H E L G O L ~ N D E R M E E R E S U N T E R S U C H U N G E N Helgol~inder M e e r e s u n t e r s . 34, 221-235 (1980)
M a t h e m a t i c a l m o d e l l i n g - a m a n a g e m e n t t o o l for
a q u a t i c e c o s y s t e m s ?
K. F e d r a
International Institute for Applied Systems Analysis;
SchloB Laxenburg, A-2361 Laxenburff, Austria
ABSTRACT: M a t h e m a t i c a l m o d e l l i n g m a y serve as a r a t i o n a l a n d powerful tool in the m a n a g e m e n t of complex ecosystems. However, e c o s y s t e m m o d e l s are drastic simplifications of the real world. As a rule they are b a s e d on a r a t h e r i n c o m p l e t e a n d s c a t t e r e d k n o w l e d g e of the system in question. Furthermore, ecological systems a n d in p a r t i c u l a r m a r i n e systems are c h a r a c t e r i s e d b y a h i g h d e g r e e of complexity, spatial a n d f u n c t i o n a l h e t e r o g e n e i t y , nonlinearity, c o m p l e x b e h a v i o u r a l features s u c h as a d p t a t i o n a n d self-organisation, a n d a c o n s i d e r a b l e stochastic e l e m e n t . N e v e r t h e - less, if m a n a g e m e n t is to b e b a s e d on predictions from m a t h e m a t i c a l m o d e l s - a n d it h a s to b e b a s e d on some kind of " m o d e l " in at least a b r o a d s e n s e - w e n e e d a n e s t i m a t e of p r e d i c t i o n accuracy in terms of the m a n a g e m e n t v a r i a b l e s a n d constraints. O n e p o s s i b l e a p p r o a c h to m o d e l u n c e r t a i n t y is a probabilistic i n t e r p r e t a t i o n of m o d e l predictions, g e n e r a t e d b y use of M o n t e - C a r l o t e c h n i q u e s . Fuzzy data sets a n d r a n g e s are used. The r e s u l t i n g m o d e l r e s p o n s e allows the d e r i v a t i o n of m e a s u r e s for m o d e l credibility. Probability distributions can b e c o m p u t e d for certain system states u n d e r (un)certain i n p u t conditions, r e p r e s e n t i n g the effects of insufficient d a t a a n d structural u n c e r t a i n t y on m o d e l - b a s e d predictions. Such analysis i n d i c a t e s t h a t p r e d i c t i o n u n c e r t a i n t y increases, not only w i t h the u n c e r t a i n t y in the data, b u t also w i t h i n c r e a s i n g " d i s t a n c e " from the empirical conditions, a n d with time. Present ecoystem m o d e l s c a n b e a tool for q u a l i t a t i v e discrimination b e t w e e n different m a n a g e m e n t alternatives, r a t h e r t h a n a c r e d i b l e m e a n s for d e t a i l e d q u a n t i t a t i v e predictions of system r e s p o n s e to a w i d e r a n g e of i n p u t conditions.
E C O S Y S T E M M O D E L L I N G I N A M A N A G E M E N T F R A M E W O R K
T h e r e is g r o w i n g a w a r e n e s s t h a t w e n e e d s o m e k i n d of e n v i r o n m e n t a l m a n a g e m e n t a n d q u a l i t y c o n t r o l i n o r d e r to " b a l a n c e " t h e i m p a c t of o u r a c t i v i t i e s o n t h e n a t u r a l e n v i r o n m e n t . M a n ' s d e s t r u c t i v e c a p a c i t y h a s r e a c h e d a l e v e l w h e r e l o c a l c a t a s t r o p h i e s a r e o n l y t h e m o s t o b v i o u s s i g n a l s of a c o n t i n u o u s p r o c e s s of g e n e r a l a n d w o r l d w i d e e n v i r o n m e n t a l d e g r a d a t i o n . T h i s m i g h t of c o u r s e b e u n d e r s t o o d a s o n l y t h e s y m p t o m s of m o r e s u b s t a n t i a l p r o b l e m s i n o u r g r o w t h - o r i e n t e d w o r l d (c.f. M e a d o w s e t al., 1972); h o w e v e r , t h e s y m p t o m s a r e i m m i n e n t a n d i m m e d i a t e d e c i s i o n s h a v e to b e m a d e .
S i n c e d e c i s i o n s a r e b e i n g m a d e , w e m i g h t a s k h o w t h i s is d o n e a n d o n w h a t s c i e n t i f i c b a s i s . D e c i s i o n s a r e m a d e o n t h e b a s i s of s o m e i m p l i c i t o r e x p l i c i t m o d e l d e v e l o p e d of t h e s y s t e m s to b e a f f e c t e d b y t h e d e c i s i o n . E n v i r o n m e n t a l m a n a g e m e n t d e c i s i o n s a r e g e n e r a l l y m a d e t o m i n i m i z e , or a t l e a s t to r e d u c e , o v e r a l l c o s t / b e n e f i t r a t i o s i n a m u l t i - o b j e c t i v e , m u l t i - u s e r s y s t e m . T h i s i n c l u d e s t h e m a i n t e n a n c e of c e r t a i n e n v i r o n m e n t a l q u a l i t y s t a n d a r d s . P a r t of a n e n v i r o n m e n t a l m a n a g e m e n t m o d e l m u s t b e a n e c o s y s t e m s s u b m o d e l , w h i c h p r e d i c t s t h e s y s t e m r e s p o n s e i n t e r m s of q u a l i t y
222 K. F e d r a
v a r i a b l e s to p o t e n t i a l m a n a g e m e n t actions. This s u b m o d e l is l i n k e d to the m a n a g e m e n t m o d e l , w h i c h looks for t h e o p t i m a l s t r a t e g y in t h e m u l t i o b j e c t i v e f r a m e w o r k . " O p t i m a l " is d e f i n e d by s o m e k i n d of g o a l function, u s u a l l y r e l a t e d to m o n e t a r y v a l u e s .
A l t h o u g h t h e r e is a v a s t a n d c o n t i n u o u s l y g r o w i n g l i t e r a t u r e on e c o l o g i c a l m o d e l - ling, this s e e m i n g l y d o e s not i n f l u e n c e m a n a g e m e n t d e c i s i o n s to a c o n s i d e r a b l e d e g r e e . In M a s o n ' s (1979) b o o k " T h e E f f e c t i v e M a n a g e m e n t of R e s o u r c e s " on the N o r t h Sea, the t e r m " e c o l o g y " is not e v e n m e n t i o n e d ; r e g u l a t i o n of p o l l u t i o n , o n e of the chapters, is t r e a t e d as a legal, e c o n o m i c , a n d p o l i t i c a l (in a v e r y n a r r o w sense) p r o b l e m .
