biomass (g45) causes large changes in the recycling efficiencies. Sensitivities of cycled throughflows (Table 7b) for Waide's model show that amount of Ca cycled through nutrients (X3) changes the most due to
SENSITIVITY
OF
CYCLING
MEASURES
DERIVED
FROM
ECOLOGICAL
FLOW
61
ANALYSIS
TABLE 9
Sensitivity values for Finn Ca model (a) Sensiti~tiesofre~clingefficiencies
i,j OP,E1/Og~jt OREz/Ogijt ORE3/Ogijt OgE,~/Og,j O~s/~g~'j i,j Ol~x/Ogijt p tp O~2/Og~j tt Oe.E3/Og~j O~4/Og~j OPd~5/~g0
1,5
!
4,1
4,2
4,3
4,5
5,4
0.61 0.65
1.10 0.75
1.07 1.16
0.00 0.00
0.73 0.75
0.73 0.75
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.61 0.34
0.56 0.56
1.00 1.00
1.00 1.00
0.00 0.00
1.00 0.00
0.69 0.69
1,5 0.78 0.78
Pl
2,1
0.66 0.66
2,1
4,1
4,2
4,3
4,5
5,4
0.74 0.78
0.76 0.52
0.62 0.67
0.00 0.00
0.60 0.61
0.89 0.91
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.71 0.71
0.67 0.67
0.70 0.70
0.58 0.58
0.00 0.00
0.82 0.82
0.85 0.85
(b) Sensitivities of cycled throughflows
i,j
1,5
2,1
4,1
4,2
4,3
4,5
5,4
OT~l/3g~jt t OT~2/Og~j
35.05 29.16 0.00 46.01 21.06 1,5
32.30 28.33 0.00 42.40 34.75 2,1
57.82 32.73 0.00 75.90 62.20 4,1
56.40 50.72 0.00 75.90 62.20 4,2
0.00 0.00 0.00 0.00 0.00 4,3
38.38 32.73 0.00 75.90 0.00 4,5
38.38 32.73 0.00 52.60 43.11 5,4
41.29 34.35
38.85 34.07
40.22 22.77
32.62 29.33
0.00 0.00
31.45 26.82
46.84 39.94
t
0 T~3/ 0 gis
3T~/Oggit OTcs/~g,'./ i,j pr 3T~I/O&j tt 3T~2/agij tt 0 Tc3/ 0gij ~T~4/Ogsj tt OTcs/~g~j II
0.00
0.00
0.00
0.00
0.00
0.00
0.00
54.20 44.42
41.00 41.79
52.80 43.27
43.90 35.98
0.00 0.00
62.20 50.97
64.19 52.60
4,1
4,2
4,3
4,5
5,4
(c) Sensitivities of cycling indices
i,j acI/OgijI i,j 0 cI/~g~j
1,5 0.51 1,5
¢t
0.39
!
2,1 0.53 2,1 0.38
1I
0.89 4,1 0.33
0.95 4,2 0.32
0.00 4,3 0.00
0.57 4,5 0.39
0.65 5,4 0.46
c h a n g e s in gij a n d gij- F o r Jordan's m o d e l , the a m o u n t o f C a c y c l e d ? P! t h r o u g h soil (X3) c h a n g e s the m o s t d u e to c h a n g e s in gij a n d gij. L i k e w i s e , in the F i n n m o d e l , the a m o u n t o f C a c y c l e d t h r o u g h available nutrients ( X 5) ! P! c h a n g e s the m o s t d u e to c h a n g e s in g~j a n d g~j. T a b l e s 7c, 8c, a n d 9c s h o w sensitivities o f c y c l i n g i n d i c e s ca'. F o r W a i d e ' s m o d e l c h a n g e s in the p r o p o r t i o n o f o u t f l o w f r o m nutrients i n t o v e g e t a t i o n (g~3) a n d the p r o p o r t i o n o f i n f l o w to nutrients f r o m litter (g32) w o u l d c a u s e
62
R.W. BOSSERMAN
the greatest changes in CL The normalized flows can be converted to actual flows by multiplying gij by fq. If g~ increases by 0.001, then cz would increase by 0.00095; an increase of 0.001 in g;3 corresponds to an increase of 0.048 kg Ca ha-1 year-1. For Jordan's model the proportion of flow into soil from canopy (g;3) and the proportion of flow out of soil into wood (g4'3) cause the greatest change in the cycling index. For Finn's model, the proportion of flow intro nutrients from forest floor (g~2), the proportion of flow into nutrients from above ground biomass would cause the greatest change in the cycling indices if changed (g51). CONCLUSIONS Flow analyses provide tools for examining influence as it is propagated through an ecosystem network. In a complex ecosystem indirect effects may exert a large influence through long causal paths and cycles; small causes can be magnified by the system network. Patten et al. (1976) and other have argued that this extended environment of an ecosystem is at the basis of many ecological phenomena. The sensitivity analyses described here further refines the usefulness of inflow-outflow analyses for studying ecosystems. Causes can be traced through the ecosystem network, and the degree of influence can be ascertained. Sensitivity to change is greatest in networks which are highly connected and have large amounts of feedback. Such sensitivity arises as a system level property of complex systems; studies of inflow-outflow sensitivity will provide meaningful insight into the behavior of ecosystems. ACKNOWLEDGEMENTS I would like to thank Thomas B. Burns and two anonymous reviewers for their comments on this manuscript. APPENDIX Measurement of condition of invertible matrix B BB -1
=
I
(OB-1B/~bij) = (~I/abij) = 0 = (~B-1/~bij)B -1 + B(aB-a/~bij) B-l(OB/Obij)n -1 + OB-1/~bij = 0
B-X/ b,j
=
SENSITIVITY OF CYCLING
MEASURES
DERIVED
FROM ECOLOGICAL
FLOW ANALYSIS
63
Sensitivity of recycling efficiency
OREk/Ogij =
O[(nkk --
1)/nkk]Ogij
= 3[1 - (1/nl, k)]/O&j
= 3 (1/nkk)/Ogij = - (o,, ~#og,,j)/,G
Sensitivity of cycled throughflow O T ~ k / O g i j - - OTc~/Ogiz = 0
a(~
T~)/Og,j - O ( ~ k T~)/Og,z = 0
Tk OREk/Ogij "+ REk OTk/Ogij = Tk 3REk/Ogiz -- REk ÙTk/Ogi~ Tk ( OREk/Ogij -- OREk/Ogiz ) -k- _nE~( OTk/Og, j Tk(OREk/Ogij-- OREk/Ogi~.) + Tk(aREk/Ogij
-
0
- O T k / O giz ) =
0
(because the throughflow is constant)
O)= TkS(i , J)kk/nkk " 2
Sensitivity of cycling indices
CI/~ gi j = ~ ( TSTc/ TST ) /O gi j = (0 TSTc/Ogij)TST =(k=~OT~JOgiJ) TST = (k~=l(OREk Tk)/Ogij) TST
=
, S)kk/nkk) /TST k=l
REFERENCES Astor, P.H., Patten, B.C. and Estberg, G.N., 1976. The sensitivity substructure of ecosystems. In: B.C. Patten (Editor), Systems Analysis and Simulation in Ecology, 4. Academic Press, New York, pp. 389-429. Bosserman, R.W., 1981. Sensitivity techniques for inflow-outflow flow analyses. In: W.J. Mitsch, R.W. Bosserman and J.M. Klopatek (Editors), Energy and Ecological Modelling. Developments in Environmental Modelling, 1. Elsevier, Amsterdam, pp. 653-660.
