laws 1986 aquacultural-engineering

7/28/2019 Laws 1986 Aquacultural-Engineering

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Aq uacu l tural Engineer ing 5 (198 6) 161 - 170

U n s t r u c t u r e d F o o d W e b s : a M o d e l f o r A q u a c u l t u r e

E d w a r d A . L a w sDepartment of O ceanography, University of Haw aii, Hono lulu, Hawaii 96822, USA

A B S T R A C TP r e d ic t io n s d e r iv e d f r o m t h e u n s t n t c t u r e d f o o d w e b m o d e l d e v e l o p e db y l s a a c s (1 97 2, 1 9 7 3 ) a n d l a t e r m o d i f i e d b y L a n g e a n d H u r l e y ( 19 75 )c a n b e d e r i v e d s t ra i g h t fo r w a r d l y f r o m a s i m p l e b o x m o d e l i n v o l v in gp l a n t b i o m a s s , h e t e r o t r o p h b i o m a s s , a n d d e tr i tu s . D i f fe r e n t i a l e q u a -t i o n s r a t h e r t h a n m a t r i c e s a r e u s e d t o d e s c r i b e t h e t r a n s f e r o f m a t e r i a l sb e t w e e n t h e s e b o x e s . B y r e c a s t i n g t h e s e e q u a t i o n s , s e v e r a l r e s t r i c t i o n sp r e s e n t i n th e o r i g i n a l m o d e l a s w e l l a s in t h e L a n g e a n d H u r l e yr e f o r m u l a t i o n c a n b e re l a xe d . T h e m o d e l w a s t h e r ef o r e g e n e r a l i z e d toa l l o w f o r (1 ) a d i s t in c t i o n b e t w e e n o r g a n i c i n p u t s d u e t o p l a n t p h o t o -s y n t h e s i s a n d t o a p p l i e d f e e d s , (2 ) a d i f fe r e n c e i n t h e t u r n o v e r r a te s o ft h e p l a n t s , h e t e r o t r o p h s , a n d d e t r i tu s , a n d (3 ) t h e e f fe c t o f h a r v e s t i n g ac a sh c ro p . T h i s m o r e g e n e r a l m o d e l i s u s e d t o p r e d i c t t h e p o t e n t i a lb i o m a s s o f h e t e r o tr o p h s i n a f r e s h w a t e r p r a w n p o n d , a n d t o p r e d i c t t h eo u t c o m e o f s i m p l e m a n a g e r i a l m a n i p u l a t i o n s o n t he sy st em .

I N T R O D U C T I O NI n a q u a li ta t iv e s e n s e , o u r u n d e r s t a n d i n g o f a q u a t i c f o o d w e b s h a sc h a n g e d d r a m a t i c a l l y i n t h e p a s t 2 0 y e a r s . S i m p l e l i n e a r f o o d c h a i nm o d e l s h a v e g i v e n w a y t o th e c o n c e p t o f a c o m p l e x f o o d w e b . I t is n o wg e n e r a l l y re c o g n i z e d t h a t b a c t e r i a a n d p r o t o z o a n s p l a y a cr it ic a l r o l ei n r e c y c l in g o r g a n i c s , a n d t h a t t h e f e e d i n g r e l a ti o n s h i p s o f m a n ya q u a t i c o r g a n i s m s a r e s u c h a s t o p r e c l u d e a s s i g n i n g p a r t i c u l a ro r g a n i s m s t o p a r t ic u l a r t r o p h i c l e ve ls . W h i l e s o m e s i m p l e s y s te m s m a yb e r e a s o n a b l y w e ll d e s c r i b e d w i t h i n t h e c o n t e x t o f p l a n t -h e r b i v o r e - c a r n i v o r e m o d e l s , a m o r e g e n e r a l t h e o r y i s n e e d e d t od e s c r i b e m a n y a q u a t ic f o o d w e b s .161Aquacu lmra l Eng ineer ing 0144-8609/86/S03.50 - Elsev ier Ap plied SciencePublishers Ltd, E ngland, 1986 . Printed in G reat Britain