Of c o u r s e t h e r e are s o m e e x c e p t i o n a l e x a m p l e s of e f f e c t i v e d i a l o g u e s b e t w e e n d e c i s i o n m a k e r s a n d e c o l o g i s t s or e c o s y s t e m a n a l y s t s (Clark et al., 1979), but r a r e l y so for the m a r i n e e n v i r o n m e n t . S c i e n t i s t s m a y w e l l b e a s k e d to c o n t r i b u t e t h e i r k n o w l e d g e , but m o r e often t h a n not t h e i r r e p o r t s are l o c k e d a w a y in s o m e d r a w e r s r a t h e r t h a n a f f e c t i n g t h e m a n a g e m e n t d e c i s i o n s . This is a v e r y u n s a t i s f a c t o r y situation, not l e a s t for the ecologist. For a d i s c u s s i o n of this d i l e m m a s e e e.g. Biswas (1975).
To r e t u r n to t h e a b o v e - d e s c r i b e d e c o s y s t e m m o d e l n e e d e d for r a t i o n a l e c o s y s t e m m a n a g e m e n t , t h e m a n a g e r a n d d e c i s i o n m a k e r w o u l d l i k e to h a v e a m o d e l he c a n u n d e r s t a n d , w h i c h c a n b e p r o v e n to b e r e l i a b l e (cf. the d i s c u s s i o n transcripts in Russel, 1975) - a n d h e w o u l d l i k e to h a v e it r i g h t now.
D A T A A N D M O D E L S
In o r d e r to b u i l d a m a t h e m a t i c a l m o d e l - w h i c h is o n l y a special, formal, case of b u i l d i n g a s c i e n t i f i c t h e o r y - w e u s u a l l y o r g a n i z e t h e e m p i r i c a l i n f o r m a t i o n a v a i l a b l e in s u c h a w a y t h a t part of it is u s e d to s t r u c t u r e t h e m o d e l , i.e. to s p e c i f y the e l e m e n t s of the m o d e l a n d t h e i r relations. A n o t h e r part of t h e i n f o r m a t i o n is u s e d as i n p u t (in a c y b e r n e t i c sense) to t h e m o d e l structure: this c o m p r i s e s the coefficients, w h i c h q u a n t i t a - t i v e l y d e s c r i b e the i n t e r n a l r e l a t i o n s . In t h e c a s e of a d y n a m i c m o d e l , this also i n c l u d e s t h e forcings, i m p o r t s a n d exports, w h i c h r e p r e s e n t t h e b o r d e r - c r o s s i n g r e l a t i o n s c o n n e c t - i n g the m o d e l w i t h its e n v i r o n m e n t , a n d f i n a l l y t h e i n i t i a l state of the m o d e l , a s s i g n i n g s t a r t i n g v a l u e s to all of t h e e l e m e n t s . A t h i r d part of the a v a i l a b l e i n f o r m a t i o n is t h e n u s e d to c o m p a r e t h e m o d e l r e s p o n s e w i t h w h a t w e h a v e o b s e r v e d in the r e a l w o r l d system, a n d b y r e c u r s i v e a d j u s t m e n t s of t h e m o d e l s t r u c t u r e a n d c o e f f i c i e n t s w e a t t e m p t to i m p r o v e the m o d e l p e r f o r m a n c e . H o w e v e r , t h e a s s i g n m e n t of a n y p i e c e of i n f o r m a t i o n to a n y of t h e a b o v e g r o u p s is arbitrary. S t r u c t u r e a n d c o e f f i c i e n t s c a n n o t b e e s t i m a t e d i n d e p e n d e n t l y , a n d an o p e r a t i o n a l m o d e l consists of m a n y m o r e s i m p l i f y i n g a s s u m p - tions t h a n p r o p e r t i e s d i r e c t l y m e a s u r a b l e in the r e a l - w o r l d system.
M o d e l l i n g - a m a n a g e m e n t tool? 223 n e t w o r k s a n d the f r i c t i o n l e s s p e n d u l u m . All t h e s e i n c o m p a t i b i l i t i e s c a n b e o v e r c o m e only w i t h n u m e r o u s a r b i t r a r y a s s u m p t i o n s , o f t e n e n o u g h i m p l i c i t l y h i d d e n in a m o d e l formulation. T h e i n f o r m a t i o n w e h a v e at h a n d is of a j i g s a w p u z z l e structure, a n d at b e s t w e c a n d e d u c e fuzzy patterns, s e m i q u a n t i t a t i v e r e l a t i o n s , r a n g e s , a n d c o n s t r a i n t c o n d i - tion, u n l e s s w e b l i n d l y b e l i e v e in n u m b e r s .
Clearly, u n d e r t h e a b o v e c o n s t r a i n t s t h e t r a d i t i o n a l d e t e r m i n i s t i c t e c h n i q u e s , u s i n g m e a n s , a v e r a g e s , a s s u m p t i o n s on h o m o g e n e i t y a n d error distributions, a n d a firm b e l i e f in n u m b e r s h a v e to b e q u e s t i o n e d . F o r c i n g e c o l o g i c a l s y s t e m s into a m a t h e m a t i c a l f r a m e w o r k , d e v e l o p e d for v a s t l y d i f f e r e n t systems, for the s a k e of e a s e a n d e l e g a n c e of t h e analysis, s e e m s to m e n o t o n l y a futile b u t also d a n g e r o u s l i n e of work. As a c o n s e q u e n c e , m a n y m o d e l b a s e d p r e d i c t i o n s on e c o s y s t e m s are e i t h e r trivial or false, or at b e s t c o m p u t e r i z e d i n t u i t i o n of the analyst. T h e r e f o r e w e s h o u l d not b e s u r p r i s e d to s e e e c o s y s t e m s a n a l y s i s w i t h o n l y m e a g e r i m p a c t o n e n v i r o n m e n t a l d e c i s i o n m a k i n g a n d p u b l i c r e a s o n i n g .
