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Ecological buffer capacity
Ecological Modelling, 3 ( 1 9 7 7 ) 3 9 - - 6 1 © Elsevier Scientific P u b l i s h i n g C o m p a n y , A m s t e r d a m - - P r i n t e d in T h e N e t h e r l a n d s
39
ECOLOGICAL BUFFER CAPACITY SVEN ERIK JORGENSEN and HENNING MEJER *
Danmarks Farmaceutiske Hc~jskole, Universitetsparken, Copenhagen (Denmark) • KCbenhavns Teknikum, Prinsesse Charlottesgade, Copenhagen (Denmark) ( R e c e i v e d 29 April 1 9 7 6 )
ABSTRACT J~Srgensen, S.E. a n d Mejer, H., 1 9 7 7 . Ecological b u f f e r capacity. Ecol. Modelling, 3: 39--61. A n e w c o n c e p t - - t h e ecological b u f i e r c a p a c i t y - - has b e e n i n t r o d u c e d t o e x p r e s s t h e r e s p o n s e of a n e c o s y s t e m t o c h a n g e s in t h e loading. E i t h e r t h e relative or a b s o l u t e s t a b i l i t y gives this sort of i n f o r m a t i o n . By m e a n s o f this c o n c e p t , it has b e e n a t t e m p t e d t o s h o w h o w c o m p l i c a t e d a m o d e l m u s t be t o give a n a c c e p t a b l e d e s c r i p t i o n o f t h e r e s p o n s e t o c h a n g e s in t h e p h o s p h o r u s loadings. It was f o u n d t h a t t h e ecological b u f f e r c a p a c i t y increases w i t h increasing m o d e l diversity, e i t h e r e x p r e s s e d b y m e a n s o f t h e S h a n n o n i n d e x or t h e n u m b e r o f s t a t e variables, b u t since it is o f i m p o r t a n c e t o i n c l u d e t h e m o s t essential mass flows, t h e exergy is m o r e s u i t a b l y r e l a t e d t o t h e b u f f e r c a p a c i t y t h a n t h e diversity. C o n s e q u e n t l y , t h e e x e r g y can be. used as a n e x p r e s s i o n for t h e b u f f e r c a p a c i t y , t h a t is, as a n e x p r e s s i o n for t h e r e s p o n s e o f a n e c o s y s t e m t o c h a n g e s in t h e driving f u n c t i o n s . B o t h t h e b u f f e r c a p a c i t y a n d t h e exergy can b e u s e d to select t h e r e q u i r e d s t a t e variables for a m o d e l . I n t e r c h a n g i n g P7 q (= p h o s p h o r u s c o n c e n t r a t i o n in b o x j at t h e r m o d y n a m i c e q u i l i b r i u m ) w i t h pO ( = p h o s p h o r u s c o n c e n t r a t i o n in b o x j a t s t e a d y s t a t e ) in t h e e x e r g y e x p r e s s i o n , s e e m s t o give a useful L i a p u n o v f u n c t i o n for t h e c o n s i d e r e d set o f models.
INTRODUCTION
It has been pointed out by Elton (1958) that more complex ecosystems might be expected to be more stable than simpler ones -- a simple twospecies ecosystem model tends to be unstable. However, May (1971, 1973) has stressed that this is valid only if models involving more species with a greater degree of food-web linkage, exhibit greater stability than simpler ones. Recent studies by May (1971, 1973) and Gardner and Ashby (1970) have suggested that this is not the case and May has concluded that a comparison between simple few-species models with analogously simple multispecies models generally shows that the latter are less stable. Many present-day ecosystem models give a fairly good description of the annual cycles in the ecosystem. Most of these models are also stable with realistic values of the driving functions, but unfortunately only very few models have been validated under changed conditions of the ecosystem.
40
However, the purpose of the models used for pollution control, is to give a realistic description of the response of the ecosystem under changed conditions, in terms of either increased or decreased impact on the ecosystem. This paper is concerned with the relation between the response of an ecosystem and its complexity. STABILITY
Liapunov (1892) invented the stability theory and defined an equilibrium state to be, stable if, following displacement from equilibrium, the systems subsequent behaviour is restricted to a b o u n d e d region of state space. Holling (1973) has pointed out that this classical stability concept has only theoretical interest. A dynamic balance between the maintenance and dissipation of structure produces non-zero ecosystem steady states, which are stable (Waide, 1975). Around this steady state exist areas within which displacements from the steady state are followed by reversion to the original condition (Lewontin, 1970; Holling, 1973). As pointed out by Webster et al. (1974), the ecologist's relevant question is n o t "are ecosystems stable?", but rather " h o w stable are ecosystems?". The ecologist focuses upon relative rather than absolute stability. However, relative stability is n o t -- mathematically or ecologically -- well defined. Relative stability concerns an ecosystem's response to displacement from a steady state, and t w o aspects have been identified (Patten et al., 1967; Child et al., 1972; Holling, 1973; Marks, 1974). The first one deals with the resistance of an ecosystem to displacement. An ecosystem that is easily displaced from a steady state has low resistance, whereas a system that is difficult to displace, has high resistance. The second aspect concerns return to a steady state -- or resilience. An ecosystem returning rapidly to its original condition is more resilient than one that responds slowly or with oscillations. The relative stability has been analyzed for general linear ecosystem models and has been related to nutrient recycling (Webster et al., 1974; Halfon, 1976). Closed-loop nutrient cycling has a great influence on the homeostasis of the system (Waide et al., 1974). A complete theory of ecosystem stability is still lacking and in particular the relation between the complexity of the ecosystem (or model) and its stability needs further investigation. ECOLOGICAL B U F F E R CAPACITY
The ecologist working with ecosystem models with the goal to predict changes in the ecosystem conditions by changing one or more driving functions, is concerned a b o u t the response of the system. Fig. 1 shows the response of Lake Fure to increased phosphorus loading. As can be seen in this figure, the concentration of phosphorus in the water phase is, initially, almost unchanged. The increased phosphorus loading does
41
IJg PO4-P'I-1
6001
0 o
500-
h00-
300"
o
" ~
200"
100. ::
0
192930
A
h5
; 50
"
o
55
o
60
65
=, 70
73
Fig. 1. T h e p h o s p h o r u s c o n c e n t r a t i o n in L a k e F u r e p l o t t e d a g a i n s t t i m e ( y e a r s ) . D u r i n g t h e p e r i o d c o n s i d e r e d t h e p h o s p h o r u s l o a d i n g is i n c r e a s e d .
n o t influence the phosphorus concentration in the water phase, because the ecosystem has a storage capacity in the sediment (see Fig. 2). However, as seen in Fig. 1, when the buffer capacity in the sediment is expended, the phosphorus concentration rapidly increases. For the user of ecosystem models for environmental impact statements and predictions, it seems very valuable to a t t e m p t to answer the question: "What will be the response of the ecosystem to changed (increased or decreased) loadings?" For this purpose it seems useful to introduce the concept ecological buffer capacity as: AL where ASV is a change in the considered state variable caused by a given change of the loading ( A t ) . To find ~ it is necessary to understand what m a n y authors call the sensitivity of the state variable (see e.g. Halfon, 1976): OSV 0L - sensitivity as a function of the loading. The ecological buffer capacity is f o u n d as L 2 -- L 1
1
42
-10000 "5000 #oo oo%ooo
I
APATIT
-looo
C.al0(P04)6(OH)2(s)
-500
~
;S0
6"
I
"T
• '
I
8"
"0,5
0,1 '0,05
9-
CALCIT
" ~
CoC03( s )
" ~
O,01 '0,005
105
6
7
8
10
9
11
12
13 pH
3,001
Fig. 2. The diagram explains, w h y the sediment has a buffer capacity. The conditions under which caleit is transformed to hydroxypatit, are given. STATISTICAL ANALYSIS SYSTEM PLOT OF EX VS
BETA
/*O.00O00O00
E : 3.13 • / 3 - 2 . 9 0 r : 0.98 F : 5178 3 2 . 00000000 +
A AA A
A
B
AA
A ABAB B
B
AAC
AB
2 & . 000000O0 +
CA B AB
BB
AB
BA
EX A
BAAA
B 8 B
16. 00000000 + A A AA
ABA A
BB B
BB B B
8.0O00000O +
A A A A
A
AAA
A AAA
B
B
B BB
BB B
# JZ 0,000O00O0 +
TG
, .........
+ ..................
0.70000000
+ .................
2.70OO(X~O
LEGEND:A = 1 OBS , B = 2 O8S~ ETC.
+ .................
¢. 7 0~00O0
+ ................
6.7 O0(XX~O
+ . . . . . . . . . . . . . . .
8.70000000
+
....
10.70000000
BETA
Fig. 3. Plot of exergy (E) versus buffer capacity (~). 180 points are plotted. A regression analysis and an F-test were performed.
A~
STATISTICAL ANALYSIS SYSTEM PLOT
OF
EX VS
"-JtO
BETA1
40. 00000000 +
E : 6.21
3 2. 00000000 +
C C
C
C C
C
C C
C
c~
c
C
C C
C
A
C
C
• ~'-5.22
r : 0.77
F : 257
2 4. 0000000O +
EX
16. 00000000 4-
C
8.
00000000 +
C
A A A A A A
C
C
C C
C
C C
C C
C
A IRI 0.00000000 +
Z . . . . . . . . . . . . . . . . .+. . . . . . . . i ........ + . . . . . . . . . . . . . . . .+. . . . . . . . . . . . . . . .+. . . . . . . . . . . . . . .-40.85000(]00
1.45000000
2.0 S 000000
LEGEND:A= 1 0 B S , B - 2 0 B S , ETC.
2.65000000
÷
3.25000000
3.850(~
BETA1
Fig. 4. Plot of exergy (E) versus buffer capacity (~'). 180 points are plotted. A regression analysis and an F-test were performed. STATISTICAL ANALYSIS SYSTEM PLOT
OF
H VS BETA
2.40000000 +
H : 0.099./'9 + 0.575 r : 0.50 F : 59
A
1. 90000000 +
A
A
AA
A
A
A A
A
A A
A A A A A
A
A AA A A AA
A
A AA A
A
A A A
A
AA
A
A
A
A
A
,B
AA A
A
A
A
AA
AA A
A A A
B
A
A
A
A
0. 90000000 +
A A
A
AA
A A
1. 40000000 4-
A AA
A
A
A
A AA
AA A A A A
A A
A A A
AA
AA AA A
A A
AA
A~A A A
A A A A4
A
I
A 0 • 40000000 +
I
i
I I i
, P i - 0.10000000
B R
+ /
I ........