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162 E.A. LawsI n 1 9 7 2 J o h n I s aa c s p r o p o s e d t h e u s e o f a s o - c a l l e d " u n s t r u c t u re d

f o o d w e b ' m o d e l t o d e s c r i b e t h e f l ux e s o f o r g a n i c m a t e ri a l in m a r i n ef o o d w e b s . I s a ac s ( 1 9 7 3 ) c o n t r a s t e d h i s m o d e l w i t h t h e c la s si c al s t ru c -t u r e d f o o d w e b m o d e l s i n w h i c h h e t e r o t r o p h s w e r e c o n s i d e r e d t oo c c u p y r e la t iv e l y w e l l - d e f in e d p o s i t i o n s w i th i n a f i n it e s e q u e n c e o ft r o p h i c s te p s . I n Is a a cs ' u n s t r u c t u r e d f o o d w e b m o d e l , e a c h m o i e t y o ff o o d w a s c o n s i d e r e d t o p a ss t h r o u g h a n i n f in i te n u m b e r o f s te p s a n dc o n v e r s io n s , a n d a t ea c h s u c h s t e p / c o n v e r s i o n w a s c o n s i d e r e d to b et r a n s f o r m e d i n t o li vi n g b i o m a s s , d e a d b i o m a s s , o r n o n - r e c o v e r a b l em a t e r ia l (e.g . C O , ). T h i s a p p r o a c h c o m p l e t e l y a v o i d e d t h e a s si g n m e n to f pa r t i cu l a r o r gan i s ms t o pa r t i cu l a r t r oph i c l eve l s . I n f ac t , s i nce ag i v e n h e t e r o t r o p h m i g h t c o n s is t o f a n a s s e m b l a g e o f o r g a n i c m o i e t ie sw h i c h h a d p a s s e d t h r o u g h a n u m b e r o f tr a n s f o rm a t i o n s r a n g in g f r o ms o m e s m a l l i n t e g e r t o i n f i n i t y , t h e p o s i t i o n o f a g i v e n o r g a n i s m i nI s a a c s ' f o o d w e b w o u l d i n g e n e r a l b e s p r e a d o u t o v e r a n i n f i n i t en u m b e r o f f o o d w e b e l e m e n t s . I sa a c s w a s a b le t o s h o w t h a t h isu n s t r u c t u r e d f o o d w e b c o u l d a c c o u n t f o r th e d i s t r ib u t i o n o f c e r t a int r a c e e l e m e n t s i n m a r i n e f o o d w e b s , w h e r e a s a s i m p l e f o o d c h a i nm o d e l c o u l d n o t ( Is aa c s, 1 9 7 2 ). B e c a u s e o f th e c o m p l e x fl ow o f f o o da n d e n e r g y i n m a n y a q u a c u l t u r e p o n d s , i t s e e m s u s e f u l t o c o n s i d e rI sa a c s' u n s t r u c t u r e d f o o d w e b m o d e l as a m e a n s o f u n d e r s t a n d i n gs o m e o f t h e t r a n s f o r m a t i o n s t h a t o c c u r w i t h i n t h e s e p o n d s , a n d f o rp r e d i c t i n g th e e f f e ct s o f m a n a g e r i a l m a n i p u l a t i o n s o n t h e s y s te m s .