64
R.W. BOSSERMAN
Bosserman, R.W., 1983. Flow analysis sensitivities for models of energy or material flow. Bull. Math. Bio., 45: 807-826. Bullard, C.W. and Sebald, A.V., 1977. Effects of parameter uncertainty and technological change on inflow-outflow models. Rev. Econ. Stat., 59" 75-81. Christ, C.F., 1955. A Review of Inflow-Outflow Analysis in Inflow-Outflow Analysis: An Appraisal. Studies in Wealth and Income, 18. National Bureau of Economic Research, Princeton University Press, Princeton, NJ. Finn, J.T., 1976. Measures of ecosystem structure and function derived from analysis of flows. J. Theor. Biol., 56: 363-380. Finn, J.T., 1977. Flow analysis: a method for tracing flows through ecosystem models. Ph.D. dissertation. University of Georgia, Athens, GA, 568 pp. Finn, J.T., 1980. Flow analysis of models of the Hubbard Brook ecosystem. Ecology, 61: 562-571. Hannon, B., 1973. The structure of ecosystems. J. Theor. Biol., 41: 535-546. Jordan, C.F., Kline, J.R. and Sasscer, D.S., 1972. Relative stability of mineral cycles in forest ecosystems. Am. Nat., 106: 237-253. Likens, G.E., Bormann, F.H., Pierce, R.S., Eaton, J.S. and Johnson, N.M., 1977. Biogeochemistry of a Forested Ecosystem. Springer, New York, 146 pp. Patten, B.C., Bosserman, R.W., Finn, U.T. and Cale, W.G., 1976. Propagation of cause in ecosystems. In: B.C. Patten (Editor), Systems Analysis and Simulation in Ecology, 4. Academic Press, New York, pp. 457-579. Sebald, A., 1974. An analysis of the sensitivity large scale input-output models to parametric uncertainty. Doc. 122, University of Illinois, Urbana, IL, 23 pp. Sherman, J. and Morrison, W., 1950. Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat., 21: 124-127. Waide, J.B. and Webster, J.R., 1976. Engineering systems analysis: applicability to ecosystems. In: B.C. Patten (Editor), Systems Analysis and Simulation in Ecology, 4. Academic Press, New York, pp. 329-371. Waide, J.B., Krebs, J.E., Clarkson, S.P. and Setzl~r, E.M., 1974. A linear systems analysis o f the calcium cycle in a forested watershed ecosystem. Progr. Theor. Biol., 3: 261-345.
45
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
ROBERT W. BOSSERMAN Department of Biology, University of Louisville, Louisville, KY 40292 (U. S.A.) (Accepted 3 April 1989)
ABSTRACT Bosserman, R.W., 1989. Sensitivity of cycling measures derived from ecological flow analysis. Ecol. Modelling, 48: 45-64. Ecologists have developed and applied flow analyses to examine indirect interactions in ecosystem models. These methods generate measures of indirect influence such as recycling efficiencies, cycled throughflows, and cycling indices. Results of such analyses provide insight into the nature of interactions in ecosystems. This paper describes a sensitivity tectmique for examining flow structures of ecosystem models. The sensitivity technique relies on a method for determining the 'condition' of inverted matrices. Resulting sensitivity values describe effects of perturbations and errors on indirect flows, recycling efficiencies, cycling indices, cycled throughflows and other flow measures. I have applied the methods to three models of Ca flow for the Hubbard Brook forest.
INTRODUCTION M a t t e r a n d energy flows interconnect ecological c o m p o n e n t s in ecosystems, a n d give rise to n e t w o r k p h e n o m e n a i m p o r t a n t to ecology. Direct a n d indirect influences p r o p a g a t e t h r o u g h ecosystems a n d affect n u t r i e n t cycling, coevolution, trophic relationships, diffuse competition, a n d d y n a m i c behavior. Causes a n d effects which are t r a n s m i t t e d t h r o u g h an ecosystem n e t w o r k can generate n o n o b v i o u s consequences which c a n n o t be predicted b y analysis of c o m p o n e n t parts. Ecologists i n t r o d u c e d flow analyses to q u a n t i f y ecosystem characteristics ( H a n n o n , 1973; Patten, et al., 1976; Finn, 1977, 1980) such as indirect flow structure, cycling indices, a n d recycling efficiencies. These measures represent system-level properties that d e p e n d on the p a t t e r n of interactions a m o n g c o m p o n e n t s . In this paper, I present a sensitivity analysis for assessing measures derived f r o m flow analyses. 0304-3800/89/$03.50
© 1989 Elsevier Science Publishers B.V.
46
R.W. BOSSERMAN
Sensitivity techniques quantify effects of changes in model variables. Changes result from several sources: environmental variation, misconception, measurement error, and experimental manipulation. In complex models, interactions among model parameters often disguise consequences of variable change. Researchers have applied sensitivity techniques to different aspects of ecological modeling (Astor et al., 1976; Waide and Webster, 1976), but only recently have they been applied to flow analyses. In economics authors have studied effects of perturbations on input-output systems (Sherman and Morrison, 1950; Christ, 1955; Sebald, 1974; Bullard and Sebald, 1977). In ecology I derived sensitivities for flow analyses of ecosystem models (Bosserman, 1981, 1983). Here, I have extended previously published methods to include cycling indices, cycled throughflow, recycling efficiencies and pathlength as defined by Finn (1977, 1980). METHODS
Relationships for flow analyses Several relationships form the basis of the flow analyses used in ecology. They apply to compartmental models of energy/matter storage and dynamics. Flow analysis begins with the formation of flow" matrix, F; f~j depict amounts of matter or energy flow from compartments j to i. Column z of F contains all of the inflows to compartments from the environment, while row y contains all outflows from compffrtments to the environment. Summing the rows of F generates throughflow vector e', and summing the columns of F generates throughflow vector e". Element e~' of e' represents the amount of flow into compartment i; %l ! of e " represents the amount of flow out of compartment j. In a steady state model inflows equal outflows and e' equals e "T, where T represents the transpose operation. For nonsteady-state models, addition of state-variable derivatives (:~+, ~_) allows a dynamic flow analysis (Patten et al., 1976; Finn, 1980). Flow analysis may also include storage as a vector x; xj equals the amount of material in compartment j. Ecological flow analysis (Patten et al., 1976) either traces flows from compartments to model outflows or traces from model inflows to compartments. I will designate matrices by ' for the first (forward-looking) case and will designate matrices by " for the second (backward-looking) case. When the mathematical derivations for the two cases are the same, I will not use ' or
rp
Normalized flow matrices, G' and G", result from the next step in flow analysis. To form G' from F, each f~j is divided by the corresponding row
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
47
sum, e~'; to form G " from F, each f~j is divided by the corresponding column sum, e)', i.e. n !