4-- ...............
0.70000000
+ ...............
2 . 7 000O00O
LEGEND:A-10BS, B = 2 0 B S , ETC
+ .............
4. . 7 0 0 0 0 0 0 0
-I- . . . . . . . . . . . . .
6 . 7 ~
4-
.............
8.70000000
-I-- - 10.70000000
BETA
Fig. 5. Plot of Shannon's index versus buffer capacity (/3). 180 points are plotted. A regression analysis and an F-test were performed.
44
STATISTICAL ANALYSIS SYSTEM PLOT
OF
H VS
BETAI
2 , 4 00000Q0 + A A A B
A
1.90000000
H = 0.38 • /3"-0.03 r = 0.79 F :288
+
1.
A
,I
B
A A
B
A A B B A B A A B A A
A
A A A
C A A B
~0000000
B
A B A
s
H
0,90000000
A ~
B A A
B
~
A
~s~
g
A
A I I
I
A A
A
A
A
B
A
A
A
A
A
0..~ 0000000 +
I
C R 0.10000000 + i ........
+ ...............
+ . . . . . . . . . . . . . .
0 . 8 5 ~ LESEND:A=10BS,
1 . 45000000 B = 2 0 8 S a ETC.
+ . . . . . . . . . . . . . .
2.05000000
+ . . . . . . . . . . . . . .
2.65000000
+ . . . . . . . . . . . . .
3 . 25000(X)0
+
....
3 . II 5000000
BETA1
Fig. 6. Plot of Shannon's index versus buffer capacity (~'): 180 points are plotted. A regression analysis and an F-test were performed.
w h e r e L2 - - L1 = AL = t h e c h a n g e o f t h e loading. This paper deals w i t h t h e r e l a t i o n b e t w e e n t h e e c o l o g i c a l buffer c a p a c i t y and t h e c o m p l e x i t y . T h e n u m b e r o f state variables that can be c o n s i d e r e d in a n y m o d e l is l i m i t e d , since t h e research e f f o r t t h a t can be i n v e s t e d in o n e p r o b l e m is also l i m i t e d . M o d e l s will a l w a y s be simple in c o m p a r i s o n w i t h t h e c o m p l e x i t y o f t h e e c o s y s t e m . E c o l o g i c a l m o d e l s are o f t e n d e v e l o p e d t o f o c u s a t t e n t i o n o n certain aspects, and this will lead t o a s e l e c t i o n o f the state variables n e e d e d . T h e c o n c e p t o f e c o l o g i c a l b u f f e r c a p a c i t y is c o n s i d e r e d as a t o o l t o select a s u f f i c i e n t n u m b e r o f state variables a m o n g t h o s e relevant t o t h e q u e s t i o n s that t h e m o d e l a t t e m p t s t o answer. A set o f e u t r o p h i c a t i o n m o d e l s c o n sidering o n l y p h o s p h o r u s as n u t r i e n t will be e x a m i n e d t o try t o illustrate these concepts. THE EXAMINATION 20 d i f f e r e n t m o d e l s describing t h e c y c l i n g o f p h o s p h o r u s in a lake are c o n sidered. T h e m o d e l s are s h o w n in Table I. Table II is a list o f state variables,
TABLE I V e r s i o n = 0 (case = 0, 1, 2, 3, 4)
i's = ( P i n - Ps) • ( Q / V )
iPs = (Pin - - P s ) " ( Q / V ) -- ( P a - - Ra) ' P a /~a = (Pa - - R a - - ( Q / V ) ) " Pa
Ps = (Pin - - P s ) " ( Q / V ) - - (//a - - R a ) " Pa + Kd ' P d Pa = (Pa - - R a - - M a - - ( Q / V ) ) . Pa Pd = M a ' Pa - - ( g d + ( Q / V ) ) . Pd
iDs = (Pin - - Ps) " ( Q / V ) - - (//a - - R a ) " Pa + K d " P d + R z " Pz Pa = (~a - - R a - - M a - - ( Q / V ) ) . Pa - - ( l / Y ) " P z " Pz i°d = M a " Pa - - ( g d + ( Q / V ) ) . Pd + (Mz + L " /,lz) • Pz Pz =
(Pz - - R z - - M z
/~s = (Pin - - P s ) "
--(Q/V))"Pz
( Q / V ) - - (Pa - - R a ) " Pa + K d " P d + R z " Pz + R~z " P'z
Pa = (Ua - - R a - - M a - - ( Q / V ) ) . Pa -- ( l / Y ) •/~z" Pz ~6d = M a " Pa - - ( g d + ( Q / V ) ) . P d + (Mz + L . / 2 z ) - P z - - (P'z - - M ' z ) " P'z /Sz = (Pz - - R z - - M z - - ( Q / V ) ) " Pz Pz = (t2'z - - R z -- M z - - ( Q / V ) ) . Pz
V e r s i o n = 1 (case = 0, 1, 2, 3, 4)
i's = ( P i n - - P s ) " ( Q / V ) + a s . K , , . P~*d i'sed = - - K x " Psed
Ps = ( P i n - - P s ) " ( Q / V ) -
(/2a-- R a ) "Pa + a s ' K x P s e d
i'a = (/la - - R a - - S a - - ( Q / V ) ) .
i'~d -- 0.4 s ~ .
Pa
P~/as - g x - P ~ d
/~s = (Pin - - P s ) " ( Q / V ) - - (/2 a - - R a ) - Pa + K d " P d + a s " K x " Psed J~a = (Pa - - R a - - Ma - - Sa - - ( Q / V ) ) " Pa i'd = M a . P a i~d
-
-
(~:,~ + ( Q / V ) )
-- 0 . 4 S a . P ~ / a s - - g x
. Pd
. P~d
Ps = (Pin - - Ps) " ( Q / V ) - - (Pa - - R a ) "Pa + K d " P d + R z " Pz + a s " K x ' Psed Pa = (/'ta - - R a - - S a - - ( Q / V ) ) i'd = - - ( g d
+ (Q/V))"Pd
" Pa - - ( l / Y ) • Pz " Pz
+ (Mz + L " P z ) " P z
/~z = (Uz - - R z - - Mz - - ( Q / V ) ) . P z Psed = 0.4 S a - P a / a s - - g x • Psed
P s = (Pin - - P s ) " ( Q / V ) - - (//a - - R a ) " Pa = (Pa - - R a - - S a - - ( V / V ' ) ) I'd = - - ( g d + ( Q / V ) ) " P d
" Pa - - ( l / Y ) " U z " P z
+ (Mz + L " / / z )
i'z = (Pz - - R z - - M z - - ( Q / V ) ) • P z • t
P~ : (U'z
t
-
R~ -- Mz
Pa + K d ' P d + R z " Pz + Rz " P'z + a s " K x " Psed
t
F
-- ( Q/ V) ) • _P~
Psed = 0.4 S a • Pa/as -- K x • Psed
" P z - - (/-t'z - - M ' z ) " P'z
V e r s i o n = 2 (0, 1, 2, 3, 4 ) ]~s = ( P i n - - P s ) " i~sed = - - g x
(Q/V) + Qd
"Psed
Jai = ( a s / a i ) g x
" Psed - - Q d / a i
/~s = ( P i n - - P s ) " ( Q / V ) - - ( # a - - R a ) " Pa + Q d Pa = (/2a - - R a - - Sa - - ( Q / V ) ) "
Pa
]~sed = 0 . 4 S a • P a / a s - - K x • Psed Pi = ( a s / a i ) K x " Psed - - Q d / a i
Ps = ( P i n - - P s ) " ( Q / V ) - - (Pa - - R a ) " P a + Q d + K d " P d ba = (pa -
R ~ - - M~ - - S a - -
(Q/V))
i6d = M a " Pa - - ( g d + ( Q / V ) ) "
- P~
Pd
i6se d = 0 . 4 S a - P a / a s - - g x • Psed iai = ( a s / a i ) g x
" Psed - - V d / a i
Ps = ( P i n - - P s ) "
(Q/V)--
(Pa - - R a ) " P a + K d " P d + R z " Pz + Q d
Pa = (~a - - R a - - Sa - - ( Q / V ) ) . Pd = - - ( g d
+ (Q/V)).
Pd+
Pa - - ( l / Y )
•/.t z • P z
(Mz + L •/.tz) - Pz
Pz = (Uz - - R z - - M z - - ( V ] Y ) ) .
Pz
Psed = 0 . 4 S a - P a / a s . K x • Psed Pi = ( a s / a i ) K x
" Psed
--
Qd/ai
Ps = ( P i n - - P s ) " ( Q / V ) - - (Ua - - R a ) " P a + K d " P d + Pa = (Pa - - R a - - S a - - ( Q / V ) ) Pd = - - ( g d
+ (Q/V))"
Pz = (/'tz - - R z - - M z t
~
" Pa - - ( l / Y )
R z " Pz + R ' z " P'z + Q d
- Uz " Pz
P d + (Mz + L • P z ) " Pz - - (U'z - - M ' z ) " P'z -- (Q/V)).
Pz
¢
t
P z = (U'z - - R z - - M z - - ( Q / V ) ) . P z Psed = 0 . 4 S a • P a / a s - - K x . Psed iP = ( a s / a i ) K x • Psed - - Q d / a i
pp. 45--52
V e r s i o n = 3 (case = 0, 1, 2, 3, 4) /~s = (Pin - - Ps) " ( Q / V ) + Q d + a b " (Qb + Qds) Psed = - - K x " Psed i)i = ( a s / a i ) K x
" Psed -- Qd/ai
/)b = - - ( Q b + Qds) /)s = (Pin - - P s ) " ( Q / V ) - -
(Pa - - R a ) "Pa + Qd + a b " (Qb + Qds)
Pa = (/2a - - R a - - Sa - - ( V / V ) ) .