T H E M E T H O D O L O G Y O F T H E A P P R O A C H
T h e r e s e e m s to b e no s c i e n t i f i c w a y to p r e d i c t t h e f u t u r e of a s y s t e m u n l e s s w e h a v e , b a s e d on a w e l l e s t a b l i s h e d theory, sufficient i n f o r m a t i o n on all t h e p r o c e s s e s a n d m e c h a n i s m s d e t e r m i n i n g the s y s t e m ' s e v o l u t i o n . C l e a r l y , w e are far from t h a t p o s i t i o n in e c o s y s t e m s r e s e a r c h , a n d it m i g h t w e l l b e q u e s t i o n e d , w h e t h e r s u c h a p o s i t i o n c a n b e r e a c h e d in p r i n c i p l e . To predict, h o w e v e r , t h e f u t u r e s t a t e of a system, w e h a v e to e x t r a p o l a t e our p r e s e n t k n o w l e d g e in t i m e as w e l l as in state s p a c e - the latter to e s t i m a t e the systems" r e s p o n s e u n d e r c o n d i t i o n s not y e t e x p e r i e n c e d . T h i s is not too b a d a s t r a t e g y - in fact o n e c o u l d say it is a b a s i c s t r a t e g y of l i v i n g systems; h o w e v e r , this s t r a t e g y is b a s e d on s o m e m o s t s t r i n g e n t a s s u m p t i o n s : first, the p r e s e n t , d e s c r i p t i v e m o d e l of the s y s t e m m u s t b e g o o d e n o u g h ; second, t h e " d i s t a n c e " of p r o j e c t i o n in t i m e as w e l l as in state s p a c e m u s t b e r e l a t e d to t h e u n p r e d i c t a b l e (in a d e t e r m i n i s t i c sense) e l e m e n t of v a r i a b i l i t y in t h e system. This s t r a t e g y of p r e d i c t i o n r e q u i r e s a l a r g e n u m b e r of a s s u m p t i o n s , a n d to i m p r o v e the s c i e n t i f i c q u a l i t y - a n d c r e d i b i l i t y - of m o d e l - b a s e d p r e d i c t i o n s , t h e s e a s s u m p t i o n s h a v e to b e m a d e explicit. W h e r e v e r p o s s i b l e , t h e effect of such a s s u m p t i o n s on m o d e l p r e d i c t i o n s s h o u l d b e e x p l o r e d , so that u n c e r t a i n t y a b o u t t h e system in q u e s t i o n w i l l b e an e x p l i c i t p a r t of the n u m e r i c a l analysis. S u c h a n a l y s i s no l o n g e r results in o n e s i n g l e a n s w e r - a s i n g l e n u m b e r , p r e t e n d i n g a most u n r e a l i s t i c k i n d of p r e c i s i o n - but a r a n g e of p o s s i b l e a n s w e r s . T h e p r o b a b i l i t y d i s t r i b u t i o n s w i t h i n s u c h r a n g e s w i l l p a r t l y d e p e n d on the q u a l i t y of o u r k n o w l e d g e , b u t also on t h e d e g r e e of e x t r a p o l a t i o n in t i m e a n d state space, as w i l l b e s h o w n b e l o w .
224 K. F e d r a
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RESPONSE SCENARIO 1 2 INPUT SCENARIO 1 2
Fig. 1. Schematic summary of the approach, a two-dimensional example. (A) raw input-data space, defined by empirical ranges; (B) model response corresponding to A, indicating the empirical range of response patterns (compare Figs, 2, 3}; (C) input-data space sub-region corresponding to the empirical range in B (compare Fig. 4); (D) Input scenarios, representing e.g. management alternati-
ves; (E) model response to D (compare Fig, 5)
Instead, o n e c a n u s e a m o r e fuzzy d e s c r i p t i o n of the e m p i r i c a l s y s t e m b e h a v i o u r in t e r m s of r a n g e s for e a c h of t h e o b s e r v e d v a r i a b l e s (Fig. 1 B). A n y such r a n g e is of course m o r e r e l i a b l e t h a n the m o r e or less a r b i t r a r y a s s i g n m e n t of o n e s i n g l e n u m b e r for a n y m e a s u r e d e s c r i b i n g t h e system, a n d thus less critical from t h e c r e d i b i l i t y p o i n t of v i e w : e v e n if o b j e c t i v e c r i t e r i a for t h e e x a c t d e l i m i t a t i o n of such r a n g e s are l a c k i n g , a n d a g a i n s o m e w h a t a r b i t r a r y a s s u m p t i o n s h a v e to b e m a d e , it s e e m s m o r e l i k e l y that an i n t e r s u b - j e c t i v e a g r e e m e n t c a n b e a c h i e v e d o n a r a n g e for a p a r t i c u l a r v a l u e , r a t h e r t h a n on o n e s i n g l e n u m b e r . E a c h of t h e s e r a n g e s , u s e d as a c o n s t r a i n t c o n d i t i o n o n t h e a l l o w a b l e r e s p o n s e p a t t e r n of a m o d e l , w h i c h r e p r e s e n t s the e m p i r i c a l r a n g e of s y s t e m b e h a v i o u r , n o w a l l o w s a n i n f i n i t e set of r e s p o n s e . But t h e c o m b i n a t i o n of m a n y c o n s t r a i n i n g r a n g e s a n d r e l a t i o n s q u i c k l y r e d u c e s t h e a l l o w a b l e patterns. T h e r e is still a n i n f i n i t e n u m b e r of t h e m , of course, b u t by a s u f f i c i e n t n u m b e r of c o n s t r a i n t c o n d i t i o n s w e can d e l i m i t a m e a n i n g f u l r e g i o n in t h e r e s p o n s e s p a c e of the m o d e l .