n
I
g,j = L : / e i = f i J E f,j
and
tp
cp
gij = fij/ej = f i J E fij)
i=1
j=l
The following equations relate normalized flow matrices to outflows and inflows (Patten et al., 1976; Finn, 1977, 1980): e' = e ' G ' + y + :~+
(1)
e " = G " e " + ~ - :~_
(2)
where e', y, e", and z are as defined above, :~ is a vector of positive time derivatives of storages, and :f_ is a vector of negative time derivatives of storages. After rearranging the above equations: e'=(y+
:~+)(I - G ' ) -1
e " = (I -- G " ) - a ( z
- k_)
(3) (4)
H a n n o n (1973) called matrix ( I - G ' ) - a a "structure matrix", while Patten et al. (1976) called (I - G ' ) - 1 and (I - G ") - a ,, transitive closure matrices". (I - G , )~j -1 shows how m u c h flow from c o m p a r t m e n t j to i is required to generate a unit of flow to model outfows. (I - G " ' ~,,j- 1 shows how m u c h flow from c o m p a r t m e n t j to i is generated by a unit of flow from model inflows. In the discussion below, N ' and N " designate the transitive closure matrices ( I - G ' ) -1 and ( I - G " ) -1, respectively. Finn (1976, 1977, 1980) derived cycling indices, recycling efficiencies, cycled throughflow and other measures from N ' and N " (described below).
Condition of matrix The 'condition' of a matrix B refers to the a m o u n t of change that occurs in B -1 when element bij is varied. Condition can be expressed as 0 B - a / 0 b i j (derivations which are shown in the Appendix):
~)B-1/~)bij = - [ n - l ( O B / 0 b i j ) B -1]
(5)
Matrix OB/Obij contains all zeros except for the /jth element which is 1.0; therefore, sensitivity matrix, 0B-1/Sb~i, results from multiplying the ith column of B - 1 by the j t h row of B -1. (0B-1/0b~j)kl shows how m u c h (B-1)kl changes when b~j is changed. If bij of B is increased by an a m o u n t e to form B*, then B * - I equals B -1 +e(OB-1/3bij). Table 1 demonstrates the sensitivity analysis for a hypothetical matrix. Tables l a and l b show matrices B and B -1 and Tables l c and l d show matrices B* and B *-a. Matrix B* (Table lc) differs from B in that b13 has been increased by 0.0001 (i.e., b~'3 - b13 = 0.0001). Postmulti-
1 2 3 4
-0.2263 -0.2263 -0.0453 0.0000
1.0000 -1.0000 0.0000 0.0000
(c) B * 1 2 3 4
(e) ~ B - 1 / ~ b ] 3
1.0000 -1.0000 0.0000 0.0000
(a) B 1 2 3 4
1
-0.2263 -0.2263 -0.0453 0.0000
0.0000 1.0000 -0.2000 0.0000
0.0000 1.0000 -0.2000 0.0000
2
-1.1317 - 1.1317 -0.2263 0.0000
- 0.2999 0.0000 1.0000 0.0000
-0.30000 0.0000 1.0000 0.0000
3 (b) B -1 1 2 3 4
-1.0638 -1.0638 -0.2128 0.0000
(~ B - 1 - B * - I 1 2 3 4
(d) B* - l -0.7000 1 0.0000 , 2 -0.8000 3 1.0000 4
-0.70000 0.0000 -0.8000 1.0000
4
2.263X 10 - s 2.263X10 -5 4.53 X 10 - 6 0.00
1.0638 1.0638 0.2128 0.0000
1.0638 1.0638 0.2128 0.0000
1
2.263X10 - s 2 . 2 6 3 x 10 -3 4.53 x 1 0 -6 0.00
0.0638 1.0638 0.2128 0.0000
0.0638 1.0638 0.2128 0.0000
2
0.9999 0.9999 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000
4
1 . 1 3 1 7 x 1 0 4 1.0638 x 1 0 - 4 1.1317 x 10 - 4 1.0638 X 10 - 4 2.263 x 10 - s 2.128 x l 0 s 0.00 0.00
0.3190 0.3190 1.0638 0.0000
0.3191 0.3191 1.0638 0.0000
3
Demonstration of sensitivity analysis with arbitrary, invertible matrix in which p a r a m e t e r b13 is increased by 0.0001
TABLE 1
oo
SENSITIVITY OF CYCLING
MEASURES
DERIVED
FROM
ECOLOGICAL
FLOW ANALYSIS
49
plying column 1 of B-1 by row 3 of B-1 produces 0B-1/Obl3 (Table le). The difference between B -1 and B * - 1 (Table If) equals - 0 . 0 0 0 1 × c)B- 1/()b13 .
Sensitivity matrices for flow analyses 'Condition' analyses apply to the relationships between the normalized matrices G and the transitive closure matrices N = (I - G)-1. In the following discussion, ' and " analyses parallel one another. Sensitivity matrices S(i, j ) = 0 N / a ( N - 1 ) i j = 0 N / 0 ( I - G)i j can be determined. Elements of ( I - G) take nonpositive values except along the main diagonal. A change in any element of G will cause an opposite change in ( I - G), so S(i, j ) = 0 N / 0 ( I - G)~j = -ON/Ogij. Changes in the negative elements of ( I - G ' ) and (I - G ' ) cancel the negative sign that occurs in equation (5). Elements of G ' or G " indicate the fraction of flow to come from or go to each part of the model. Each row or column in G ' or G " must sum 1.0, i.e. t/
Y'. gi'j= 1.0 j=l or n it
E gij = 1.0
i=1
The fact that the row sums or column sums are constrained to equal 1.0 ! means that a single element cannot be changed by itself. If one element g~j I! or &j is changed then another element (or several elements) in the same row (for G ' ) or column (for G " ) must be changed to maintain the constraint. The most reasonable elements to change are those that represent inflow z or ! ,,¢ outflow y of the model (i.e., either &z or gyj). Sensitivity can therefore be calculated as S'(i, j ) - S'(i, z) or S ' ( i , j ) - S ' ( y , j ) . S ' ( i , z) contains all zeros, except for the column that corresponds to inflow z, and S " ( y , j ) contains all zeros except for the row that corresponds to outflow y. Other combinations of elements in the row or column can also be changed, as long as the row sum in G ' or the column sum in G " equals 1.0. For very small changes in elements, this constraint m a y not be important. M a t r i c e s SPL' a n d SPL" can be formed, where: n
s.L;j= E s'(i, J)k, k,l=l
and n
SPLi'2 = Y'~ S " ( i, j ) kt k,l=l
50
R.W. BOSSERMAN
Elements of SPL' and Sl'L" show the total change in N ' and N " due to ? changes in gij and g~'j.