Pa
Psed = 0.4 Q s / a s - - K x • P s e d Pi = (as/ai)Kx
" Psed -- Qd/ai
/~b = ( S a " Pa - - Q s ) / a b - - (Qb + Q d s ) /;s = (Pin - - P s ) " ( Q / V ) - - (Pa - - R a ) " Pa + g d " P d + Vd + ab " (Qb + Qds) /~a = ( P a -
Ra - - Ma - - Sa - - ( Q / V ) )
~Sd = M a " P a - - ( g d
• Pa
+ (Q/V))"Pd
Psed = 0.4 Q s / a s - - K x • P s e d Pi = (as/ai)Kx
" Psed -- Qd/ai
/~b = (Sa" Pa - - Q s ) / a b - - (Vb + Vds) Ps = (Pin - - P s ) " ( Q / V ) Pa = ( # a - - R a - - S a - Pd= --(Kd + (Q/V))
(Pa - - R a ) " P a + K d " P d + Rz ' Pz + ab " (Qb + Qds) + Qd -- (l/Y).
(Q/V))-Pa
P z " Pz
• P d + (Mz + L • P z ) " P z
P z = (Pz - - R z - - M z - - ( Q / V ) ) .
Pz
Psed = 0.4 V s / a s - - K x • P s e d Pi = (as/ai)Kx
" Psed - - Q d / a i
Pb = (Sa" P a - - Q s ) / a b - - (Qb + Qds)
Ps = (Pin - - P s ) " ( Q / V ) -
(Pa - - R a ) "
Pa = (/Za - - R a - - Sa - - ( Q / V ) ) . Pd= --(gd + (Q/V)).
P d + (Mz + L " U z ) " P z - - (P'z - - M ' z ) " P'z
Pz = (Pz - - R z - - Mz - - ( Q / V ) )
• Pz
P'z = (P'z - - R ' z - - M ' z - - ( Q / V ) ) .
Pz
r
Psed = 0.4 Q s / a s - - K x - P s e d Pi = (as/ai)Kx
e a + K d " P d + R z " Pz + R'z" P'z + Qd + ab " (Qb - - Qds)
P a - - ( l / Y ) • Uz" Pz
" Psed -- Qd/ai
Pb = (Sa" Pa - - V s ) / a b - - (Qb + Qds)
53 TABLE II State variables [g P/m 3 ] Symbol
Name
Ps Pa Pd P~ P~
Soluble inorganic P in water P bound in algae P in detritus P bound in algal-grazing zooplankton P bound in detritus-feeding zooplankton Exchangeable P bound in dry matter of sediment P in interstitial water in sediment P in uppermost thin layer of biologically active sediment
Psed
Pi Pb
and Tables III, IV and V define constructions used for rates, flows between sediment and water, and total phosphorus concentration in lake and sediment. Tables VI and VII provide data for thermodynamic equilibrium values and steady-state values for the state variables. The equilibrium values are based upon free energies of living matter as estimated by Morowitz (1968). Tables VIII and IX state parameter values used. The models show an increasing complexity. Version 0, case 0, 1, 2, 3, 4 does not include sediment. Versions 1, 2 and 3 consider the sediment in increasing detail (see J~brgensen et al., 1975). Case 0 considers only soluble phosphorus in the water body, while cases 1, 2, 3 and 4 include phytoplankton, detritus, algal-grazing zooplankton and detritus-feeding zooplankton, respectively. TABLE III Rates [24 h - 1 ] Name
Expression *
Decay rate of detritus
Kd = K~0 "WT
Respiration rate of algae
R a = R2a0 " f T
Respiration rate of zooplankton
R z = R2z O" f T
Respiration rate of detritus-feeding zooplankton
Rz = Rz20' f T
Growth rate of algae
Pa = f~a " f T " K p + Ps
Growth rate of zooplankton Growth rate of detritus-feeding zooplankton * f T = 1-03T--293
Ps [ Pd P'z =firz " f T " g a + p d
Pa - - K s
54 T A B L E IV F l o w s b e t w e e n s e d i m e n t and w a t e r [g p / m 3 / 2 4 h / Name
Expression
F l o w o f P released biologically f r o m the u p p e r m o s t t h i n layer (db ) o f s e d i m e n t
(0 exp[0.203(T-- 273)] Qb = / 0.563
F l o w o f P released b y d i f f u s i o n from sediment to water phase
Version
1,000d b
Sedimentation of P to the e x c h a n g e a b l e (40%) and n o n e x c h a n g e a b l e (60%) s e d i m e n t
5,687
'0
3 0 1
Qd = as " Kx ' Psed
l'2(Pi--Ps)--l'7 1,000 • d
F l o w o f P released b y d e s o r p tion from the uppermost t h i n layer (db ) o f s e d i m e n t
0--2
Pb
T 28O
2--3
Qds = - - 0 . 6 0 In Ps - - 2.27
0--2 3
/Sa" Pa Qs = ( m a x ( 0 , S a ' P a - - q s )
0--2 3
0
(
Sources: K a m p - N i e i s e n ( 1 9 7 5 ) a n d J~brgensen e t al. (1975)•
TABLE V V o l u m e ratios and t o t a l P - c o n c e n t r a t i o n Symbol
Name
Definition
ab
V o l u m e ratio: biological layer/lake v o l u m e
db a b = DMU • ~ -
ai
V o l u m e ratio: interstitial w a t e r / l a k e v o l u m e
a i = (1 - - DMU) " ~ -
V o l u m e ratio : dry s e d i m e n t / l a k e v o l u m e
Gs
Total m e a n P - c o n c e n t r a t i o n
Pt
d S
a s = DMU • d_~s d P t = Ps + Pa + Pd + Pz + P" + as" Psed + ai ' Pi + ab " Pb *
* Only in t h e m o s t general case are all t e r m s non-zero• By i n s p e c t i o n o f t h e differential e q u a t i o n s it is seen t h a t :
bt
(Pin - - Ps - - Pa - - Pd - - Pz - - P'z )" Q -- 0.6 Qs
(la)
in t h e m o s t general case. N o t e t h a t ( l a ) d e g e n e r a t e s t o /~t = ( P i n - - P t ) " Q
if version = 0 (no s e d i m e n t ) .
(lb)
55 Table VI T h e r m o d y n a m i c equilibrium concentrations: p ~ P~q
Value
[g P/m s ]
Based u p o n
Pseq
Pin
b o u n d a r y conditions
peq
10 - 5 0
Morowitz (1968)
P~q
10 - 4 6
Morowitz (1968)
peq
10 - 5 0
Morowitz ( 1968 )
p~eq eq Psed
10--50 10 ---46
Morowitz (1968)
peq P~q
Pin + 1 27 10 - 4 6
Q~q = 0 A Pseq = Pin Morowitz (1968)
Morowitz (1968)
The extreme of this spectrum of models is then a lake considered to be just a tank w i t h o u t chemical and biological reactions, and a model with eight compartments including a three-compartment sediment submodel. The 20 models are intercalibrated to give the same steady-state concentrations TABLE VII Steady-state concentrations: pO [g p / m 3 ] P~j
Value
Base.d u p o n
p0
depends o n case and version
calibration
p0
0.1
choice
p0
0.05
choice
p0
0.01
choice
Pz0
0.01
choice
ps0ed
( 400 or:
choice
~ 400 - - a b " pO
choice and mass balance
(ve ion%)
p0
p0
60.0 (
choice 2.27 + 0.6 In Pin
{ db" --~9~ ~ /
I, or: 5687 *
- fT
calibration calibration
* The expression is used for case 0, the~literal, which is the value of the expression w h e n Pin = 1.5, d b = 0.0018 and T = 283.15, is used in all other cases, fT stands for: e x p [ 0 . 2 0 3 ( T - 273)]
56 TABLE VIII Parameters n o t found by calibration Symbol d db ds DMU Ka Kp Ks Ma Me Mz Pin Q R Ra20 T V Y L
Name Average depth of water phase Thickness of biologically active layer Thickness of upper layer of sediment containing exchangeable P Fraction of dry matter in upper layer Michaelis-constant in zooplankton-feeding expressions Michaelis-constant for P-uptake in algae Steele-constant in grazing expression Mortality of algae Mortality of zooplankton Mortality of detritus-feeding zooplankton Concentration of soluble P in inflow Flow of water through the lake Gas constant Respiration rate of algae at 20°C Mean temperature Volume of water body Yield factor Loss factor
Value
Unit
10
m
1.8 X 10 - 3
m
0.1
m
0.1--0.9 5 X 10 -3
g P/m 3
0.02
g P/m 3
5 × 10 - 3 0 *, 0.025 **, 0.05 J" 0.01
g P/m 3 24 h - 1 24 h - 1
0.01
24 h--1
1.5 5,000 0.2683 x 10 - 3 0.35 283.15 450,000 0.67
g P/m 3 m3/24 h kJ/K/g P 24 h - 1 K m3
1/Y-
1
* cases 3 and 4, version > 0 ** cases 3 and 4, version = 0 and case 2, version > 0 t" case 2, version O.