T H E B A S I C M O D E L

I n h is o ri g in a l m o d e l I s aa c s ( 1 9 7 2 ) a s s u m e d t h a t f o o d w a s c o n v e r t e dt o l i v i ng t i s sue w i t h an e f f i c i ency K l , t o i r r e t r i ev ab l e f o r m s ( e.g . C O 2)w i t h a n e f f ic i e n c y K 2 , a n d t o n o n - l i v i n g b u t r e t r i e v a b l e f o r m s ( i.e .d e t r i tu s ) w i t h a n e f f ic i e n c y K 3. I f t h e s e a r e e x p r e s s e d i n f r a c ti o n a l f o r m ,t h e n K i + K 2 + K 3 = 1 . F i g u r e 1 i s a d i a g r a m m a t i c r e p r e s e n t a t i o n o fI sa a c s' u n s t r u c t u r e d f o o d w e b . N e w f o o d is a s s u m e d t o b e i n t r o d u c e da t t h e p o i n t ( 0, 0 ). T h e f l o w o f f o o d is e i t h e r t o t h e r i g h t o r d o w n . F l o wt o t h e r ig h t p r o d u c e s l iv i ng b i o m a s s ; f lo w d o w n p r o d u c e s n o n - l iv i n gb u t r e t r i e v a b l e m a t e r ia l . A c r it ic a l a s s u m p t i o n o f th e o r i g i n a l m o d e lw a s t h a t t h e t i m e i n t e rv a l b e t w e e n s t e p s i n t h e m a t r ix w a s t h e s a m e f o ra ll s t eps . L a t e r I w i l l s ho w t ha t t h i s a s s um pt i o n ca n be r e l axed . If t hes ys t em is in s t eady s t a t e an d i f t he b i om as s a t p o i n t (0 , 0 ) is un i ty , t hen


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UnstrucR~red ood webs: a mo de l o r aquaculture 16 3nL I V I N G S T E P S K 1 n ~

0 1 , 5 "o " " ' " ] I/ / ..,/~ ,,'~ / / 1 / / / / / / / I/ / 1 . .i / / / s / /

/ / * ' ~' " I / s s/ , * / // I , ,/ / I / / I / " I . / " I F /, I/ / / " 2 / . 3 ~ 4 / " 5 / " 6 7s / / / / / / i / /

I / / / I / / 1 / /: / , i // / 2 / / 3 / / 4 5 : , , 6 , , 7 , , "0 / / / " 3 / ' 6 / / I 0 , '/ 1 5 / / I / /

/ ' / : 1 / , J '/ / / / , ,// I ,* / / / :. / 3 , / 6 ,, I 0 / / 1 5... 2 1 , /

0 / / / " 4 / , ' 10 , / 2 0 / 3 5/ , s / 1 / // / / // / , / // / // 4 . " , I 0 " 2 0 / 3 5 / "

D / " : I , , / / 5 / / / " 1 5 : / 3 5 1" s / ~ / / s .

/ / / / 1 5 / / 3. 5 / "0 / / / " 6 / , / 21/ s/ / / 1 / / s ,

L / 6 / " 2 1 / / o , ,, = F OR C O N S T R U C T I O N O F M A r R I X" r e c o v e r a b l e

Token a s t h e ~ u m b e r o F ~ o r h s\ ' ~ \ l eo d mg t o t h e po,nt,

Fig. I . Isaacs"unstructured food web matrix. In the original version, l iving biomassin the form of p lan t biomass was envisione d as being introduced at po int (0, 0). Con-versions to living biom ass corresp ond to m oving to the right. Co nversion s to detrituscorrespond to mo ving dow n. E ac h diagonal l ine passes through p oints representingbiomass wh ich has undergone an e qual n um ber of transformations since being intro-duced into the system. Th e num ber o f transformations associated with each diagonalline is indicated by the nu m ber a t both en ds of the diagonal. Th e expressions alongthe top o f the m atrix are the total bioma ss of l iving organisms along each diagon al int h e s t e a d y s t at e . T h e e x p r e s s i o n s a t t h e l e ft h a v e a si m i la r m e a n i n g f o r th e d e t r it u s . K /i s t h e f r a c t i o n o f b i o m a s s c o n v e r t e d i n t o l iv i n g b i o m a s s a t e a c h t r a n s f o r m a t i o n . K_~ i st h e f ra c t io n o f b i o m a s s s i m i l a r ly l e a d i n g t o d et r it u s . T h e n u m b e r o f p o s s i b l e p a t h w a y sl e a d i n g t o l iv i n g b i o m a s s a t a p a r t ic u l a r la t t ic e p o i n t i s i n d i c a t e d b y a n u m b e r b e l o wa n d t o t h e rig h t o f t h e la t t ic e p o i n t . T h e n u m b e r o f p a t h w a y s l e a d in g t o d e t r it u s a t ap a r t i c u l a r l a t ti c e p o i n t i s i n d i c a t e d b y a n u m b e r a b o v e a n d t o t h e le ft o f th e l a t t ic e

p o i n t .