Relationships between sensitivity analysis and power series The relationship between the power series of G and transitive closure matrix N demonstrates the mechanism of the sensitivity analysis. Patten et al. (1976) used the equality: oo
E G i= ( l - G) -1 i=1
to trace flow through paths of increasingly longer lengths. G 2 shows the proportion of material that has traveled along two-length paths, G 3 the proportion of material that has traveled along three-length paths, and in general, G" the proportion along n-length paths. The product series can also be used to trace the effects of parameter modification along the multilength paths in a model. If g~ij and gi~ are increased by e and - e , respectively, to form (], then ~ i = 1 G ' = ( I - G ) -a. The difference between ~o¢ i = l G~ and Eoo (], traces the effect of modification through the model network. G - (~ i=1 shows the difference between one-length paths, G 2 - (~2 the difference between modified and unmodified two-length paths, etc. The network structure of the model governs the propagation of the modified flows. The series E~i = 1 (G ~ - (~i) converges to the sensitivity matrix S(i, j ) - S(i, z).
Sensitivity of flow pathlengths The sums of the rows and the columns of the transitive closure matrices N ' and N " represent the average pathlengths that material or energy have traveled to the outflow or from the inflow (Finn, 1979, 1980). YPL can be defined as a vector of pathlengths from compartments to outflows where elements Y P L k = ~nj=l nkj' is the average pathlength from compartment k to the model outflows; Zl'L is a vector of pathlengths from the inflow to compartments, where element zPL k = ~"~=1 nik "' is the average pathlength from the model inflow to compartment k. The total sum of all elements in the transitive closure matrices is the average pathlength from inflow to outflow, YPL, or to outflow from inflow, ZPL. Each row sum of S'(i, j) represents the change that occurs in the average pathlength from compartment k to the model outflow when gi'j changes; i.e. OYeLI,/Ogi'j = ~t"=l S'(i, J)kt. Likewise, each column sum of S"(i, j ) represents the amount of change that occurs in the average pathlength from the It model inflow to compartment k when g~j changes; i.e. O Z P L k / O g i j It = Y~7<=l Sit( i, J)kl" SYPLk can be defined for compartment k to be the matrix
SENSITIVITY
OF CYCLING
MEASURES
DERIVED
FROM
ECOLOGICAL
FLOW
ANALYSIS
51
of pathlength sensitivities where each element SYPLij is 0 YPLk/Ogi'j. SZPL k can be defined to be the matrix of pathlength sensitivities where each tf element SZPL~j is 3 ZPLk/3g~j.
Sensitivity of recycling efficiencies Finn defined the recycling efficiency of compartment k, REk, as: p
?
t?
pt
REk= [nkk-- 1]/nkk = [nkk-- 1]/nkk = 1 -- (1/nkk)
(6)
Recycling efficiencies reflect the degree to which material which has passed through a compartment once will return and pass through again. The sensitivity of REk with respect to a change in an element of G' or G " (derived in Appendix) is:
3 RF,k/Og,j= --(Onkk/Og, j)/n2kk =
-
S(i, j ) / n 2 k
(7)
The sensitivity of the recycling index for compartment k when the sensitivity due to a change in the inflow is removed is:
[s(i,
s(i,
(8)
Sensitivity of cycled throughflow The amount of flow that passes through a compartment, Tk, can be partitioned into a cycled part, T~, and a noncycled part (T k - T~k); Tc~ can be determined by the relationship:
T~, = REk(Tk)
(9)
Sensitivity of the cycled throughflow for compartment k with respect to changes in gij can be calculated as follows:
OTck/Ogij
-
OTck/Ogiz = Tk(OREk/Ogij ) = Tk
S(i
,
"
z
(lO)
Tk and Tck for each compartment can be added to get the total system
throughflow (TST) and total system cycled throughflow (rsrc) through the entire model; i.e. TST = E'k=l Tk, and TSTo = Y~7,=aT~. Sensitivities of the total cycled throughflow, TST¢, equal the sums of sensitivities of compartmental cycled throughflows: n
O rSTc/Og,j = Y'~ OTc~/Og,y k=l
(11)
52
R.W. B O S S E R M A N
Sensitivity of cycling index Finn (1976, 1980) defined a cycling index as follows: n
CI= rSrc/rSr=
n
ELk/ E rk k=l
(12)
k=l
r s r will be assumed to be constant in these derivations because a change in gij will be balanced by a change in g/z or giy. The sensitivity of ci with respect to changes in g,j (derivation in Appendix) is:
OCI/OgiJ=O(TSTc/TST)/OgiJ= [ ~ ~
(13)
Therefore, after substituting from equation (10) the sensitivity of the cycling index cI with respect to the element gij is:
14, APPLICATION TO THE FOREST CALCIUM MODELS The flow analysis sensitivities strongly depend on both the model structure and the values of flows among compartments. Different researchers often will generate different models of the same system. Sensitivity analyses can provide perspectives into the differences between various models and can generate questions about the differences between models. In this paper, the sensitivity analysis has been applied to three models of Ca flow through the Hubbard Brook experimental forest; each model was developed by a different group of authors. Finn (1976, 1980) did the original flow analyses for this model. Figure 1 shows the diagrams for these model.
Waide et al. 's model of Hubbard Brook temperate forest Waide et al. (1974) constructed a model of Ca flow through a temperate hardwood forest at the Hubbard Brook experimental forest. The model contains vegetation (X1), litter (X2), and available nutrients (X3), soil and rock minerals (X4). A single cycle contains vegetation, litter and available nutrients, while no cycle contains mineral soil. Flow matrix F (Table 2a), which consists of flows of Ca (kg ha -a year -1) to compartment i from compartment j, shows that inflows and outflows are balanced. Tables 2b and 2c show transitive closure matrices N ' and N " . The fifth row and column depict inflows from the environment, while the sixth row and
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
o.
Waide
b. Jordan
53
Model
Model
c. Finn Model
Fig. 1. Three compartmental models of Ca flow through Hubbard Brook experimental forest: (a) model developed by Waide et al. (1974), (b) model developed by Jordan et al. (1972), (c) model developed by Finn (1980). All models were calibrated by Finn (1976, 1980).
column depict outflows from the model. N ' and N " also contain average pathlengths from compartments to outflows and to compartments from inflows, respectively (row sums and column sums of N ' and N " ) .
Jordan mineral model for temperate and tropical forests Jordan et al. (1972) developed a general mineral model for temperate and tropical forests (Fig. lb), which Finn (1976) calibrated with Ca data from Hubbard Brook. This model contains canopy (X1), litter (X2), soil (X3), and Wood (X4). Cycles interconnect all four of the compartments. Flow matrix F (Table 3a) shows that the model is at steady state where inflows balance outflows. Tables 3b and 3c) also contain transitive closure matrices N ' and N PP.