o
(except Ps), when driving functions (loading, temperature and water flow) are held constant. T h e f o l l o w i n g s i t u a t i o n is s i m u l a t e d : B y u s i n g a l a k e m o d e l , i t is p o s s i b l e t o p r e d i c t t h e c o n s e q u e n c e s o f a c h a n g e d p h o s p h o r o u s l o a d i n g (Pin). T h e s t e a d y - s t a t e v a l u e s o f s o l u b l e p h o s p h o r u s i n t h e l a k e (p0) a n d t h e t o t a l p h o s . p h o r u s in l a k e a n d s e d i m e n t (P~t) as a c o n s e q u e n c e o f t h e l o a d i n g a r e c a l c u lated. Two kinds of buffer capacities have been considered:
/3 = P~t
and
/3' - P i n
T h e q u e s t i o n is: h o w a r e / 3 a n d / 3 ' d e p e n d e n t u p o n t h e c o m p l e x i t y o f t h e m o d e l ? W e d o k n o w t h a t e v e n t h e m o s t c o m p l i c a t e d m o d e l is a s i m p l i f i e d d e s c r i p t i o n o f t h e e c o s y s t e m , b u t as w e c a l i b r a t e t o o b t a i n e.g. t h e r i g h t
57 T A B L E IX Parameters f o u n d by calibration (DMU = 0.93) Symbol
Name
Kd~°
Degradation rate of detritus at 20°C Degradation rate of P in sediment Specific growth rate of p h y t o p l a n k t o n at 20°C Specific growth rate of algalgrazing z o o p l a n k t o n at 20°C Specific growth rate of detritus-feeding z o o p l a n k t o n at
gx ~a hz Pz R20 Rrz20
Sa qs
20°C Respiration rate of algalgrazing zooplankton at 20°C Respiration rate of detritus-feeding zooplankton at 20°C Settling rate of phytoplankton Upper limit of flow to biologically active layer (db)
Value
Unit
Based u p o n
0.04--0.12 *
24 h - 1
/)d = 0 **
2 × 10 - 3
24 h - 1
~5i = 0
0.37--0.74 *
24 h - 1
iSa = 0
0.72
24 h - 1
Pd = 0 1"
0.09--0.16 *
24 h - 1
/~d = 0 **
0.62
24 h - 1
/~z = 0
0.05--0.12 * 0.18
24 h - 1 24 h - 1
~5' .Z ~ 0 )Pt.= 0
10 - 4
g p/m3/24 h /as~0 s e d + aiPi
.
i~b = 0 * Depending on case and version. ** and the assumption that (a) only half of the m o r t a l i t y takes place in the water phase, w h e n s e d i m e n t is present, and that (b) the net f l o w through detritus-feeding z o o p l a n k t o n (when present) is one f o u r t h o f the flow f r o m Pd to Ps. j- and the assumption that half of algal m o r t a l i t y flow is interpreted as z o o p l a n k t o n m o r t a l i t y plus loss during grazing.
p h y t o p l a n k t o n value, the model will of course give the right level of phytoplankton under unchanged conditions. If we have no occasion to validate the model under changed conditions, which is most often the case, the calibration will n o t be able to tell us which model gives the best description under changed conditions. However, if the ecological buffer capacity is changed by increased complexity, it seems necessary to choose at least a model with a certain complexity to get a reasonably correct response under changed conditions. Shannons index H and the exergy E, are used as measurements for the complexity. The exergy measures the mechanical energy equivalent of the distance from thermodynamic equilibrium. The mechanical energy equivalent is defined as the m a x i m u m a m o u n t of entropy-free energy, which can be produced upon establishment of equilibrium with the surroundings of the system. As there is an important link between concepts of information theory and the exergy, it seems to be a useful measurement of the complexity. H and E are defined here as shown in Table X.
58 TABLE X S o m e indices of stability and c o m p l e x i t y Name
Definition
Buffer capacities:
j3 = ~Pt ; ~t = Pin Ps
Shannon-index:
H = .
•i
Unit
Pt
aj • ~t" i°g2 ai "Pj '
where Pt -- ~ a j
bit
• P~
i Exergy
E = R " T" ~ ai .(Pi . ln P1 --(Pi--PTq)) j ~q
kJ/m 3
T i m e derivative:
i= R. T.~ai.
i
A Liapunov function:
L =R. T ' ~ a j "
i i . ln Py
Pp (J'j "ln--~--(P~-" l,o~) Pj
kJ/m3/24 h
kJ/m 3
T i m e derivative:
i = R " T "~_/ aj " i~, " ln ~
kJ/m3/24 h
(a i } is a set of v o l u m e ratios (only different f r o m 1 for s e d i m e n t variables).
Note the inequalities:
/3 > 1, ~' > 1. 0 < H < log 2 n (n = n u m b e r o f state variables)
E> 0 (E=O*~foralljPj=P~) L > 0 (L
0*=~foralljPy=P)).
The questions which this examination attempts to answer can be summarized as follows: (1) H o w will ~ and ~' vary with increasing compexity of the model, using (a) Shannons index, (b) exergy, as a measurement of the complexity? (2) Is it possible to determine the complexity the model must have to give a reasonably good description of the response under changed conditions? RESULTS
Figs. 1--4 give the results of the examination; corresponding values of and fi' versus E and H are plotted and a regression analysis is performed. In order to obtain a reasonable number of points between points belonging to models with and w i t h o u t sediment, the sediment was " d i l u t e d " for
59
each of the 20 models in nine steps by assigning the values 0.1, 0.2 .... , 0 . 9 to the relative amounts of dry matter in the sediment (DMU). This gives 20 X 9 = 180 points. Only ~ versus E shows a good correlation (r = 0.98). Inclusion of the sediment in the model gives a significant rise in the buffer capacities and the exergy, as the sediment has a large storage capacity for phosphorus. Furthermore, the buffer capacities increase with an increasing number of state variables, the Shannon index and the exergy. ABSOLUTE STABILITY
An a t t e m p t has been made to find a Liapunov function for the models examined. Inspired by the exergy expression, the Liapunov function shown in Table X was considered. Since L always is non-negative, it follows from Liapunov's theory that stability exists if the time derivative L < 0 outside steady state. Computer experiments show that all the models considered witho u t zooplankton are globally stable; inclusion of one or two types of zooplankton gives self-sustained limit cycles. DISCUSSION
When a model is applied as a tool for prognosis, it is of importance that it gives the correct buffer capacity needed to meet the changes in the driving functions. An increasing diversity, expressed by either the number of state variables or by use of Shannon's index, gives an increasing buffer capacity in most cases, b u t as the diversity does n o t express the significance of the mass flows by ignoring the fact that living systems are a long way away from thermodynamic equilibrium, it does n o t seem to be a good expression for the buffer capacity. This is reflected in the calculations as a low correlation between H and ~ ( or fi'). However, the exergy, which does consider the mass flows is a far better expression for the buffer capacity. As can be seen, it is important to include the reactions in the sediment, since a considerable a m o u n t of the mass flow (in the case studied the flow of phosphorus) goes through the sediment, while it is of less significance to include one or two zooplankton species. Consequently, it is of importance to set up a submodel for the mud-water exchange of phosphorus, giving a good description of the actual mass flow, such as has been attempted by the work of Kamp-Nielsen (1975), J~brgensen et al. {1975) and Jacobsen and J~brgensen (1975). It is of importance to note, that all the models give the same level of p h y t o p l a n k t o n , detritus, zooplankton etc., and it is n o t possible to distinguish between one model and another on the basis of measurements of these state variables. The value of ~a found by calibration increases with increasing model complexity, b u t the value in all cases is considerably lower than the usual value applied {see e.g. Chen and Orlob, 1974; J~brgensen, 1976). The reason is that
60 the Pa applied here includes the influence of light, including an integration over the depth. It has been calculated, that a pa of 0.6/24 h here would have been close to 2.0/24 h if the usual light expression (see J~brgensen, 1976) had been applied, and this value is in accordance with the literature values (see also Patten et al., 1975). By application of a Liapunov function it was possible to show that increased complexity does n o t mean increased absolute stability. The more complicated models are rather less stable. On the basis of the examination carried out it seems to be possible to select the state variables needed in a model built to answer certain questions. The selection should be based upon an examination of the actual state variables and their contribution to the exergy or buffer capacity. This conclusion will, however, require more case studies before it can be used as a general principle, although the principle has been discussed generally above. CONCLUSION A situation where models of increasing complexity were calibrated to give the same level of p h y t o p l a n k t o n , detritus, zooplankton etc. was considered. The buffer capacity, fl or fi', of the different models was found and it is seen that fi (and less pronounced ~') increases with increasing complexity. In other words, it seems necessary to have a model of a certain complexity to be able to describe a response to changes in the driving functions. It is of great importance to include the essential mass flows. When phosphorus is considered, the sediment plays an essential role -- at least in shallow lakes. Although increasing diversity {expressed by means of the Shannon index or the number of state variables) gives an increasing buffer capacity, it does not correlate well with the buffer capacity, as it does not take the mass flows into consideration. However the exergy, which measures the distance from t h e r m o d y n a m i c equilibrium gives an excellent expression for the buffer capacity, and the exergy should, rather than the diversity, be used as an expression for the response of an ecosystem to changes in the driving functions. The exergy or the ecological buffer capacity are useful concepts for selection of state variables for a given set of models. By application of a Liapunov function it was shown t h a t the absolute stability was unchanged or decreased with increased complexity of the model. However, it must n o t be forgotten that the model is a rough simplification of the real world and that the models examined do n o t contain too m a n y feed-back mechanisms, which might increase the stability considerably.
REFERENCES Chen, C.W. and Orlob, G.T., 1974. Ecological Simulation for Aquatic Environments. Off. Water Resourc. Res. C-2044 WRE-0500.