1

2

, - ,_ ~'t-x Ab , , -t / ' }

e r ~

r vu , ~ "OL )


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164 E. A . L awsi t i s s t r a i g h t f o r w a r d t o s h o w t h a t t h e b i o m a s s a t (0 , 1 ) i s K 1 a n d t h eb i o m a s s a t ( 1 , 0 ) i s K 3.P r o c e e d i n g w i t h t h e a n a l y s is , w e n o t e t h a t t h e d i a g o n a l l in e s i n F ig .1 p a s s t h r o u g h p o i n ts , e a c h o f w h i c h r e p r e s e n t b i o m a s s w h i c h h asg o n e t h r o u g h t h e s a m e n u m b e r o f s te p s si n c e b e i n g in t r o d u c e d a tp o i n t ( 0, 0 ). W i t h t h e e x c e p t i o n o f p o i n t s l y i n g a l o n g t h e ( 0 , n ) a n d( n , 0 ) l in e s , e a c h p o i n t i n t h e m a t r i x r e p r e s e n t s a c o m b i n a t i o n o f b o t hl iv i n g b i o m a s s a n d d e t r it u s . F o r e x a m p l e , l i v in g b i o m a s s a t p o i n t ( 1 , 1 )is p r o d u c e d b y a t r a n s f o r m a t i o n o f b i o m a s s f r o m p o i n t ( 1, 0 ); d e t r it u sa t p o i n t ( l , 1 ) is p r o d u c e d b y t ra n s f o r m i n g b i o m a s s f r o m p o i n t (0 , 1).T h e t o t al n u m b e r o f p a t h w a y s l e a d i n g t o th e p r o d u c t i o n o f l iv i n gb i o m a s s a n d d e t r it u s a t a ~ v e n p o i n t i s e a si ly c o u n t e d , a n d t h e m e t h o do f c a l c u l a t i o n is in d i c a t e d i n F i g . 1. I f t h e n u m b e r s o f p a t h w a y s l e a d i n gt o d e t ri t u s a n d l iv i n g b i o m a s s a t p o i n t ( n - 1 , m ) a r e a a n d b , r e s p e c -t iv e ly , t h e n t h e n u m b e r o f p a t h w a y s l e a d i n g t o d e t r i tu s a t p o i n t (n , m )is a + b . S i m i l ar ly , i f t h e n u m b e r s o f p a t h w a y s l e a d i n g t o d e t r it u s a n dl iv i n g b i o m a s s a t p o i n t ( n , m - 1 ) a r e c a n d d , r e s p e c t i v e l y , t h e n t h en u m b e r o f p a t h w a y s l e a d i n g to l i v in g b i o m a s s a t p o i n t (n , m ) is c + d .G i v e n t h e a s s u m p t i o n o f s t e a d y s t at e , it is s t r a i g h t f o r w a r d t o c a lc u -l at e th e a m o u n t o f li v in g b i o m a s s a n d d e t r i t u s a t e a c h p o i n t i n th em a t r i x r e l a t i v e t o t h e a m o u n t a t p o i n t ( 0 , 0 ) b y r e m e m b e r i n g t h a t t h ee f f ic i e n c y o f li v in g b i o m a s s p r o d u c t i o n is K ~ , a n d t h e e f f ic i e n c y o fd e t r i tu s p r o d u c t i o n is K 3. T h e s u m o f a ll t h e l i v in g b i o m a s s a n d o f al lt h e d e t r i t u s a l o n g e a c h d i a g o n a l l i n e f o r m s a s i m p l e g e o m e t r i c s e r i e s ,a s i n d i c a t e d i n F i g . 1 , a n d o n e c a n t h e r e f o r e e a s i l y c a l c u l a t e t h ea m o u n t o f l iv i n g b i o m a s s a n d d e t r i t u s i n t h e s y s t e m . I f o n e a s s ig n s t ot h e b i o m a s s a t p o i n t ( 0 , 0 ) th e s y m b o l M 0, t h e n t h e t o t a l l iv i n g b i o m a s sM I i n th e s y s t e m i n a d d i t i o n t o t h a t c o n t a i n e d i n M 0 is