54
R.W. BOSSERMAN
TABLE 2 Flow and transitive closure matrices for Waide Ca model (a) Flow matrix F (kg Ca m-1 year-1) X1 X2 X3 X4 Inflow Outflow
X1
X2
X3
0.00 50.03 0.00 0.00
0.00 0.00 50.21 0.00
47.69 0.00 0.00 0.00
0.00 0.00
0.00 0.08
0.00 11.32
X4
Inflow
Outflow
0.00 0.00 8.80 0.00
2.34 0.26 0.00 9.00
0.00 0.00 0.00 0.00
0.00 0.20
0.00 0.00
0.00 0.00
(b) Transitive closure matrix N ' S1
X1 X2 X3 X4
X2
5.18 5.15 4.38 0.00
4.20 5.18 4.41 0.00
S3
4.94 4.91 5.18 0.00
X4
0.74 0.73 0.77 1.00
Row
sum
16.16 17.07 15.76 2.00
(c) Transitive closure matrix N " Xl
X1 X2 X3 X4 Column sum
X2
S3
X4
5.18 5.18 5.17 0.00
4.18 5.18 5.17 0.00
4.19 4.19 5.18 0.00
4.09 4.09 5.06 1.00
16.53
15.53
14.56
15.24
Finn's model of Hubbard Brook forest F i n n (1980) d e v e l o p e d a t h i r d m o d e l o f C a f l o w t h r o u g h a H u b b a r d B r o o k forest w i t h d a t a f r o m L i k e n s et al. (1977). T h i s m o d e l (Fig. l c ) consists of a b o v e g r o u n d b i o m a s s (X1), forest f l o o r ( X 2 ) , m i n e r a l soil (X3) , b e l o w g r o u n d b i o m a s s ()(4), a n d a v a i l a b l e n u t r i e n t p o o l ()(5); it differs f r o m W a i d e ' s a n d J o r d a n ' s m o d e l s b e c a u s e it is n o t in s t e a d y state; t h e r e f o r e , d e r i v a t i v e s of s t a t e v a r i a b l e s are n o n z e r o . Cycles c o n n e c t a b o v e g r o u n d b i o m a s s , b e l o w g r o u n d b i o m a s s , forest floor, a n d a v a i l a b l e n u t r i e n t s , w h i l e n o cycles include m i n e r a l soil. F l o w m a t r i x F ( T a b l e 4a) i n c l u d e s t i m e d e r i v a t i v e s of s t a t e v a r i a b l e s (:~+ = 0 if d x k / d t <_O, .it+ = d x ~ / d t if d x k / d t > 0; :~_ = 0 if d x k / d t >__O, At_ = d x ~ / d t if d x k / d t < 0) ( F i n n , 1980). F l o w a n d sensitivity a n a l y s e s p r o c e e d as in the s t e a d y s t a t e case, e x c e p t t h a t r o w a n d c o l u m n s u m s u s e d to f o r m G ' a n d G " i n c l u d e :~+ a n d : ~ .
55
SENSITIVITY OF CYCLING MEASURES DERIVED FROM ECOLOGICAL FLOW ANALYSIS
TABLE 3 Flow and transitive closure matrices for Jordan Ca model (a) Flow matrix F (kg Ca m -2 year -1) X1 X2 X3 X4 Inflow Outflow
)(1
X2
X3
Xa
Inflow
Outflow
0.00 49.90 13.00 0.00
0.00 0.00 49.90 0.00
0.00 0.00 0.00 59.30
59.30 0.00 0.00 0.00
3.60 0.00 39.50 0.00
0.00 0.00 0.00 0.00
0.00 0.00
0.00 0.00
0.00 43.10
0.00 0.00
0.00 0.00
0.00 0.00
(b) Transitive closure matrix N' X1
X1 X2 X3 X4
X2
2.38 2.38 1.46 1.46
1.09 2.09 1.16 1.16
X3
2.24 2.24 2.38 2.38
X4
2.24 1.38 2.38
Row
sum
7.95 8.95 6.48 7.48
(c) Transitive closure matrix N "
s~
x2
x3
x4
X1 X2 X3 X4
2.38 1.89 2.38 1.38
1.38 2.09 2.38 1.38
1.38 1.09 2.38 1.38
2.38 1.89 2.38 2.38
Column sum
8.03
7.23
6.23
9.03
RESULTS
T r a n s i t i v e closure matrices for W a i d e ' s m o d e l (Tables 2b a n d 2c) s h o w that vegetation (X1), litter (X2), a n d available n u t r i e n t s (X3) o c c u r in C a cycles while soil (X4) does n o t (n n = / ' / 2 2 = / / 3 3 = 5.18, /144 = 1.00). F o r J o r d a n ' s m o d e l (Tables 3b a n d 3c), nij's v a r y b e t w e e n 2.09 a n d 2.38, indicating that all c o m p a r t m e n t s cycle Ca. Cycling in F i n n ' s m o d e l lies b e t w e e n the o t h e r two, as s h o w n b y n , ( T a b l e s 4b a n d 4c) w h i c h v a r y b e t w e e n 1.00 a n d 3.26. T a b l e 5 c o n t a i n s REk, T~k, TST, TSTc, a n d cI for e a c h model. G r e a t e s t REk'S Occur in W a i d e ' s m o d e l ( T a b l e 5a), lowest REp's in J o r d a n ' s m o d e l a n d i n t e r m e d i a t e REk'S in F i n n ' s m o d e l J o r d a n ' s m o d e l differs f r o m the o t h e r two m o d e l s b e c a u s e m i n e r a l soil (X3) lies inside C a cycles a n d t h e r e f o r e has a n o n z e r o recycling efficiency, T~'s show p a t t e r n s similar to REk'S; m i n e r a l soil in the J o r d a n m o d e l has high cycled t h r o u g h f l o w , while m i n e r a l soil in
56
R.W. BOSSERMAN
TABLE 4 Flow a n d transitive closure matrices for the F i n n Ca model (a) Flow matrix F (kg Ca m -2 year - 1 )
X1 X2 X3 X4 X5 Input Output - .~_
X1
X2
X3
X4
X5
Input
Output
~+
0.00 40.70 0.00 0.00 6.70
0.00 0.00 0.00 0.00 42.40
0.00 0.00 0.00 0.00 21.10
52.80 3.20 0.00 0.00 3.50
0.00 0.00 0.00 62.20 0.00
0.00 0.00 0.00 0.00 2.20
0.00 0.10 0.20 0.00 13.70
0.00 0.00 21.30 0.00 0.00
0.00 0.00 5.40
0.00 0.10 1.40
0.00 0.20 0.00
0.00 0.00 2.70
2.20 13.70 0.00
0.00 0.00 0.00
0.00 0.00 0.00
(b) Transitive closure matrix N ' X1
X1 X2 X3 X4 X5
2.98 2.90 0.00 1.98 1.98
X2
1.82 2.82 0.00 1.82 1.82
X3
0.91 0.91 1.00 0.91 0.91
S 4