61 Child, G.I. and Shugart, H.H., 1912. Frequency response analysis of magnesium cycling in a tropical forest ecosystem. In: B.C. Patten (Editor), System Analysis and Simulation in Ecology, Vol. III. Academic Press, New York, N.Y., pp. 103--135. Elton, C., 1958. The Ecology of Invasions by Animals and Plants. Methuen, London. Gardner, M.R. and Ashby, W.R., 1970. Connectance of large dynamic (cybernetic) systems: critical values for stability. Nature, 228: 784. Halfon, E., 1976. Relative stability of ecosystem linear models. Ecol. Modelling, 2 : 2 7 9 .... 296. Holling, C.S., 1973. Resilience and stability of ecological systems. Annu. Rev. Ecol. Syst., 4 : 1--24. J~brgensen, S.E., 1976. A eutrophication model for a lake. Ecol. Modelling, 2: 1 4 7 - 1 6 5 . Jacobsen, O.S. and J~brgensen, S.E., 1975. A submodel for nitrogen release from sediment. Ecol. Modelling, 1 : 147--151. J~brgensen, S.E., Kamp-Nielsen, L. and Jacobsen, O.S., 1975. A submodel for anaerobic mud-water exchange of phosphate. Ecol. Modelling, 1: 133--146. Kamp-Nielsen, L., 1975. A kinetic approach to the aerobic sediment--water exchange of phosphorus in Lake Esrom. Ecol. Modelling, 1: 153--160. Lewontin, R.C., 1970. The units of selection. Annu. Rev. Ecol. Syst., 10: 1--18. Liapunov, M.A., 1892. Probl~me G~n~rale de la Stabilit~ du Mouvement. Kharkov. Reprinted as Annals of Mathematical Study No. 17. Princeton University Press, Princeton, N.J. Marks, P.C., 1974. The role of Pin Cherry in the maintenance of stability in northern hardwood ecosystem Ecol. Monogr., 44: 73---88. May, R.M., 1971. The stability in model ecosystems. Proc. Ecol. Soc., Australia, 6: 18--56. May, R.M., 1973. Stability and Complexity in Model Ecosystems. Princeton Univ. Press, Princeton, N.J. Morowitz, H.J., 1968. Energy Flow in Biology. Academic Press, London, New York, N.Y., 179 pp. Patten, B.C. and Witkamp, M., 1969. Systems analyses of 134Cs kinetics in terrestrial microcosmos. Ecology, 48 : 813--824. Patten, B.C., Egloff, D.A. and Richardson, T.H., 1975. Total ecosystem model for a cove in Lake Texoma. In: B.C. Patten (Editor), System Analysis in Ecology, Vol. III. Academic Press, New York, N.Y., pp. 2 0 6 - 4 2 3 . Waide, J.B., Krebs, J.E., Clarkson, S.P. and Setzler, E.M., 1974. A linear systems analysis of the calcium cycle in a forested watershed ecosystem. Prog. Theor. Biol., 3: 261--345. Webster, J.R., Waide, J.B. and Patten, B.C., 1974. Nutrient recycling and stability. In: F.G. Howell (Editor), Proceeding of the Symposium on Mineral Cycling in Southeastern Ecosystem. Augusta, Ga.
39
ECOLOGICAL BUFFER CAPACITY SVEN ERIK JORGENSEN and HENNING MEJER *
Danmarks Farmaceutiske Hc~jskole, Universitetsparken, Copenhagen (Denmark) • KCbenhavns Teknikum, Prinsesse Charlottesgade, Copenhagen (Denmark) ( R e c e i v e d 29 April 1 9 7 6 )
ABSTRACT J~Srgensen, S.E. a n d Mejer, H., 1 9 7 7 . Ecological b u f f e r capacity. Ecol. Modelling, 3: 39--61. A n e w c o n c e p t - - t h e ecological b u f i e r c a p a c i t y - - has b e e n i n t r o d u c e d t o e x p r e s s t h e r e s p o n s e of a n e c o s y s t e m t o c h a n g e s in t h e loading. E i t h e r t h e relative or a b s o l u t e s t a b i l i t y gives this sort of i n f o r m a t i o n . By m e a n s o f this c o n c e p t , it has b e e n a t t e m p t e d t o s h o w h o w c o m p l i c a t e d a m o d e l m u s t be t o give a n a c c e p t a b l e d e s c r i p t i o n o f t h e r e s p o n s e t o c h a n g e s in t h e p h o s p h o r u s loadings. It was f o u n d t h a t t h e ecological b u f f e r c a p a c i t y increases w i t h increasing m o d e l diversity, e i t h e r e x p r e s s e d b y m e a n s o f t h e S h a n n o n i n d e x or t h e n u m b e r o f s t a t e variables, b u t since it is o f i m p o r t a n c e t o i n c l u d e t h e m o s t essential mass flows, t h e exergy is m o r e s u i t a b l y r e l a t e d t o t h e b u f f e r c a p a c i t y t h a n t h e diversity. C o n s e q u e n t l y , t h e e x e r g y can be. used as a n e x p r e s s i o n for t h e b u f f e r c a p a c i t y , t h a t is, as a n e x p r e s s i o n for t h e r e s p o n s e o f a n e c o s y s t e m t o c h a n g e s in t h e driving f u n c t i o n s . B o t h t h e b u f f e r c a p a c i t y a n d t h e exergy can b e u s e d to select t h e r e q u i r e d s t a t e variables for a m o d e l . I n t e r c h a n g i n g P7 q (= p h o s p h o r u s c o n c e n t r a t i o n in b o x j at t h e r m o d y n a m i c e q u i l i b r i u m ) w i t h pO ( = p h o s p h o r u s c o n c e n t r a t i o n in b o x j a t s t e a d y s t a t e ) in t h e e x e r g y e x p r e s s i o n , s e e m s t o give a useful L i a p u n o v f u n c t i o n for t h e c o n s i d e r e d set o f models.
INTRODUCTION
It has been pointed out by Elton (1958) that more complex ecosystems might be expected to be more stable than simpler ones -- a simple twospecies ecosystem model tends to be unstable. However, May (1971, 1973) has stressed that this is valid only if models involving more species with a greater degree of food-web linkage, exhibit greater stability than simpler ones. Recent studies by May (1971, 1973) and Gardner and Ashby (1970) have suggested that this is not the case and May has concluded that a comparison between simple few-species models with analogously simple multispecies models generally shows that the latter are less stable. Many present-day ecosystem models give a fairly good description of the annual cycles in the ecosystem. Most of these models are also stable with realistic values of the driving functions, but unfortunately only very few models have been validated under changed conditions of the ecosystem.
40
However, the purpose of the models used for pollution control, is to give a realistic description of the response of the ecosystem under changed conditions, in terms of either increased or decreased impact on the ecosystem. This paper is concerned with the relation between the response of an ecosystem and its complexity. STABILITY
Liapunov (1892) invented the stability theory and defined an equilibrium state to be, stable if, following displacement from equilibrium, the systems subsequent behaviour is restricted to a b o u n d e d region of state space. Holling (1973) has pointed out that this classical stability concept has only theoretical interest. A dynamic balance between the maintenance and dissipation of structure produces non-zero ecosystem steady states, which are stable (Waide, 1975). Around this steady state exist areas within which displacements from the steady state are followed by reversion to the original condition (Lewontin, 1970; Holling, 1973). As pointed out by Webster et al. (1974), the ecologist's relevant question is n o t "are ecosystems stable?", but rather " h o w stable are ecosystems?". The ecologist focuses upon relative rather than absolute stability. However, relative stability is n o t -- mathematically or ecologically -- well defined. Relative stability concerns an ecosystem's response to displacement from a steady state, and t w o aspects have been identified (Patten et al., 1967; Child et al., 1972; Holling, 1973; Marks, 1974). The first one deals with the resistance of an ecosystem to displacement. An ecosystem that is easily displaced from a steady state has low resistance, whereas a system that is difficult to displace, has high resistance. The second aspect concerns return to a steady state -- or resilience. An ecosystem returning rapidly to its original condition is more resilient than one that responds slowly or with oscillations. The relative stability has been analyzed for general linear ecosystem models and has been related to nutrient recycling (Webster et al., 1974; Halfon, 1976). Closed-loop nutrient cycling has a great influence on the homeostasis of the system (Waide et al., 1974). A complete theory of ecosystem stability is still lacking and in particular the relation between the complexity of the ecosystem (or model) and its stability needs further investigation. ECOLOGICAL B U F F E R CAPACITY
The ecologist working with ecosystem models with the goal to predict changes in the ecosystem conditions by changing one or more driving functions, is concerned a b o u t the response of the system. Fig. 1 shows the response of Lake Fure to increased phosphorus loading. As can be seen in this figure, the concentration of phosphorus in the water phase is, initially, almost unchanged. The increased phosphorus loading does
41
IJg PO4-P'I-1
6001
0 o
500-
h00-
300"
o
" ~
200"
100. ::
0
192930
A
h5
; 50
"
o
55
o
60
65
=, 70
73
Fig. 1. T h e p h o s p h o r u s c o n c e n t r a t i o n in L a k e F u r e p l o t t e d a g a i n s t t i m e ( y e a r s ) . D u r i n g t h e p e r i o d c o n s i d e r e d t h e p h o s p h o r u s l o a d i n g is i n c r e a s e d .
n o t influence the phosphorus concentration in the water phase, because the ecosystem has a storage capacity in the sediment (see Fig. 2). However, as seen in Fig. 1, when the buffer capacity in the sediment is expended, the phosphorus concentration rapidly increases. For the user of ecosystem models for environmental impact statements and predictions, it seems very valuable to a t t e m p t to answer the question: "What will be the response of the ecosystem to changed (increased or decreased) loadings?" For this purpose it seems useful to introduce the concept ecological buffer capacity as: AL where ASV is a change in the considered state variable caused by a given change of the loading ( A t ) . To find ~ it is necessary to understand what m a n y authors call the sensitivity of the state variable (see e.g. Halfon, 1976): OSV 0L - sensitivity as a function of the loading. The ecological buffer capacity is f o u n d as L 2 -- L 1
1
42
-10000 "5000 #oo oo%ooo
I
APATIT
-looo
C.al0(P04)6(OH)2(s)
-500
~
;S0
6"
I
"T
• '
I
8"
"0,5
0,1 '0,05
9-
CALCIT
" ~
CoC03( s )
" ~
O,01 '0,005
105
6
7
8
10
9
11
12
13 pH
3,001
Fig. 2. The diagram explains, w h y the sediment has a buffer capacity. The conditions under which caleit is transformed to hydroxypatit, are given. STATISTICAL ANALYSIS SYSTEM PLOT OF EX VS
BETA
/*O.00O00O00
E : 3.13 • / 3 - 2 . 9 0 r : 0.98 F : 5178 3 2 . 00000000 +
A AA A
A
B
AA
A ABAB B
B
AAC
AB
2 & . 000000O0 +
CA B AB
BB
AB
BA
EX A
BAAA
B 8 B
16. 00000000 + A A AA
ABA A
BB B
BB B B
8.0O00000O +
A A A A
A
AAA
A AAA
B
B
B BB
BB B
# JZ 0,000O00O0 +
TG
, .........
+ ..................
0.70000000
+ .................
2.70OO(X~O
LEGEND:A = 1 OBS , B = 2 O8S~ ETC.
+ .................
¢. 7 0~00O0
+ ................
6.7 O0(XX~O
+ . . . . . . . . . . . . . . .