M o d e l l i n g - a m a n a g e m e n t tool? 225
u n c e r t a i n t y , a n d are b o u n d e d b y limits of p l a u s i b i l i t y , d e d u c e d from the a v a i l a b l e e m p i r i c a l k n o w l e d g e . T h e first step i n the n u m e r i c a l a n a l y s i s is t h e n to r a n d o m l y s a m p l e this i n p u t - s p a c e region, a n d g e n e r a t e a m o d e l r e s p o n s e for e a c h of t h e s e s a m p l e i n p u t - vectors. T h e s e s a m p l e vectors a r e t h e n classified a c c o r d i n g to w h e t h e r the r e s u l t i n g m o d e l r e p o n s e is w i t h i n the r e g i o n of p l a u s i b l e or e m p i r i c a l b e h a v i o u r or n o t (Fig. 1C). Thus, b y r e f e r e n c e to t h e e x p e c t e d m o d e l b e h a v i o u r , a s e g r e g a t i o n of t h e initial, c r u d e i n p u t space is a t t e m p t e d . This is n e c e s s a r y , as t h e d e f i n i t i o n of the i n p u t s p a c e d i d n o t consider the r e l a t i o n s a n d d e p e n d e n c i e s b e t w e e n t h e i n p u t data: e v e n w i t h i n p l a u s i b l e r a n g e s for each v a l u e , u n r e a l i s t i c c o m b i n a t i o n s m a y b e s a m p l e d . H o w e v e r , as our k n o w l e d g e a b o u t t h e s e d e p e n d e n c i e s a n d c o r r e l a t i o n s is g e n e r a l l y e v e n more s p e c u l a - tive t h a n the k n o w l e d g e o n the s i n g l e v a l u e s e m p i r i c a l r a n g e s , the a-priori s p e c i f i c a t i o n of i n p u t - s p a c e structure will o n l y b e p o s s i b l e i n a few cases.
A m u l t i t u d e of b e h a v i o u r g e n e r a t i n g i n p u t - v e c t o r s c a n thus b e g e n e r a t e d . T h e i r v a r i a b i l i t y r e p r e s e n t s the u n c e r t a i n t y i n t h e b a s i c a s s u m p t i o n s , a n d a n a n a l y s i s of t h e i r structural r e l a t i o n s allows some c o n c l u s i o n s o n the a d e q u a c y of the m o d e l chosen. For predictions, w h e r e o n e or m o r e i n p u t - d a t a are c h a n g e d , the w h o l e set of i n p u t - v e c t o r s is used, as e a c h of t h e m is a p l a u s i b l e s o l u t i o n of the m o d e l a d j u s t m e n t (or c a l i b r a t i o n ) p r o c e d u r e i n the light of the b a s i c u n c e r t a i n t y (Fig. 1D). C o n s e q u e n t l y , e a c h c h a n g e i n the i n p u t c o n d i t i o n s for the m o d e l , e a c h s c e n a r i o or a l t e r n a t i v e to b e explored, results i n a m u l t i t u d e of answers, w h i c h c a n b e i n t e r p r e t e d i n a p r o b a b i l i s t i c w a y (Fig. 1E). Thus, the r e s o l u t i o n or s e n s i t i v i t y of a m o d e l i n p r e d i c t i n g t h e s y s t e m s r e s p o n s e to i n p u t c h a n g e s is a n explicit part of t h e forecast. T h e a p p r o a c h is s c h e m a t i c a l l y s u m m a r i z e d i n F i g u r e 1. For a d e t a i l e d d e s c r i p t i o n of t h e n u m e r i c a l m e t h o d s i n v o l v e d , see F e d r a (1979b).
RESULTS A N D EXPERIENCES
T h r e e s i m u l a t i o n m o d e l s w i t h l a r g e l y different structure a n d characteristics w e r e u s e d to study the a b o v e a p p r o a c h : a trophic state p r e d i c t i o n m o d e l for stratified w a t e r bodies, b a s e d o n the l a k e p h o s p h o r u s m o d e l of I m b o d e n & G~ichter (1978), a m o d i f i e d v e r s i o n of a p e l a g i c food c h a i n m o d e l (Steele, 1974), a n d a p r i m a r y p r o d u c t i o n m o d e l for shallow, u n s t r a t i f i e d w a t e r b o d i e s such as l a g o o n s or s h a l l o w lakes. In e a c h case, a n a p p r o p r i a t e data set w a s e s t a b l i s h e d from literature, s p e c i f y i n g r a n g e s for i n p u t data a n d the c o r r e s p o n d i n g e m p i r i c a l b e h a v i o u r for a g i v e n ecosystem. A l m o s t 100 000 r u n s of these three models, w e r e performed, e x t e n d i n g the s i m u l a t i o n s for u p to 10 years, i n order to o b t a i n a sufficient n u m b e r of r u n s for e a c h m o d e l / s c e n a r i o for the f i n a l a n a l y s i s .
As a c o m m o n f e a t u r e of all three models, the p e r c e n t a g e of successful r u n s i n terms of the e m p i r i c a l b e h a v i o u r c o n s t r a i n t (Figs 2, 3) w a s a r o u n d 1 to 3 %. T h e effect of i n c r e a s i n g "know.ledge" a b o u t the system, r e s u l t i n g i n a r e d u c t i o n of i n p u t d a t a r a n g e s , is s h o w n i n F i g u r e 3: i n c r e a s i n g k n o w l e d g e i n c r e a s e d the p r o b a b i l i t y for a s u c c e s s f u l r u n g i v i n g rise to the e x p e c t e d m o d e l b e h a v i o u r . T h e p e r c e n t a g e of successful r u n s w a s a b o u t 15 % for a 80 % s y m m e t r i c s u b r e g i o n of the i n i t i a l i n p u t - d a t a space.
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i n p u t d a t a c o m p r i s e p a r a m e t e r s s e n s u stricto, i m p o r t s a n d forcings, a n d i n i t i a l condi- tions. T w o e x a m p l e s of p r o j e c t i o n s of s u c h a n i n p u t - d a t a space are s h o w n i n F i g u r e 4. T h e a-posteriori d i s t r i b u t i o n s w i t h i n the o r i g i n a l r a n g e s s a m p l e d ( a s s u m i n g r e c t a n g u l a r p r o b a b i l i t y d e n s i t y f u n c t i o n s a-priori) give some i n d i c a t i o n of m o d e l s e n s i t i v i t y to the i n p u t - d a t a u n d e r the b e h a v i o u r a l constraints. Fig. 4a gives a n e x a m p l e of g o v e r n i n g p a r a m e t e r s , s i g n i f i c a n t l y correlated, w h e r e a s Fig. 4b shows r a t h e r i n d e p e n d e n t p a r a m e - ters, more or less u n i f o r m l y d i s t r i b u t e d . In g e n e r a l , the m o d e l b e h a v i o u r is m u c h more g o v e r n e d b y the c o m b i n a t i o n s a n d r e l a t i v e ratios of the i n p u t - d a t a t h a n their i n d i v i d u a l , a b s o l u t e v a l u e s . This r e s u l t s i n the m a r k e d c o r r e l a t i o n structure of the i n p u t - d a t a space for a g i v e n r a n g e of m o d e l r e s p o n s e . T h i s also i n d i c a t e s , that the d e t e r m i n i s t i c calibra- t i o n of more t h a n o n e p a r a m e t e r is a n a l m o s t futile exercise (at best a self-fulfilling tautology) as there exist no u n i q u e solutions, if t h e u n c e r t a i n t y i n the data u s e d is not a s s u m e d a b s e n t .