3.26 3.26 0.00 3.26 2.26
X5 3.26 3.26 0.00 3.26 3.26
Row
sum
11.23 12.15 1.00 10.23 9.23
(c) Transitive closure matrix N "
X1 X2 X3 X4 X5 C o l u m n sum
X1
g 2
g 3
g 4
X5
2.98 2.41 0.00 2.33 2.84 10.56
2.19 2.82 0.00 2.58 3.15 10.74
2.24 1.87 1.00 2.64 3.23 10.98
2.77 2.41 0.00 2.33 2.84 10.35
2.27 1.88 0.00 2.67 3.26 10.08
the other two models has no cycled throughflow. Jordan's model has highest TST and Waide's model lowest; Finn's model has highest TSTc and Waide's model the lowest. The cycling indices for the three models show that Waide's model has the highest cycling index (ci = 0.76). Total sensitivities which result from modifying each element gijt or gijt t for the three models are shown in Table 6; SPL'ij a n d SPL~. represent the sums of all elements in the sensitivity matrices S'(i, j) and S"(i, j), respectively. If element gi~ is changed by e, then the amount of change in all elements of N ' equals e SPL'ij; SPLPij equals the sensitivity of pathlength from model inflow tP to model outflow to changes in gi'j, while SPLij equals the sensitivity of the tt pathlength from model inflow to model outflow due to changes in g~j. Total sensitivity matrices SPL' and SPL" for Waide's model (Tables 6a, 6b) show that changes in direct flows among vegetation (X1) , litter (X2) and
SENSITIVITYOF CYCLINGMEASURESDERIVEDFROMECOLOGICALFLOWANALYSIS
57
TABLE 5 Recycling efficiencies, cycled throughflow, cycling efficiencies, and total system throughflow for forest Ca models (a) Recycling efficiencies for forest Ca models Waide model
Jordan model
Finn model
X3
0.81 0.81 0.81
0.58 0.52 0.58
X4
0.00
0.58
0.66 0.65 0.00 0.69 0.69
X1 Xz
X5 (b) Cycled throughflow, T~k (kg Ca
m -2
Waide model X1 40.4 X2 40.6 X3 47.6 X4 0.0 X5 (c) Total system throughflow,
TST" (kg
year -1) Jordan model
Finn model
36.4 26.0 59.3 34.3
31.5 25.4 0.0 41.3 60.6
Ca m
-2
year -1)
Waide model
Jordan model
Finn model
168.3
274.5
256.1
(d) Total system cycled throughflow, TST (kg Ca m
-2
year -1)
Waide model
Jordan model
Finn model
128.6
156.1
158.8
Waide model
Jordan model
Finn model
0.76
0.57
0.62
(e) Cycling index, ci
available n u t r i e n t s (X3) cause largest changes in N ' a n d N " . F l o w s to a n d f r o m m i n e r a l soil (X4) cause smaller changes in N ' a n d N " b e c a u s e the cycle that c o n n e c t s X 1, )(2, X 3 does n o t include X 4. T h e t o t a l sensitivity matrices SPL' a n d SPL" for J o r d a n ' s m o d e l ( T a b l e s 6c a n d 6d) c o n t a i n elements that are a b o u t 4 - 5 times smaller t h a n for W a i d e ' s m o d e l . P r o p o r tions of flow into soil f r o m litter (g32), to soil f r o m c a n o p y (g;1) a n d to w o o d f r o m c a n o p y (g41) m o s t c h a n g e N ' . P r o p o r t i o n s of o u t f l o w f r o m c a n o p y to soil (g43), f r o m soil to litter (g23), a n d f r o m w o o d to c a n o p y elements (g4'1) m o s t c h a n g e N " . Flows g41 a n d g23 h a v e zero values in J o r d a n ' s m o d e l ; however, the sensitivity analysis shows the a m o u n t o f c h a n g e t h a t w o u l d o c c u r if t h e y were m a d e n o n z e r o . SPL' a n d SPL" for F i n n ' s m o d e l (Tables 6e a n d 6f) c o n t a i n e l e m e n t s w h i c h are i n t e r m e d i a t e to those o f the o t h e r two models. Sensitivity m a t r i x SPL'
221.5 207.6 226.3 48.7
234.9 220.2 240.0 51.7
x2
111.1 93.6 52.3 136.1
120.2 101.3 56.6 147.2
0.0 0.0 0.0 0.0
91.5 77.1 43.0 112.0
x,
56.5 40.5 68.0 60.6
x,
x5
x~ 289.4 27O.8 252.7 265.4
X1 X2 X3 X4
60.1 54.1 46.6 67.6
x~ 55.7 50.2 43.2 62.6
x~
101.3 85.4 47.7 124.0
X1 X2 X3 X5
144.6 147.1 150.5 151.8
128.4 130.6 133.6 134.8
x~
x~
X1 X2 X3 X4
x2
x~
48.8 35.0 58.7 52.4
x~
273.9 256.3 239.1 251.1
x~
(d) SPL" for J o r d a n model
X1 X2 X3 X4
(b) SPL" for Waide model
(f) SPL" for Finn model
68.6 49.2 82.6 73.6
x2
14.7 13.7 15.0 3.2
x~
x,
60.9 43.7 73.3 65.4
x~
216.8 203.2 221.5 47.7
x3
(e) SPL' for Finn model
X1 X2 X3 X4
(c) SPL' for Jordan model
X1 X2 X3 X4
x~
(a) SPL' for Waide model
Total sensitivities for three forest Ca models
TABLE 6
x~
0.0 0.0 0.0 0.0
x3
76.2 68.6 59.1 85.6
x~
319.6 299.0 279.0 293.0
184.5 187.6 192.0 193.6
x4
52.1 46.9 40.4 58.6
x4
15.5 14.5 13.5 14.2
x4
x~ 160.8 163.5 167.2 153.5
59
SENSITIVITY OF C Y C L I N G MEASURES D E R I V E D F R O M ECOLOGICAL FLOW ANALYSIS
TABLE 7
Sensitivity values for Waide Ca model (a) Sensitivities of recycling efficiencies
i,j
1,3 t
ORE 1//~gij t ORE 2 / Ogij
0RE3/ 0gijt i
ORE 4 / Ogij
i,j
2,1
3,2
3,4
0.85
0.81
0.95
0.00
0.85
0.81
0.95
0.00
0.85
0.81
0.95
0.00
0.00
0.00
0.00
0.00
1,3
2,1
3,2
3,4
ORE1/Ogijtt 0REz / 0gijtt ORE3/Ogijt!