8.70000000
+
....
10.70000000
BETA
Fig. 3. Plot of exergy (E) versus buffer capacity (~). 180 points are plotted. A regression analysis and an F-test were performed.
A~
STATISTICAL ANALYSIS SYSTEM PLOT
OF
EX VS
"-JtO
BETA1
40. 00000000 +
E : 6.21
3 2. 00000000 +
C C
C
C C
C
C C
C
c~
c
C
C C
C
A
C
C
• ~'-5.22
r : 0.77
F : 257
2 4. 0000000O +
EX
16. 00000000 4-
C
8.
00000000 +
C
A A A A A A
C
C
C C
C
C C
C C
C
A IRI 0.00000000 +
Z . . . . . . . . . . . . . . . . .+. . . . . . . . i ........ + . . . . . . . . . . . . . . . .+. . . . . . . . . . . . . . . .+. . . . . . . . . . . . . . .-40.85000(]00
1.45000000
2.0 S 000000
LEGEND:A= 1 0 B S , B - 2 0 B S , ETC.
2.65000000
÷
3.25000000
3.850(~
BETA1
Fig. 4. Plot of exergy (E) versus buffer capacity (~'). 180 points are plotted. A regression analysis and an F-test were performed. STATISTICAL ANALYSIS SYSTEM PLOT
OF
H VS BETA
2.40000000 +
H : 0.099./'9 + 0.575 r : 0.50 F : 59
A
1. 90000000 +
A
A
AA
A
A
A A
A
A A
A A A A A
A
A AA A A AA
A
A AA A
A
A A A
A
AA
A
A
A
A
A
,B
AA A
A
A
A
AA
AA A
A A A
B
A
A
A
A
0. 90000000 +
A A
A
AA
A A
1. 40000000 4-
A AA
A
A
A
A AA
AA A A A A
A A
A A A
AA
AA AA A
A A
AA
A~A A A
A A A A4
A
I
A 0 • 40000000 +
I
i
I I i
, P i - 0.10000000
B R
+ /
I ........
4-- ...............
0.70000000
+ ...............
2 . 7 000O00O
LEGEND:A-10BS, B = 2 0 B S , ETC
+ .............
4. . 7 0 0 0 0 0 0 0
-I- . . . . . . . . . . . . .
6 . 7 ~
4-
.............
8.70000000
-I-- - 10.70000000
BETA
Fig. 5. Plot of Shannon's index versus buffer capacity (/3). 180 points are plotted. A regression analysis and an F-test were performed.
44
STATISTICAL ANALYSIS SYSTEM PLOT
OF
H VS
BETAI
2 , 4 00000Q0 + A A A B
A
1.90000000
H = 0.38 • /3"-0.03 r = 0.79 F :288
+
1.
A
,I
B
A A
B
A A B B A B A A B A A
A
A A A
C A A B
~0000000
B
A B A
s
H
0,90000000
A ~
B A A
B
~
A
~s~
g
A
A I I
I
A A
A
A
A
B
A
A
A
A
A
0..~ 0000000 +
I
C R 0.10000000 + i ........
+ ...............
+ . . . . . . . . . . . . . .
0 . 8 5 ~ LESEND:A=10BS,
1 . 45000000 B = 2 0 8 S a ETC.
+ . . . . . . . . . . . . . .
2.05000000
+ . . . . . . . . . . . . . .
2.65000000
+ . . . . . . . . . . . . .
3 . 25000(X)0
+
....
3 . II 5000000
BETA1
Fig. 6. Plot of Shannon's index versus buffer capacity (~'): 180 points are plotted. A regression analysis and an F-test were performed.
w h e r e L2 - - L1 = AL = t h e c h a n g e o f t h e loading. This paper deals w i t h t h e r e l a t i o n b e t w e e n t h e e c o l o g i c a l buffer c a p a c i t y and t h e c o m p l e x i t y . T h e n u m b e r o f state variables that can be c o n s i d e r e d in a n y m o d e l is l i m i t e d , since t h e research e f f o r t t h a t can be i n v e s t e d in o n e p r o b l e m is also l i m i t e d . M o d e l s will a l w a y s be simple in c o m p a r i s o n w i t h t h e c o m p l e x i t y o f t h e e c o s y s t e m . E c o l o g i c a l m o d e l s are o f t e n d e v e l o p e d t o f o c u s a t t e n t i o n o n certain aspects, and this will lead t o a s e l e c t i o n o f the state variables n e e d e d . T h e c o n c e p t o f e c o l o g i c a l b u f f e r c a p a c i t y is c o n s i d e r e d as a t o o l t o select a s u f f i c i e n t n u m b e r o f state variables a m o n g t h o s e relevant t o t h e q u e s t i o n s that t h e m o d e l a t t e m p t s t o answer. A set o f e u t r o p h i c a t i o n m o d e l s c o n sidering o n l y p h o s p h o r u s as n u t r i e n t will be e x a m i n e d t o try t o illustrate these concepts. THE EXAMINATION 20 d i f f e r e n t m o d e l s describing t h e c y c l i n g o f p h o s p h o r u s in a lake are c o n sidered. T h e m o d e l s are s h o w n in Table I. Table II is a list o f state variables,
TABLE I V e r s i o n = 0 (case = 0, 1, 2, 3, 4)
i's = ( P i n - Ps) • ( Q / V )
iPs = (Pin - - P s ) " ( Q / V ) -- ( P a - - Ra) ' P a /~a = (Pa - - R a - - ( Q / V ) ) " Pa
Ps = (Pin - - P s ) " ( Q / V ) - - (//a - - R a ) " Pa + Kd ' P d Pa = (Pa - - R a - - M a - - ( Q / V ) ) . Pa Pd = M a ' Pa - - ( g d + ( Q / V ) ) . Pd
iDs = (Pin - - Ps) " ( Q / V ) - - (//a - - R a ) " Pa + K d " P d + R z " Pz Pa = (~a - - R a - - M a - - ( Q / V ) ) . Pa - - ( l / Y ) " P z " Pz i°d = M a " Pa - - ( g d + ( Q / V ) ) . Pd + (Mz + L " /,lz) • Pz Pz =
(Pz - - R z - - M z
/~s = (Pin - - P s ) "
--(Q/V))"Pz
( Q / V ) - - (Pa - - R a ) " Pa + K d " P d + R z " Pz + R~z " P'z
Pa = (Ua - - R a - - M a - - ( Q / V ) ) . Pa -- ( l / Y ) •/~z" Pz ~6d = M a " Pa - - ( g d + ( Q / V ) ) . P d + (Mz + L . / 2 z ) - P z - - (P'z - - M ' z ) " P'z /Sz = (Pz - - R z - - M z - - ( Q / V ) ) " Pz Pz = (t2'z - - R z -- M z - - ( Q / V ) ) . Pz
V e r s i o n = 1 (case = 0, 1, 2, 3, 4)
i's = ( P i n - - P s ) " ( Q / V ) + a s . K , , . P~*d i'sed = - - K x " Psed
Ps = ( P i n - - P s ) " ( Q / V ) -
(/2a-- R a ) "Pa + a s ' K x P s e d
i'a = (/la - - R a - - S a - - ( Q / V ) ) .
i'~d -- 0.4 s ~ .
Pa
P~/as - g x - P ~ d
/~s = (Pin - - P s ) " ( Q / V ) - - (/2 a - - R a ) - Pa + K d " P d + a s " K x " Psed J~a = (Pa - - R a - - Ma - - Sa - - ( Q / V ) ) " Pa i'd = M a . P a i~d
-
-
(~:,~ + ( Q / V ) )
-- 0 . 4 S a . P ~ / a s - - g x
. Pd
. P~d
Ps = (Pin - - Ps) " ( Q / V ) - - (Pa - - R a ) "Pa + K d " P d + R z " Pz + a s " K x ' Psed Pa = (/'ta - - R a - - S a - - ( Q / V ) ) i'd = - - ( g d
+ (Q/V))"Pd
" Pa - - ( l / Y ) • Pz " Pz
+ (Mz + L " P z ) " P z
/~z = (Uz - - R z - - Mz - - ( Q / V ) ) . P z Psed = 0.4 S a - P a / a s - - g x • Psed
P s = (Pin - - P s ) " ( Q / V ) - - (//a - - R a ) " Pa = (Pa - - R a - - S a - - ( V / V ' ) ) I'd = - - ( g d + ( Q / V ) ) " P d
" Pa - - ( l / Y ) " U z " P z
+ (Mz + L " / / z )
i'z = (Pz - - R z - - M z - - ( Q / V ) ) • P z • t
P~ : (U'z
t
-
R~ -- Mz
Pa + K d ' P d + R z " Pz + Rz " P'z + a s " K x " Psed
t
F
-- ( Q/ V) ) • _P~
Psed = 0.4 S a • Pa/as -- K x • Psed
" P z - - (/-t'z - - M ' z ) " P'z
V e r s i o n = 2 (0, 1, 2, 3, 4 ) ]~s = ( P i n - - P s ) " i~sed = - - g x
(Q/V) + Qd
"Psed
Jai = ( a s / a i ) g x
" Psed - - Q d / a i
/~s = ( P i n - - P s ) " ( Q / V ) - - ( # a - - R a ) " Pa + Q d Pa = (/2a - - R a - - Sa - - ( Q / V ) ) "
Pa
]~sed = 0 . 4 S a • P a / a s - - K x • Psed Pi = ( a s / a i ) K x " Psed - - Q d / a i
Ps = ( P i n - - P s ) " ( Q / V ) - - (Pa - - R a ) " P a + Q d + K d " P d ba = (pa -
R ~ - - M~ - - S a - -
(Q/V))
i6d = M a " Pa - - ( g d + ( Q / V ) ) "
- P~
Pd
i6se d = 0 . 4 S a - P a / a s - - g x • Psed iai = ( a s / a i ) g x
" Psed - - V d / a i
Ps = ( P i n - - P s ) "
(Q/V)--
(Pa - - R a ) " P a + K d " P d + R z " Pz + Q d
Pa = (~a - - R a - - Sa - - ( Q / V ) ) . Pd = - - ( g d
+ (Q/V)).