MI) K IM r - - - ( 1 )K ,a n d t h e t o ta l a m o u n t o f d e t r it u s M d i n t h e s y s t e m i n a d d i t i o n t o t h a tc o n t a i n e d i n M o is

l ~ o K 3M j = ( 2 )K ,A l t h o u g h I s a a cs ' m o d e l d o e s n o t r e q u i r e th e a s s ig n m e n t o f p a r t i c u la ro r g a n i s m s t o p a r t i c u l a r t r o p h i c l e v e l s , i t i s n e v e r t h e l e s s p o s s i b l e t o


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Unstn~cmred oo d webs: a m odel for aquaculture 165der ive express ions for the potent ia l b iomass of cer ta in types oforganisms , but these express ions provide usefu l predic t ive tool s onlyto the extent tha t such organisms exi s t in and dominate the rea l eco-sys tem. I saacs (1972) prov ides the fo l lowing express ions :

St r ic t Preda tors (Mp)M p = K I M l = K , (3 )

Pa rticle Fe ed ers (),.Ipf)mpf

O m nivo res (M, ,)

II/IoKI( K2 + K3)= K ~ ( M j + M o ) = (4 )K ,

M v = K I ( M ,, + M , + M d ) = - -MoK~ ( 5 )K ,I saacs (1972) de r ived add i t i ona l expres s ions on t he a s sumpt ion t ha tM 0 cons i s ted ent i re ly of p lant s , but tha t assum pt ion c lear ly need no tapply in a pond rece iv ing appl ied feeds . I wi l l show la te r how tod i s t i ngu i sh be tween i npu t s f rom app l i ed f eeds and f rom pho to -synthes is . Ex press ion s such as eqns (3) - (5) may pro ve usefu l for pre-d ic t ing the e f fec t of var ious m anage r ia l m anip ula t ion s on the sys tem.A poten t ia l ly revea l ing app l ica t ion of I saacs ' equat ions is the s tudyof e lementa l ra t ios in organisms . Let us assume for example tha t thecoef f ic ients K~, K2, an d g 3 app ly to ca rbo n C, and tha t the coef f ic ient sK I, K~ , and K ; app ly t o som e o the r e l emen t , E . T he n accord ing t oeqn (5) the ra t io E/C in the om nivo res is re la ted to the ra t io E/ C in M0by the expres s ion K I K 2 / ( K ; K ~ ) . Simi lar express ions can be der ivedfo r E /C r a ti os in o the r g roups o f o rgan i sms .Subsequen t t o I s aacs ' two pub l i ca t i ons on t he uns t ruc tu red foodweb m ode l , Lange and Hu r l ey (1975) pub l i shed a pape r i n wh ich t heyshowed tha t t he mode l cou ld be gene ra l i zed and t he ma themat i c ss impl i f ied by the use o f an a l t e rna t ive mat r ix form ula t ion . T he ygene ra l ized t he m ode l by d ro pp ing t he co nd i t i on t ha t a ll conve r s ionsto l iv ing b iom ass a nd a l l con vers ion s to d e t r i tus o ccu r wi th the sam eef fic iency. T he y a l lowed fo r a to ta l of n ine separa te con vers io n fac tors


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166 E . A . L a w sT A B L E 1Definition of the Nine Conversion Factors Used in the Lange and

Hurley (1975) M odelS y m b o l M e a n i n g

K IK,K~KaKsKsK7KsK,~

ConversionConversionConversionConversionConversionConversionConversionConversionConversion

factor from source to livingfactor from source to dead retrievablefactor from source to irretrievablefactor from living to livingfactor from living o dead retrievablefactor from living o irretrievablefactor from dead to livingfactor from dead to de ad retrievablefactor from dea d to irretrievable

a s in d i c a t e d i n T a b l e 1. T h e y d e f i n e d W~ a s t h e a m o u n t o f b i o m a s s i nt h e pl an t s, W , a s t h e a m o u n t o f b i o m a s s i n t h e h e t e r o t r o p h s a n d W3 a st h e a m o u n t o f d e t ri tu s . T h e m a t r i x r e p r e s e n t a t i o n o f t h ei r m o d e l t h e nt a k e s t h e f o r m