This first, d e s c r i p t i v e step i n t h e a p p r o a c h r e s u l t s i n a set of input-vectors, all g i v i n g rise to the e x p e c t e d r a n g e of m o d e l b e h a v i o u r . T h e v a r i a b i l i t y of this set r e p r e s e n t s the u n c e r t a i n t y a b o u t t h e " t r u e " systems state as w e l l as the systems v a r i a b i l i t y d u r i n g the t i m e of r e f e r e n c e . For p r e d i c t i o n s of the s y s t e m s r e s p o n s e o u t s i d e this e m p i r i c a l r a n g e of b e h a v i o u r , this v a r i a b i l i t y is p r e s e r v e d b y u s i n g the w h o l e set of input-vectors. O n e or a few s e l e c t e d e l e m e n t s are c h a n g e d to r e p r e s e n t the c h a n g e i n i n p u t c o n d i t i o n s to be s t u d i e d (Fig. 1D). For the e x a m p l e s h o w n i n F i g u r e 5, n u t r i e n t i n p u t i n terms of p h o s p h o r u s for the trophic state m o d e l for a stratified w a t e r b o d y w a s v a r i e d a r o u n d a n e m p i r i c a l r a n g e of a b o u t 1 m g P / m -2 d -1, 21 l o a d i n g scenarios, r a n g i n g from 0 to 5 m g P / m -2 d -1 w e r e s i m u l a t e d over a p e r i o d of 10 years. This c o u l d b e t a k e n to represent, e.g., c h a n g e s i n r e g i o n a l d e v e l o p m e n t , w a t e r s h e d m a n a g e m e n t , l a n d use, a g r i c u l t u r a l prac- tice, s e w e r s y s t e m s i n s t a l l a t i o n s , or t h e o p e r a t i o n of w a s t e w a t e r t r e a t m e n t facilities. M o d e l r e s p o n s e i n terms of q u a l i t y v a r i a b l e s such as o r t h o p h o s p h a t e p e a k c o n c e n t r a - tions, a l g a e b i o m a s s peak, or total y e a r l y p r i m a r y p r o d u c t i o n w e r e s i m u l a t e d . The g e n e r a l p a t t e r n s o b t a i n e d s h o w e d :
(a) a n i n c r e a s e i n t h e u n c e r t a i n t y of the p r e d i c t i o n s {measured e.g. as coefficient of variation} w i t h i n c r e a s i n g u n c e r t a i n t y (wide r a n g e s ) i n the o r i g i n a l data for i n p u t s a n d e m p i r i c a l systems b e h a v i o u r . This is to b e e x p e c t e d a n d is the most straightforward r e l a t i o n ; (b) a n i n c r e a s e i n the u n c e r t a i n t y of p r e d i c t i o n s w i t h i n c r e a s i n g d e v i a t i o n from the e m p i r i c a l r a n g e s of i n p u t / b e h a v i o u r . Plots of coefficients of v a r i a t i o n s versus i n p u t s show a p a r a b o l i c shape, w i t h t h e m i n i m u m i n t h e e m p i r i c a l r a n g e ; (c) a n i n c r e a s e i n the u n c e r t a i n t y of the p r e d i c t i o n s w i t h time. A l t h o u g h the m e a n s a n d r a n g e s of the p r e d i c t e d v a r i a b l e s r e a c h e d some k i n d of s t e a d y state after s e v e r a l years of s i m u l a t i o n { d e p e n d i n g o n the p h y s i c a l characteristics of the system}, the t r a n s i e n t p e r i o d w a s c h a r a c t e r i s e d by a m a r k e d i n c r e a s e of v a r i a b i l i t y i n time. Plotting coeffcients of v a r i a t i o n versus time r e s u l t e d i n h y p e r b o l i c patterns, a p p r o a c h i n g a s y m p t o t i c a l l y a m a x i m u m l e v e l after some years. For e x a m p l e , after t e n years of s i m u l a t i o n , w i t h a l a r g e d e v i a t i o n from the e m p i r i c a l s i t u a t i o n {more t h a n t h r e e f o l d n u t r i e n t input}, e s t i m a t e s of p r i m a r y p r o d u c t i o n r a n g e d from r o u g h l y 100 to 1000 g C / m -2 y-1. This is c e r t a i n l y a r a t h e r trivial prediction.
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nutrient loading in simulation year 10
r a n g e , for all p o s s i b l e c o m b i n a t i o n s , will b e t h e n u m b e r of t h e s e s a m p l i n g p o i n t s to the the p o w e r of the n u m b e r of i n p u t s or d i m e n s i o n s . This q u i t e o b v i o u s l y l e a d s r a t h e r q u i c k l y to a s t r o n o m i c r u n n u m b e r s . A n a l y s i s of c o m p l i c a t e d m o d e l s w i t h l a r g e d a t a r e q u i r e m e n t s will b e t e c h n i c a l l y a l m o s t i m p o s s i b l e - at l e a s t u n t i l m o r e e l e g a n t m e t h o d s are d e v e l o p e d . Second, m o d e l s s h o u l d n o t c o n t a i n v a r i a b l e s or p a r a m e t e r s n o t directly m e a s u r a b l e i n the real w o r l d system. Data u n c e r t a i n t y as a m a j o r e l e m e n t i n p r e d i c t i o n u n c e r t a i n t y c a n b e r e d u c e d o n l y if the r a n g e s for e a c h of t h e n u m b e r s to b e u s e d i n the n u m e r i c a l s i m u l a t i o n c a n b e f o u n d e d o n the b a s i s of sufficient m e a s u r e m e n t s or e x p e r i m e n t a t i o n . Finally, t h e r e is also a n i m p o r t a n t c o n t r i b u t i o n n e c e s s a r y from field research: n o t o n l y h a s t h e m o d e l to b e a p p r o p r i a t e to the d a t a a v a i l a b l e , b u t also d a t a collection p r o g r a m s will h a v e to b e d e s i g n e d m u c h more i n a c c o r d a n c e w i t h the n e e d s of n u m e r i c a l a n a l y s i s , if such a n a l y s i s is to b e u s e d to a i d r e a l - w o r l d m a n a g e m e n t decisions.