Pd+
Pa - - ( l / Y )
•/.t z • P z
(Mz + L •/.tz) - Pz
Pz = (Uz - - R z - - M z - - ( V ] Y ) ) .
Pz
Psed = 0 . 4 S a - P a / a s . K x • Psed Pi = ( a s / a i ) K x
" Psed
--
Qd/ai
Ps = ( P i n - - P s ) " ( Q / V ) - - (Ua - - R a ) " P a + K d " P d + Pa = (Pa - - R a - - S a - - ( Q / V ) ) Pd = - - ( g d
+ (Q/V))"
Pz = (/'tz - - R z - - M z t
~
" Pa - - ( l / Y )
R z " Pz + R ' z " P'z + Q d
- Uz " Pz
P d + (Mz + L • P z ) " Pz - - (U'z - - M ' z ) " P'z -- (Q/V)).
Pz
¢
t
P z = (U'z - - R z - - M z - - ( Q / V ) ) . P z Psed = 0 . 4 S a • P a / a s - - K x . Psed iP = ( a s / a i ) K x • Psed - - Q d / a i
pp. 45--52
V e r s i o n = 3 (case = 0, 1, 2, 3, 4) /~s = (Pin - - Ps) " ( Q / V ) + Q d + a b " (Qb + Qds) Psed = - - K x " Psed i)i = ( a s / a i ) K x
" Psed -- Qd/ai
/)b = - - ( Q b + Qds) /)s = (Pin - - P s ) " ( Q / V ) - -
(Pa - - R a ) "Pa + Qd + a b " (Qb + Qds)
Pa = (/2a - - R a - - Sa - - ( V / V ) ) .
Pa
Psed = 0.4 Q s / a s - - K x • P s e d Pi = (as/ai)Kx
" Psed -- Qd/ai
/~b = ( S a " Pa - - Q s ) / a b - - (Qb + Q d s ) /;s = (Pin - - P s ) " ( Q / V ) - - (Pa - - R a ) " Pa + g d " P d + Vd + ab " (Qb + Qds) /~a = ( P a -
Ra - - Ma - - Sa - - ( Q / V ) )
~Sd = M a " P a - - ( g d
• Pa
+ (Q/V))"Pd
Psed = 0.4 Q s / a s - - K x • P s e d Pi = (as/ai)Kx
" Psed -- Qd/ai
/~b = (Sa" Pa - - Q s ) / a b - - (Vb + Vds) Ps = (Pin - - P s ) " ( Q / V ) Pa = ( # a - - R a - - S a - Pd= --(Kd + (Q/V))
(Pa - - R a ) " P a + K d " P d + Rz ' Pz + ab " (Qb + Qds) + Qd -- (l/Y).
(Q/V))-Pa
P z " Pz
• P d + (Mz + L • P z ) " P z
P z = (Pz - - R z - - M z - - ( Q / V ) ) .
Pz
Psed = 0.4 V s / a s - - K x • P s e d Pi = (as/ai)Kx
" Psed - - Q d / a i
Pb = (Sa" P a - - Q s ) / a b - - (Qb + Qds)
Ps = (Pin - - P s ) " ( Q / V ) -
(Pa - - R a ) "
Pa = (/Za - - R a - - Sa - - ( Q / V ) ) . Pd= --(gd + (Q/V)).
P d + (Mz + L " U z ) " P z - - (P'z - - M ' z ) " P'z
Pz = (Pz - - R z - - Mz - - ( Q / V ) )
• Pz
P'z = (P'z - - R ' z - - M ' z - - ( Q / V ) ) .
Pz
r
Psed = 0.4 Q s / a s - - K x - P s e d Pi = (as/ai)Kx
e a + K d " P d + R z " Pz + R'z" P'z + Qd + ab " (Qb - - Qds)
P a - - ( l / Y ) • Uz" Pz
" Psed -- Qd/ai
Pb = (Sa" Pa - - V s ) / a b - - (Qb + Qds)
53 TABLE II State variables [g P/m 3 ] Symbol
Name
Ps Pa Pd P~ P~
Soluble inorganic P in water P bound in algae P in detritus P bound in algal-grazing zooplankton P bound in detritus-feeding zooplankton Exchangeable P bound in dry matter of sediment P in interstitial water in sediment P in uppermost thin layer of biologically active sediment
Psed
Pi Pb
and Tables III, IV and V define constructions used for rates, flows between sediment and water, and total phosphorus concentration in lake and sediment. Tables VI and VII provide data for thermodynamic equilibrium values and steady-state values for the state variables. The equilibrium values are based upon free energies of living matter as estimated by Morowitz (1968). Tables VIII and IX state parameter values used. The models show an increasing complexity. Version 0, case 0, 1, 2, 3, 4 does not include sediment. Versions 1, 2 and 3 consider the sediment in increasing detail (see J~brgensen et al., 1975). Case 0 considers only soluble phosphorus in the water body, while cases 1, 2, 3 and 4 include phytoplankton, detritus, algal-grazing zooplankton and detritus-feeding zooplankton, respectively. TABLE III Rates [24 h - 1 ] Name
Expression *
Decay rate of detritus
Kd = K~0 "WT
Respiration rate of algae
R a = R2a0 " f T
Respiration rate of zooplankton
R z = R2z O" f T
Respiration rate of detritus-feeding zooplankton
Rz = Rz20' f T
Growth rate of algae
Pa = f~a " f T " K p + Ps
Growth rate of zooplankton Growth rate of detritus-feeding zooplankton * f T = 1-03T--293
Ps [ Pd P'z =firz " f T " g a + p d
Pa - - K s
54 T A B L E IV F l o w s b e t w e e n s e d i m e n t and w a t e r [g p / m 3 / 2 4 h / Name
Expression
F l o w o f P released biologically f r o m the u p p e r m o s t t h i n layer (db ) o f s e d i m e n t
(0 exp[0.203(T-- 273)] Qb = / 0.563
F l o w o f P released b y d i f f u s i o n from sediment to water phase
Version
1,000d b
Sedimentation of P to the e x c h a n g e a b l e (40%) and n o n e x c h a n g e a b l e (60%) s e d i m e n t
5,687
'0
3 0 1
Qd = as " Kx ' Psed
l'2(Pi--Ps)--l'7 1,000 • d
F l o w o f P released b y d e s o r p tion from the uppermost t h i n layer (db ) o f s e d i m e n t
0--2
Pb
T 28O
2--3
Qds = - - 0 . 6 0 In Ps - - 2.27
0--2 3
/Sa" Pa Qs = ( m a x ( 0 , S a ' P a - - q s )
0--2 3
0
(
Sources: K a m p - N i e i s e n ( 1 9 7 5 ) a n d J~brgensen e t al. (1975)•
TABLE V V o l u m e ratios and t o t a l P - c o n c e n t r a t i o n Symbol
Name
Definition
ab
V o l u m e ratio: biological layer/lake v o l u m e
db a b = DMU • ~ -
ai
V o l u m e ratio: interstitial w a t e r / l a k e v o l u m e
a i = (1 - - DMU) " ~ -
V o l u m e ratio : dry s e d i m e n t / l a k e v o l u m e
Gs
Total m e a n P - c o n c e n t r a t i o n
Pt
d S
a s = DMU • d_~s d P t = Ps + Pa + Pd + Pz + P" + as" Psed + ai ' Pi + ab " Pb *
* Only in t h e m o s t general case are all t e r m s non-zero• By i n s p e c t i o n o f t h e differential e q u a t i o n s it is seen t h a t :
bt
(Pin - - Ps - - Pa - - Pd - - Pz - - P'z )" Q -- 0.6 Qs
(la)
in t h e m o s t general case. N o t e t h a t ( l a ) d e g e n e r a t e s t o /~t = ( P i n - - P t ) " Q
if version = 0 (no s e d i m e n t ) .
(lb)
55 Table VI T h e r m o d y n a m i c equilibrium concentrations: p ~ P~q
Value
[g P/m s ]
Based u p o n
Pseq
Pin
b o u n d a r y conditions
peq
10 - 5 0
Morowitz (1968)
P~q
10 - 4 6
Morowitz (1968)
peq
10 - 5 0
Morowitz ( 1968 )
p~eq eq Psed
10--50 10 ---46
Morowitz (1968)
peq P~q
Pin + 1 27 10 - 4 6
Q~q = 0 A Pseq = Pin Morowitz (1968)
Morowitz (1968)
The extreme of this spectrum of models is then a lake considered to be just a tank w i t h o u t chemical and biological reactions, and a model with eight compartments including a three-compartment sediment submodel. The 20 models are intercalibrated to give the same steady-state concentrations TABLE VII Steady-state concentrations: pO [g p / m 3 ] P~j
Value
Base.d u p o n
p0
depends o n case and version
calibration
p0
0.1
choice
p0
0.05
choice
p0
0.01
choice
Pz0
0.01
choice
ps0ed
( 400 or:
choice
~ 400 - - a b " pO
choice and mass balance
(ve ion%)
p0
p0
60.0 (
choice 2.27 + 0.6 In Pin
{ db" --~9~ ~ /
I, or: 5687 *
- fT
calibration calibration
* The expression is used for case 0, the~literal, which is the value of the expression w h e n Pin = 1.5, d b = 0.0018 and T = 283.15, is used in all other cases, fT stands for: e x p [ 0 . 2 0 3 ( T - 273)]
56 TABLE VIII Parameters n o t found by calibration Symbol d db ds DMU Ka Kp Ks Ma Me Mz Pin Q R Ra20 T V Y L
Name Average depth of water phase Thickness of biologically active layer Thickness of upper layer of sediment containing exchangeable P Fraction of dry matter in upper layer Michaelis-constant in zooplankton-feeding expressions Michaelis-constant for P-uptake in algae Steele-constant in grazing expression Mortality of algae Mortality of zooplankton Mortality of detritus-feeding zooplankton Concentration of soluble P in inflow Flow of water through the lake Gas constant Respiration rate of algae at 20°C Mean temperature Volume of water body Yield factor Loss factor
Value
Unit
10
m
1.8 X 10 - 3
m
0.1
m
0.1--0.9 5 X 10 -3
g P/m 3
0.02
g P/m 3
5 × 10 - 3 0 *, 0.025 **, 0.05 J" 0.01
g P/m 3 24 h - 1 24 h - 1
0.01
24 h--1
1.5 5,000 0.2683 x 10 - 3 0.35 283.15 450,000 0.67
g P/m 3 m3/24 h kJ/K/g P 24 h - 1 K m3
1/Y-
1
* cases 3 and 4, version > 0 ** cases 3 and 4, version = 0 and case 2, version > 0 t" case 2, version O.