1.00 1.00 1.00
0.81 0.81 0.81
0.81 0.81 0.81
0.00 0.00 0.00
ORE4 /, 4(1gij,,
0.00
0.00
0.00
0.00
(b) Sensitivities of cycled throughflows
i,j
1,3
2,1
3,2
3,4
0 T~3/ 0 gu
42.35 42.57 49.95
40.58 40.79 47.86
47.44 47.69 55.96
0.00 0.00 0.00
OT~ / Ogut
0.00
0.00
0.00
0.00
l
aT~a/ 0gu ! OT~2/Ogij t
i,j it
OTcl//Ogij
¢!
Tc2//O gij t!
Tc3/ 0gu 0TeA/ Ogutt
1,3
2,1
3,2
3,4
49.95 50.21 58.92 0.00
40.37 40.58 47.61 0.00
40.43 40.64 47.69 0.00
0.00 0.00 0.00 0.00
2,1
3,2
3,4
(c) Sensitivities of cycling indices
i,j
1,3 t
OCI/Ogij i,j
0.80 1,3
tt
OCI//Ogij
0.95
0.77 2,1 0.76
0.90 3,2 0.76
0.00 3,4 0.00
shows that flows to and from mineral soil (X3) change less than those to and from other compartments. Elements of sPIY indicate that flows from mineral soil (X3) change N " less than from other compartments; flows to mineral soil (X3) are as influential as flows from the other four compartments. Recycling efficiency sensitivities for the models are shown in Tables 7a, 8a, and 9a. Only sensitivities for parameters which represent nonzero flows have been calculated. Recycling efficiencies of Waide's model change the greatest amount if proportion of inflow into nutrients from litter (g32) or if the proportion out outflow from nutrients into vegetation (g~3) are changed. In Jordan's model the greatest changes in recycling efficiencies occur if the
60
R.W. BOSSERMAN
TABLE 8
Sensitivity values for Jordan Ca model (a) Sensitivities of recycling efficiencies
i,j
1,4 t
0~1/~gij
!
Oe,Ez/~gij!
RE3/ 0 gij
RE4//~gijt i, j
0.61 0.63 0.61 0.61 1,4
tt
ORE1/~gij ~RE2/ogij "" "
Oe~3/Ogijpt tt
~RE4/~gij
2,1 0.46 0.52 0.46 0.46 2,1
3,1 0.94 0.56 0.94 0.94 3,1
3,2 0.94 1.07 0.94 0.94 3,2
4,3 0.58 0.59 0.58 0.58 4,3
0.58 0.59 0.58
0.58 0.66 0.58
0.58 0.34 0.58
0.46 0.52 0.46
1.00 1.02 1.00
0.58
0.58
0.58
0.46
1.00
(b) Sensitivities of cycled throughflows
i,j 3T~a/3gijt ~T~2/Ogi'J
1,4
2,1
3,1
3,2
0 Tc3/ 0 gij p Tc4/ 0 gij
38.64 31.38 62.90 36.43
28.90 26.04 47.04 27.24
59.30 27.89 96.54 55.91
59.30 53.44 96.54 55.91
i,j
1,4
2,1
3,1
3,2
4,3
36.43 29.58 59.30 34.34
36.43 32.83 59.30 34.34
36.43 17.13 59.30 34.34
28.90 26.04 47.04 27.24
62.90 51.08 102.40 59.30
1,4 0.31
2,1 0.10
3,1 0.78
3,2 0.64
4,3 0.21
1,4
2,1
3,1
3,2
4,3
t
tt
T~a/ 0 gij
~Tc2/agijt t ~T~3/Ogijpt 0 T~ / ~ gijt !
4,3 36.43 29.58 59.30 34.34
(c) Sensitivities of cycling indices
i,j cl/~gijt i,j c,/~g'~
0.51
0.59
0.54
0.47
1.00
proportion of inflow into soil from canopy (g31), of inflow into soil from litter (g32) or of outflow from soil into wood (g43) are changed. In Finn's model the greatest changes in recycling efficiencies occur if the proportion of inflow into available nutrients from above ground biomass (gEl), into available nutrients from forest floor (g~2), or into nutrients from below ground biomass (g~4) are changed. The proportion of outflow from nutrients to below ground biomass (g45) causes large changes in the recycling efficiencies. Sensitivities of cycled throughflows (Table 7b) for Waide's model show that amount of Ca cycled through nutrients (X3) changes the most due to
SENSITIVITY
OF
CYCLING
MEASURES
DERIVED
FROM
ECOLOGICAL
FLOW
61
ANALYSIS
TABLE 9
Sensitivity values for Finn Ca model (a) Sensiti~tiesofre~clingefficiencies
i,j OP,E1/Og~jt OREz/Ogijt ORE3/Ogijt OgE,~/Og,j O~s/~g~'j i,j Ol~x/Ogijt p tp O~2/Og~j tt Oe.E3/Og~j O~4/Og~j OPd~5/~g0
1,5
!
4,1
4,2
4,3
4,5
5,4
0.61 0.65
1.10 0.75
1.07 1.16
0.00 0.00
0.73 0.75
0.73 0.75
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.61 0.34
0.56 0.56
1.00 1.00
1.00 1.00
0.00 0.00
1.00 0.00
0.69 0.69
1,5 0.78 0.78
Pl
2,1
0.66 0.66
2,1
4,1
4,2
4,3
4,5
5,4
0.74 0.78
0.76 0.52
0.62 0.67
0.00 0.00
0.60 0.61
0.89 0.91
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.71 0.71
0.67 0.67
0.70 0.70
0.58 0.58
0.00 0.00
0.82 0.82
0.85 0.85
(b) Sensitivities of cycled throughflows
i,j
1,5
2,1
4,1
4,2
4,3
4,5
5,4
OT~l/3g~jt t OT~2/Og~j
35.05 29.16 0.00 46.01 21.06 1,5
32.30 28.33 0.00 42.40 34.75 2,1
57.82 32.73 0.00 75.90 62.20 4,1
56.40 50.72 0.00 75.90 62.20 4,2
0.00 0.00 0.00 0.00 0.00 4,3
38.38 32.73 0.00 75.90 0.00 4,5
38.38 32.73 0.00 52.60 43.11 5,4
41.29 34.35
38.85 34.07
40.22 22.77
32.62 29.33
0.00 0.00
31.45 26.82
46.84 39.94
t
0 T~3/ 0 gis
3T~/Oggit OTcs/~g,'./ i,j pr 3T~I/O&j tt 3T~2/agij tt 0 Tc3/ 0gij ~T~4/Ogsj tt OTcs/~g~j II
0.00
0.00
0.00
0.00
0.00
0.00
0.00
54.20 44.42
41.00 41.79
52.80 43.27
43.90 35.98
0.00 0.00
62.20 50.97
64.19 52.60
4,1
4,2
4,3
4,5
5,4
(c) Sensitivities of cycling indices
i,j acI/OgijI i,j 0 cI/~g~j
1,5 0.51 1,5
¢t
0.39
!