D I S C U S S I O N
232 K. F e d r a
t h a n w e w o u l d l i k e t h e m to be, h a s b e e n d i s c u s s e d (e.g. H e d g p e t h , 1977). Also, critical e v a l u a t i o n s h a v e b e e n m a d e of i m p l e m e n t a t i o n s of s y s t e m s a n a l y s i s a n d m o d e l l i n g (Hilborn, 1979) a n d of t h e f a i l u r e s w h i l e i m p l e m e n t i n g m o d e l l i n g into t h e m a n a g e m e n t a n d d e c i s i o n - m a k i n g p r o c e s s (Watt, 1977). H o w e v e r , for r a t i o n a l e n v i r o n m e n t a l m a n - a g e m e n t , the i s s u e is v e r y d e f i n i t e l y on t h e s i d e of h a v i n g a m o d e l , e v e n a c r u d e one, a g a i n s t h a v i n g no m o d e l at all (Biswas, 1975) It s h o u l d b e s t r e s s e d t h a t e s t i m a t e s of m o d e l u n c e r t a i n t y , a n d in p a r t i c u l a r p r e d i c t i o n u n c e r t a i n t y , a r e s e e n as a n e s s e n t i a l c r i t e r i o n for m o d e l a p p l i c a t i o n a n d i m p l e m e n t a t i o n . S o m e r e c e n t effort has b e e n d e v o t e d to t h e e s t i m a t i o n of m o d e l u n c e r t a i n t y a n d c r e d i b i l i t y (e.g. Di Toro & v a n Straten, 1979; v a n S t r a t e n & d e Boer, 1979; F e d r a , 1979b; R e c k h o w , 1979; F e d r a et aL, 1980), a n d the e x p l i c i t i n c l u s i o n of u n c e r t a i n t y in m o d e l l i n g s t u d i e s (e.g. W a l t e r s , 1975; S p e a r & H o r n b e r g e r , 1978), p a r t l y in a n a t t e m p t to e x p l i c i t l y p r e d i c t p r o b a b i l i t i e s for c e r t a i n e v e n t s or s y s t e m s t a t e s (Reichel & Dyck, 1979; J a m e s et al., 1979). Also, on the i m p l e m e n t a t i o n side, s t r a t e g i e s to c o p e w i t h u n c e r t a i n t y a n d n o n - u n i q u e a n s w e r s are b e i n g d e v e l o p e d ( H o l l i n g , 1978; I I A S A 1979).
W i t h r e g a r d to t h e m a r i n e e n v i r o n m e n t , c o m p a r a t i v e l y f e w e c o s y s t e m m o d e l s h a v e b e e n p u b l i s h e d (for s o m e m o r e r e c e n t e x a m p l e s s e e e.g. O ' B r i e n & W r o b l i n s k i , 1976; W a l s h & H o w e , 1976; V i n o g r a d o v & M e n s h u t k i n , 1977; K r e m e r & Nixon, 1978). To m y k n o w l e d g e , n o n e of t h e m c o u l d c l a i m to b e u s e d in a m a n a g e m e n t context; but a d m i t t e d l y t h e s e m o d e l s h a v e b e e n d e s i g n e d for d i f f e r e n t p u r p o s e s . T h e s i t u a t i o n is s o m e w h a t d i f f e r e n t w h e n l o o k i n g at l a k e s , rivers, a n d e s t u a r i e s , w h e r e s o m e successful w o r k h a s b e e n c a r r i e d out a n d (partly) i m p l e m e n t e d (e.g. C h e n & O r l o b , 1975; B i g e l o w & d e H a v e n , 1977; B i g e l o w et al., 1977; B i g e l o w et al., 1978; I A H S - A I S H , 1978; J o r g e n s e n , 1979; S c a v i a & Robertson, 1979). This m i g h t p a r t l y b e a t t r i b u t e d to t h e fact that m a r i n e s y s t e m s a r e g e n e r a l l y l a r g e s c a l e s y s t e m s . H e r e s p a t i a l p a t t e r n s p l a y a n i m p o r t a n t role (cf. S t e e l e , 1978), b u t t h e s e are difficult to assess.
M a r i n e m o d e l s a r e often r e s t r i c t e d to s e l e c t e d p h y s i c o - c h e m i c a l f e a t u r e s of the e n v i r o n m e n t (e.g. G o l d b e r g et al., 1977), or to s e l e c t e d c o m p a r t m e n t s , as in f i s h e r i e s m o d e l s . E x t e n d i n g s u c h f i s h e r y m o d e l s t o w a r d s m u l t i - s p e c i e s a n d e c o s y s t e m m o d e l s ( A n d e r s e n & Ursin, 1977) r e s u l t s in a r a t h e r c o m p l e x m o d e l w i t h h i g h d a t a r e q u i r e m e n t s , h i g h d i m e n s i o n a l i t y a n d n u m e r o u s c o e f f i c i e n t s - a n d the c o n s e q u e n t " e d u c a t e d g u e s s - w o r k " . W i t h o u t d o u b t , such m o d e l s m a y p l a y a n i m p o r t a n t role in t h e d e v e l o p m e n t of e n v i r o n m e n t a l s c i e n c e . But it m a y w e l l b e q u e s t i o n e d w h e t h e r t h e y c a n r e a s o n a b l y b e a p p l i e d to r e a l - w o r l d m a n a g e m e n t p r o b l e m s , as t h e i r c o m p l e x i t y a n d h i g h d i m e n s i o n a l - ity is p r o h i b i t i v e to a n a n a l y s i s of t h e i r s e n s i t i v i t y to u n c e r t a i n t y .
M o d e l l i n g - a m a n a g e m e n t t o o l ? 2 3 3 d a t a . T h i s r a t h e r a i m s a t a r e c o n s i d e r a t i o n of t h e q u e s t i o n s w e m i g h t r e a s o n a b l y e x p e c t a m a t h e m a t i c a l m o d e l to a n s w e r , a n d c o n s e q u e n t l y t h e d e s i g n of s u c h m o d e l s . M o d e l s for m a n a g e m e n t a p p l i c a t i o n s w i l l h a v e to b e c o m p a r a t i v e l y s i m p l e ; a t t e m p t s h a v e to b e m a d e to c a p t u r e n a t u r a l c o m p l e x i t y i n a m a n n e r o t h e r t h a n a d d i n g m e c h a n i s t i c d e t a i l (see e.g., F e d r a , 1 9 7 9 a ; S t r a s k r a b a , 1979).