o
(except Ps), when driving functions (loading, temperature and water flow) are held constant. T h e f o l l o w i n g s i t u a t i o n is s i m u l a t e d : B y u s i n g a l a k e m o d e l , i t is p o s s i b l e t o p r e d i c t t h e c o n s e q u e n c e s o f a c h a n g e d p h o s p h o r o u s l o a d i n g (Pin). T h e s t e a d y - s t a t e v a l u e s o f s o l u b l e p h o s p h o r u s i n t h e l a k e (p0) a n d t h e t o t a l p h o s . p h o r u s in l a k e a n d s e d i m e n t (P~t) as a c o n s e q u e n c e o f t h e l o a d i n g a r e c a l c u lated. Two kinds of buffer capacities have been considered:
/3 = P~t
and
/3' - P i n
T h e q u e s t i o n is: h o w a r e / 3 a n d / 3 ' d e p e n d e n t u p o n t h e c o m p l e x i t y o f t h e m o d e l ? W e d o k n o w t h a t e v e n t h e m o s t c o m p l i c a t e d m o d e l is a s i m p l i f i e d d e s c r i p t i o n o f t h e e c o s y s t e m , b u t as w e c a l i b r a t e t o o b t a i n e.g. t h e r i g h t
57 T A B L E IX Parameters f o u n d by calibration (DMU = 0.93) Symbol
Name
Kd~°
Degradation rate of detritus at 20°C Degradation rate of P in sediment Specific growth rate of p h y t o p l a n k t o n at 20°C Specific growth rate of algalgrazing z o o p l a n k t o n at 20°C Specific growth rate of detritus-feeding z o o p l a n k t o n at
gx ~a hz Pz R20 Rrz20
Sa qs
20°C Respiration rate of algalgrazing zooplankton at 20°C Respiration rate of detritus-feeding zooplankton at 20°C Settling rate of phytoplankton Upper limit of flow to biologically active layer (db)
Value
Unit
Based u p o n
0.04--0.12 *
24 h - 1
/)d = 0 **
2 × 10 - 3
24 h - 1
~5i = 0
0.37--0.74 *
24 h - 1
iSa = 0
0.72
24 h - 1
Pd = 0 1"
0.09--0.16 *
24 h - 1
/~d = 0 **
0.62
24 h - 1
/~z = 0
0.05--0.12 * 0.18
24 h - 1 24 h - 1
~5' .Z ~ 0 )Pt.= 0
10 - 4
g p/m3/24 h /as~0 s e d + aiPi
.
i~b = 0 * Depending on case and version. ** and the assumption that (a) only half of the m o r t a l i t y takes place in the water phase, w h e n s e d i m e n t is present, and that (b) the net f l o w through detritus-feeding z o o p l a n k t o n (when present) is one f o u r t h o f the flow f r o m Pd to Ps. j- and the assumption that half of algal m o r t a l i t y flow is interpreted as z o o p l a n k t o n m o r t a l i t y plus loss during grazing.
p h y t o p l a n k t o n value, the model will of course give the right level of phytoplankton under unchanged conditions. If we have no occasion to validate the model under changed conditions, which is most often the case, the calibration will n o t be able to tell us which model gives the best description under changed conditions. However, if the ecological buffer capacity is changed by increased complexity, it seems necessary to choose at least a model with a certain complexity to get a reasonably correct response under changed conditions. Shannons index H and the exergy E, are used as measurements for the complexity. The exergy measures the mechanical energy equivalent of the distance from thermodynamic equilibrium. The mechanical energy equivalent is defined as the m a x i m u m a m o u n t of entropy-free energy, which can be produced upon establishment of equilibrium with the surroundings of the system. As there is an important link between concepts of information theory and the exergy, it seems to be a useful measurement of the complexity. H and E are defined here as shown in Table X.
58 TABLE X S o m e indices of stability and c o m p l e x i t y Name
Definition
Buffer capacities:
j3 = ~Pt ; ~t = Pin Ps
Shannon-index:
H = .
•i
Unit
Pt
aj • ~t" i°g2 ai "Pj '
where Pt -- ~ a j
bit
• P~
i Exergy
E = R " T" ~ ai .(Pi . ln P1 --(Pi--PTq)) j ~q
kJ/m 3
T i m e derivative:
i= R. T.~ai.
i
A Liapunov function:
L =R. T ' ~ a j "
i i . ln Py
Pp (J'j "ln--~--(P~-" l,o~) Pj
kJ/m3/24 h
kJ/m 3
T i m e derivative:
i = R " T "~_/ aj " i~, " ln ~
kJ/m3/24 h
(a i } is a set of v o l u m e ratios (only different f r o m 1 for s e d i m e n t variables).
Note the inequalities:
/3 > 1, ~' > 1. 0 < H < log 2 n (n = n u m b e r o f state variables)
E> 0 (E=O*~foralljPj=P~) L > 0 (L
0*=~foralljPy=P)).
The questions which this examination attempts to answer can be summarized as follows: (1) H o w will ~ and ~' vary with increasing compexity of the model, using (a) Shannons index, (b) exergy, as a measurement of the complexity? (2) Is it possible to determine the complexity the model must have to give a reasonably good description of the response under changed conditions? RESULTS
Figs. 1--4 give the results of the examination; corresponding values of and fi' versus E and H are plotted and a regression analysis is performed. In order to obtain a reasonable number of points between points belonging to models with and w i t h o u t sediment, the sediment was " d i l u t e d " for
59
each of the 20 models in nine steps by assigning the values 0.1, 0.2 .... , 0 . 9 to the relative amounts of dry matter in the sediment (DMU). This gives 20 X 9 = 180 points. Only ~ versus E shows a good correlation (r = 0.98). Inclusion of the sediment in the model gives a significant rise in the buffer capacities and the exergy, as the sediment has a large storage capacity for phosphorus. Furthermore, the buffer capacities increase with an increasing number of state variables, the Shannon index and the exergy. ABSOLUTE STABILITY
An a t t e m p t has been made to find a Liapunov function for the models examined. Inspired by the exergy expression, the Liapunov function shown in Table X was considered. Since L always is non-negative, it follows from Liapunov's theory that stability exists if the time derivative L < 0 outside steady state. Computer experiments show that all the models considered witho u t zooplankton are globally stable; inclusion of one or two types of zooplankton gives self-sustained limit cycles. DISCUSSION
When a model is applied as a tool for prognosis, it is of importance that it gives the correct buffer capacity needed to meet the changes in the driving functions. An increasing diversity, expressed by either the number of state variables or by use of Shannon's index, gives an increasing buffer capacity in most cases, b u t as the diversity does n o t express the significance of the mass flows by ignoring the fact that living systems are a long way away from thermodynamic equilibrium, it does n o t seem to be a good expression for the buffer capacity. This is reflected in the calculations as a low correlation between H and ~ ( or fi'). However, the exergy, which does consider the mass flows is a far better expression for the buffer capacity. As can be seen, it is important to include the reactions in the sediment, since a considerable a m o u n t of the mass flow (in the case studied the flow of phosphorus) goes through the sediment, while it is of less significance to include one or two zooplankton species. Consequently, it is of importance to set up a submodel for the mud-water exchange of phosphorus, giving a good description of the actual mass flow, such as has been attempted by the work of Kamp-Nielsen (1975), J~brgensen et al. {1975) and Jacobsen and J~brgensen (1975). It is of importance to note, that all the models give the same level of p h y t o p l a n k t o n , detritus, zooplankton etc., and it is n o t possible to distinguish between one model and another on the basis of measurements of these state variables. The value of ~a found by calibration increases with increasing model complexity, b u t the value in all cases is considerably lower than the usual value applied {see e.g. Chen and Orlob, 1974; J~brgensen, 1976). The reason is that
60 the Pa applied here includes the influence of light, including an integration over the depth. It has been calculated, that a pa of 0.6/24 h here would have been close to 2.0/24 h if the usual light expression (see J~brgensen, 1976) had been applied, and this value is in accordance with the literature values (see also Patten et al., 1975). By application of a Liapunov function it was possible to show that increased complexity does n o t mean increased absolute stability. The more complicated models are rather less stable. On the basis of the examination carried out it seems to be possible to select the state variables needed in a model built to answer certain questions. The selection should be based upon an examination of the actual state variables and their contribution to the exergy or buffer capacity. This conclusion will, however, require more case studies before it can be used as a general principle, although the principle has been discussed generally above. CONCLUSION A situation where models of increasing complexity were calibrated to give the same level of p h y t o p l a n k t o n , detritus, zooplankton etc. was considered. The buffer capacity, fl or fi', of the different models was found and it is seen that fi (and less pronounced ~') increases with increasing complexity. In other words, it seems necessary to have a model of a certain complexity to be able to describe a response to changes in the driving functions. It is of great importance to include the essential mass flows. When phosphorus is considered, the sediment plays an essential role -- at least in shallow lakes. Although increasing diversity {expressed by means of the Shannon index or the number of state variables) gives an increasing buffer capacity, it does not correlate well with the buffer capacity, as it does not take the mass flows into consideration. However the exergy, which measures the distance from t h e r m o d y n a m i c equilibrium gives an excellent expression for the buffer capacity, and the exergy should, rather than the diversity, be used as an expression for the response of an ecosystem to changes in the driving functions. The exergy or the ecological buffer capacity are useful concepts for selection of state variables for a given set of models. By application of a Liapunov function it was shown t h a t the absolute stability was unchanged or decreased with increased complexity of the model. However, it must n o t be forgotten that the model is a rough simplification of the real world and that the models examined do n o t contain too m a n y feed-back mechanisms, which might increase the stability considerably.
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