2,1 0.53 2,1 0.38
1I
0.89 4,1 0.33
0.95 4,2 0.32
0.00 4,3 0.00
0.57 4,5 0.39
0.65 5,4 0.46
c h a n g e s in gij a n d gij- F o r Jordan's m o d e l , the a m o u n t o f C a c y c l e d ? P! t h r o u g h soil (X3) c h a n g e s the m o s t d u e to c h a n g e s in gij a n d gij. L i k e w i s e , in the F i n n m o d e l , the a m o u n t o f C a c y c l e d t h r o u g h available nutrients ( X 5) ! P! c h a n g e s the m o s t d u e to c h a n g e s in g~j a n d g~j. T a b l e s 7c, 8c, a n d 9c s h o w sensitivities o f c y c l i n g i n d i c e s ca'. F o r W a i d e ' s m o d e l c h a n g e s in the p r o p o r t i o n o f o u t f l o w f r o m nutrients i n t o v e g e t a t i o n (g~3) a n d the p r o p o r t i o n o f i n f l o w to nutrients f r o m litter (g32) w o u l d c a u s e
62
R.W. BOSSERMAN
the greatest changes in CL The normalized flows can be converted to actual flows by multiplying gij by fq. If g~ increases by 0.001, then cz would increase by 0.00095; an increase of 0.001 in g;3 corresponds to an increase of 0.048 kg Ca ha-1 year-1. For Jordan's model the proportion of flow into soil from canopy (g;3) and the proportion of flow out of soil into wood (g4'3) cause the greatest change in the cycling index. For Finn's model, the proportion of flow intro nutrients from forest floor (g~2), the proportion of flow into nutrients from above ground biomass would cause the greatest change in the cycling indices if changed (g51). CONCLUSIONS Flow analyses provide tools for examining influence as it is propagated through an ecosystem network. In a complex ecosystem indirect effects may exert a large influence through long causal paths and cycles; small causes can be magnified by the system network. Patten et al. (1976) and other have argued that this extended environment of an ecosystem is at the basis of many ecological phenomena. The sensitivity analyses described here further refines the usefulness of inflow-outflow analyses for studying ecosystems. Causes can be traced through the ecosystem network, and the degree of influence can be ascertained. Sensitivity to change is greatest in networks which are highly connected and have large amounts of feedback. Such sensitivity arises as a system level property of complex systems; studies of inflow-outflow sensitivity will provide meaningful insight into the behavior of ecosystems. ACKNOWLEDGEMENTS I would like to thank Thomas B. Burns and two anonymous reviewers for their comments on this manuscript. APPENDIX Measurement of condition of invertible matrix B BB -1
=
I
(OB-1B/~bij) = (~I/abij) = 0 = (~B-1/~bij)B -1 + B(aB-a/~bij) B-l(OB/Obij)n -1 + OB-1/~bij = 0
B-X/ b,j
=
SENSITIVITY OF CYCLING
MEASURES
DERIVED
FROM ECOLOGICAL
FLOW ANALYSIS
63
Sensitivity of recycling efficiency
OREk/Ogij =
O[(nkk --
1)/nkk]Ogij
= 3[1 - (1/nl, k)]/O&j
= 3 (1/nkk)/Ogij = - (o,, ~#og,,j)/,G
Sensitivity of cycled throughflow O T ~ k / O g i j - - OTc~/Ogiz = 0
a(~
T~)/Og,j - O ( ~ k T~)/Og,z = 0
Tk OREk/Ogij "+ REk OTk/Ogij = Tk 3REk/Ogiz -- REk ÙTk/Ogi~ Tk ( OREk/Ogij -- OREk/Ogiz ) -k- _nE~( OTk/Og, j Tk(OREk/Ogij-- OREk/Ogi~.) + Tk(aREk/Ogij
-
0
- O T k / O giz ) =
0
(because the throughflow is constant)
O)= TkS(i , J)kk/nkk " 2
Sensitivity of cycling indices
CI/~ gi j = ~ ( TSTc/ TST ) /O gi j = (0 TSTc/Ogij)TST =(k=~OT~JOgiJ) TST = (k~=l(OREk Tk)/Ogij) TST
=
, S)kk/nkk) /TST k=l
REFERENCES Astor, P.H., Patten, B.C. and Estberg, G.N., 1976. The sensitivity substructure of ecosystems. In: B.C. Patten (Editor), Systems Analysis and Simulation in Ecology, 4. Academic Press, New York, pp. 389-429. Bosserman, R.W., 1981. Sensitivity techniques for inflow-outflow flow analyses. In: W.J. Mitsch, R.W. Bosserman and J.M. Klopatek (Editors), Energy and Ecological Modelling. Developments in Environmental Modelling, 1. Elsevier, Amsterdam, pp. 653-660.
64
R.W. BOSSERMAN
Bosserman, R.W., 1983. Flow analysis sensitivities for models of energy or material flow. Bull. Math. Bio., 45: 807-826. Bullard, C.W. and Sebald, A.V., 1977. Effects of parameter uncertainty and technological change on inflow-outflow models. Rev. Econ. Stat., 59" 75-81. Christ, C.F., 1955. A Review of Inflow-Outflow Analysis in Inflow-Outflow Analysis: An Appraisal. Studies in Wealth and Income, 18. National Bureau of Economic Research, Princeton University Press, Princeton, NJ. Finn, J.T., 1976. Measures of ecosystem structure and function derived from analysis of flows. J. Theor. Biol., 56: 363-380. Finn, J.T., 1977. Flow analysis: a method for tracing flows through ecosystem models. Ph.D. dissertation. University of Georgia, Athens, GA, 568 pp. Finn, J.T., 1980. Flow analysis of models of the Hubbard Brook ecosystem. Ecology, 61: 562-571. Hannon, B., 1973. The structure of ecosystems. J. Theor. Biol., 41: 535-546. Jordan, C.F., Kline, J.R. and Sasscer, D.S., 1972. Relative stability of mineral cycles in forest ecosystems. Am. Nat., 106: 237-253. Likens, G.E., Bormann, F.H., Pierce, R.S., Eaton, J.S. and Johnson, N.M., 1977. Biogeochemistry of a Forested Ecosystem. Springer, New York, 146 pp. Patten, B.C., Bosserman, R.W., Finn, U.T. and Cale, W.G., 1976. Propagation of cause in ecosystems. In: B.C. Patten (Editor), Systems Analysis and Simulation in Ecology, 4. Academic Press, New York, pp. 457-579. Sebald, A., 1974. An analysis of the sensitivity large scale input-output models to parametric uncertainty. Doc. 122, University of Illinois, Urbana, IL, 23 pp. Sherman, J. and Morrison, W., 1950. Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat., 21: 124-127. Waide, J.B. and Webster, J.R., 1976. Engineering systems analysis: applicability to ecosystems. In: B.C. Patten (Editor), Systems Analysis and Simulation in Ecology, 4. Academic Press, New York, pp. 329-371. Waide, J.B., Krebs, J.E., Clarkson, S.P. and Setzl~r, E.M., 1974. A linear systems analysis o f the calcium cycle in a forested watershed ecosystem. Progr. Theor. Biol., 3: 261-345.