C l e a r l y , c o n s i d e r i n g t h e a b o v e , t h e a n s w e r s o b t a i n a b l e f r o m t h e m o s t s o p h i s t i c a t e d m o d e l a r e h a r d l y r e l i a b l e i n q u a n t i t a t i v e t e r m s . A l t h o u g h t h e c o m p u t e r w i l l of c o u r s e p r o d u c e n u m b e r s , it is n o t t h e i r a r i t h m e t i c p r e c i s i o n , b u t t h e i r m e a n i n g w h i c h m a t t e r s . A s t h e a n s w e r w i l l a l w a y s b e s o m e n u m b e r , t h e q u e s t i o n m u s t b e p u t i n a n a p p r o p r i a t e w a y . T h i s a m o u n t s to t h e f o r m u l a t i o n of a l t e r n a t i v e h y p o t h e s e s , w h i c h t h e n c a n b e t e s t e d b y m e a n s of t h e c o m p u t e r . T h i s is s o m e w h a t c o m p a r a b l e t o h y p o t h e s i s t e s t i n g i n m a t h e m a t i c a l s t a t i s t i c s . It is n o t t h e a c t u a l v a l u e of t h e t e s t s t a t i s t i c w h i c h m a t t e r s , b u t w h e t h e r it f o r c e s u s t o r e j e c t o u r h y p o t h e s i s or n o t . I n t e r m s of m a n a g e m e n t a p p l i c a t i o n of e c o s y s t e m m o d e l s , t h i s w o u l d m e a n a m o r e or l e s s q u a l i t a t i v e d i s c r i m i n a t i o n b e t w e e n s e v e r a l m a n a g e m e n t a l t e r n a t i v e s , s o m e k i n d of r a n k i n g a c c o r d i n g t o p e r f o r m a n c e c r i t e r i a of t h e p o s s i b l e s t r a t e g i e s . T h i s of c o u r s e r e q u i r e s a r a t h e r c a r e f u l a n d e l a b o r a t e f o r m u l a t i o n of t h e m a n a g e m e n t a l t e r n a t i v e s a n d p e r f o r m a n c e c r i t e r i a ; a n d i t h a s t o b e u n d e r s t o o d t h a t a m a t h e m a t i c a l m o d e l c a n o n l y b e a t o o l for e c o s y s t e m m a n a g e m e n t , if t h e b a s i c e n v i r o n m e n t a l a n d m a n a g e m e n t p r o b l e m is w e l l f o r m u l a t e d for t h e n u m e r i c a l a n a l y s i s .
Acknowledgements.
The r e s e a r c h d e s c r i b e d in this report was s u p p o r t e d by the r e s e a r c h g r a n t No. 3905 of the Austrian Fonds zur F 6 r d e r u n g der w i s s e n s c h a f t l i c h e n Forschung. T h e a u t h o r is i n d e b t e d to B. S c h w e e g e r from IIASA C o m p u t e r Services for his introductory h e l p w i t h c o m p u t e r graphics, and to I. Fisher for his linguistic help.L I T E R A T U R E C I T E D
In a n effort to m a k e this b i b l i o g r a p h y as current as possible, works are i n c l u d e d w h i c h are p r e s e n t l y a v a i l a b l e only as p u b l i c a t i o n s of the I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis (IIASA). These may b e o b t a i n e d from: D o c u m e n t s a n d Publications, I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis, Schlof~ Laxenburg, A-2361 Laxenburg, Austria.
Andersen, K. P. & Ursin, E., 1977. A m u l t i s p e c i e s e x t e n s i o n of t h e Beverton a n d Holt theory of fishing, with accounts of p h o s p h o r u s circulation a n d p r i m a r y production. - M e d d r Danm. Fisk. - og Havunders. N. S. 7, 319-435.
Bigelow, J. H. & De Haven, J.C., 1977. Protecting a n estuary from floods - policy analysis of the Oosterschelde. Vol. V: A n a e r o b i c c o n d i t i o n s a n d r e l a t e d ecological d i s t u r b a n c e s . - RAND, R- 2121/5-NETH.
Bigelow, J. H., Bolten, J. G. & De Haven, J. C., 1977. Protecting a n estuary from floods - a policy analysis of the Oosterschelde. Vol. 4: A s s e s s m e n t of a l g a e blooms, a p o t e n t i a l ecological disturbance. - RAND, R-2121/4-NETH,
Bigelow, J. H., Dzitzer, C. & Peeters, J. H., 1978. Protecting a n estuary from Floods - a policy analysis of the Oosterschelde. Vol. 2: A s s e s s m e n t of l o n g - r u n ecological b a l a n c e s : a n a d d e n d u m (calibration data). - RAND, R-2121/3A-NETH.
Bigelow, J. H., De Haven, J.C., Dzitzer, C., Eilers, P. & Peeters, J. C. H., 1977. Protecting a n estuary from floods - a policy analysis of the Oosterschelde. Vol. 3: A s s e s s m e n t of l o n g - r u n ecological balances. - RAND, R-2121/3-NETH.
234 K. F e d r a
C h e n , K. W. & Orlob, G. T., 1975. Ecological s i m u l a t i o n for a q u a t i c e n v i r o n m e n t s . In: Systems analysis a n d s i m u l a t i o n in ecology. Ed. b y B. C. Patten. Acad. Press, N e w York, 3, 475-587. Clark, W. C., Jones, D. D. & Holling, C. S., 1979. Lessons for ecological policy design: a case study of
e c o s y s t e m m a n a g e m e n t . - Ecol. Model1. 7, 1-53.
Di Toro, D. M. & v a n Straten, G., 1979. U n c e r t a i n t y in the p a r a m e t e r s a n d predictions of p h y t o p l a n k - ton models. I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis, Laxenburg, 34 pp. (WP-79-27). Fedra, K., 1979a. M o d e l l i n g biological p r o c e s s e s in the a q u a t i c e n v i r o n m e n t (with special reference
to adaptation). I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis, Laxenburg, 56 pp. (WP-79- 20).
Fedra, K., 1979b. A stochastic a p p r o a c h to m o d e l u n c e r t a i n t y : a lake m o d e l l i n g example. Interna- t i o n a l Institute for A p p l i e d Systems Analysis, L a x e n b u r g , 46 pp. (WP-79-63).
Fedra, K., v a n Straten, G. & Beck, M. B., 1980. U n c e r t a i n t y a n d A r b i t r a r i n e s s in ecosystems m o d e l l i n g : a lake m o d e l l i n g e x a m p l e . I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis, Laxenburg, 39 pp. (WP-80-87).
Goldberg, E. D., McCave, I. N., O'Brien, J. J. & Steele, J. M. (Eds), 1977. M a r i n e modelling. Wiley- Interscience, N e w York, 895 pp.
H e d g p e t h , J. W., 1977. M o d e l s a n d m u d d l e s . Some p h i l o s o p h i c a l observations. - Helgoltinder wiss. M e e r e s u n t e r s . 30, 92-104.
Hilborn, R., 1979. Some failures a n d successes in a p p l y i n g systems analysis to ecological systems. - J. appl. Systems A n a l y s i s 6, 25-31.
Holling, C. S., (Ed.), 1978. A d a p t i v e e n v i r o n m e n t a l a s s e s s m e n t a n d m a n a g e m e n t . Wiley, Chiches- ter, 377 pp.
IAHS-AISH, 1978. M o d e l l i n g the w a t e r quality of the hydrological cycle. P r o c e e d i n g s of the B a d e n Symposium, Sept. 1978. IAHS-AISH Publ. 125, 1-382.
Imboden, D. & Gtichter, R., 1978. A d y n a m i c lake m o d e l for trophic state prediction. - Ecol. Modell. 4, 77-98.
I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis, 1979. A d a p t i v e e n v i r o n m e n t a l a s s e s s m e n t a n d m a n a g e m e n t : c u r r e n t progress a n d prospects for the approach. S u m m a r y report of the first Policy S e m i n a r 18-21 J u n e , 1979. I n t e r n a t i o n a l Institute for A p p l i e d Systems Analysis, Laxenburg, CP- 79-9, 123 pp.
J a m e s , L D., Bowles, D. S., J a m e s , R. W. & Canfield, R, V., 1979. Estimation of w a t e r surface e l e v a t i o n p r o b a b i l i t i e s a n d a s s o c i a t e d d a m a g e s for the G r e a t Salt Lake. U t a h State University, W a t e r Resources P l a n n i n g Series UWRL/P-79/03, 182 pp.
J e r g e n s e n , S.E. (Ed.), 1979. State-of-the-art in ecological m o d e l l i n g . P r o c e e d i n g s of the Conference on Ecological M o d e l l i n g , C o p e n h a g e n , D e n m a r k , 28 A u g u s t - 2 S e p t e m b e r 1978. I n t e r n a t i o n a l Society for Ecological M o d e l l i n g , 891 pp.
Kremer, J. N. & Nixon, S. W., 1978. A coastal m a r i n e ecosystem: s i m u l a t i o n a n d analysis. Springer, Berlin, 217 pp.
Mason, C. M. (Ed.), 1979. T h e effective m a n a g e m e n t of resources. T h e i n t e r n a t i o n a l politics of the N o r t h Sea. F r a n c e s Pinter, London, 268 pp.
Meadows, D. H., Meadows, D. I., Renders, J. & Behrens, W. W., 1972. T h e limits to growth. Universe Books, N e w York, 205 pp.
O'Brien, J. J. & Wroblisnki, J. S., 1976. A s i m u l a t i o n of the m e s o s c a l e distribution of t h e lower m a r i n e trophic levels off West Florida. In: Systems analysis a n d s i m u l a t i o n in ecology. Ed. by B. C. Patten. Acad. Press, N e w York, 4, 63-110.
Reck_how, K. H., 1979. Empirical l a k e m o d e l s for p h o s p h o r u s : d e v e l o p m e n t , applications, limita- tions a n d u n c e r t a i n t y 193-222. In: Perspectives on lake ecosystem modeling. Ed. by D. Scavia & A. Robertson. A n n Arbor Science Publ., A n n Arbor, Mich., 183-221 pp.
Reichel, F. & Dyck, S., 1979. Stochastische g e o h y d r a u l i s c h e B e r e c h n u n g s v e r f a h r e n fiir die Projek- t i e r u n g der T a g b a u e n t w t i s s e r u n g - N o t w e n d i g k e i t , Ziele, Verfahren. - N e u e B e r g b a u t e c h n . 9 (1), 6-10.
Scavia, D. & Robertson, A. (Eds.), 1979. Perspectives o n l a k e ecosystem m o d e l i n g . Arm Arbor Science, Publ., A n n Arbor, Mich. 326 pp.
Spear, R. C. & H o r n b e r g e r , G. M,, 1978. E u t r o p h i c a t i o n in Peel Inlet: a n analysis of b e h a v i o u r a n d sensitivity of a poorly d e f i n e d system. - Cres. Rep. AS/P 24, 99 pp.
M o d e l l i n g - a m a n a g e m e n t t o o l ? 235
Steele, J. H. (Ed.), 1978. Spatial patterns in p l a n k t o n communities. P l e n u m Press, N e w York, 470 pp. (NATO C o n f e r e n c e Series, Ser. 4: M a r i n e Sciences, Vol. 3).
Straskraba, M., 1979. Natural control m e c h a n i s m s in m o d e l s of aquatic ecosystems. - E c o l . Modell. 6, 305-321.
van Straten, G, & de Boer, B., 1979. Sensitivity to u n c e r t a i n t y in a p h y t o p l a n k t o n - o x y g e n m o d e l for lowland streams. International Institute for A p p l i e d Systems Analysis, Laxenburg, WP-79-28, 21 pp.
Vinogradov, M. E. & Menshutkin, V. V., 1977. The m o d e l i n g of o p e n s e a ecosystems. In: Marine modelling. Ed. by E. D. Goldberg, J. N. McCave, J. J. O'Brien & J. M. Steele. Wiley-Interscience, N e w York, 891-921.
Watsh, J. J. & Howe, S. O., 1976. Protein from the sea: a c o m p a r i s o n of the s i m u l a t e d n i t r o g e n a n d carbon productivity of the Peru u p w e l l i n 9 ecoystem. In: Systems analysis a n d simulation in ecology. Ed. by B. C. Patten. Acad~ Press, N e w York, 4, 47-61.
Waiters, C. J., 1975. Optimal h a r v e s t strategies for salmon in relation to e n v i r o n m e n t a l variability and uncertain production parameters. - J. Fish. Res. Bd Can. 32, 1777-1784.