1 0 0AI7V= K t K4 K7K2 K5 Ks

W I% Wl= K , W I + K ~ W . _ + K T W 3K 2 W I + K s IV ,_ + K s W 3

( 6 )

T h e m a t r i x A o p e r a t i n g o n t h e c o l u m n v e c t o r W f r o m t h e le ftp r o j e c t s t h e v a l u e s o f W ~ , W2, an d W3 f r o m t i m e , t , t o t i m e , t + A t .L a n g e a n d H u r l e y ( 1 9 7 5 ) r e q u i r e d t h a t K t + K 2 + K 3 = K 4 + K 5 + K 6= K 7 + K8 + K 9 = 1. T h e s e c o n d i t i o n s i n e f f e c t r e q u i r e t h a t t h e t u r n -o v e r r a t e s o f Wt , W2, an d W3 b e i d e n t i c a l , a c o n d i t i o n s i m i l a r t o I s a a c s '( 1 9 7 2 ) r e q u i r e m e n t t h a t t h e t i m e i n t e r v a l s b e t w e e n a l l s t e p s i n t h ef o o d w e b m a t r i x b e t h e s a m e . U s i n g t h e L a n g e a n d H u r l e y ( 1 9 7 5 )a p p r o a c h , i t i s s t r a i g h t f o r w a r d t o d e r i v e m a n y r e l a t i o n s h i p s s i m i l a r t ot h o s e o b t a i n e d b y I s a ac s ( 1 9 7 2 , 1 9 7 3 ) , b u t w i t h n i n e t ra n s f e r c o ef fi -c i en t s ra t h e r t h a n t h re e . F o r e x a m p l e , L a n g e a n d H u r l e y ( 1 9 7 5 ) s h o wt h a t t h e s t e a d y - s t a t e p o t e n t i a l b i o m a s s o f s t r i c t p r e d a t o r s i s g i v e n b yt h e e x p r e s s i o n

- K 4 K I ( K 8 - 1 ) + K 4 K , _ K 7M p = M o ( K a - 1 ) ( K s - 1 ) - K s K 7 ( 7 )


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Unstr~wtured foo d webs." a model fo r aquacul ture 167T h i s m o d e l s e e m s p o t e n t ia l l y u s e fu l fo r s t u d y i n g f o o d w e b s i n a q u a -c u l t u r e , b u t I f e e l s e v e r a l m o d i f i c a t i o n s i n t h e m o d e l w o u l d b e d e s i r -a b l e . F i rs t , it w o u l d b e u s e f u l t o d i s t in g u i s h b e ~ ' e e n o r g a n i c i n p u t sd u e t o p l a n t p h o t o s y n t h e s i s a n d o r g a n i c i n p u t s d u e t o a p p l i e d f e e d s .S e c o n d , o n e w o u l d l i k e t o r e l a x t h e r e s t r i c t i o n t h a t t h e t u r n o v e r r a t e so f W l , ~ , an d B'~ be i den t i ca l . F i na l l y , on e s h ou l d a l l ow f o r the e f f ec to f h a r v e s t i n g t h e c a s h c r o p .

A c t u a l l y t h e u n s t r u c t u r e d f o o d w e b d e s c r i b e d w i t h t h e u s e o fm a t r L x e q u a t i o n s b y L a n g e a n d H u r l e y ( 1 9 7 5 ) c a n b e e q u a l l y w e l ld e s c r i b e d w i t h t h e u s e o f d i f f e r e n t i a l e q u a t i o n s , a n d i n o r d e r t og e n e r a l i z e th e m o d e l I p r o p o s e t h e fo l l o w i n g e q u a t i o n s

dW~ 1 ( M , - ( C , + C , + (7 3) W ,)d t r s )d }~q 1- = - ( C a W 2 + C , W ~ + C , W 3 - ( C , + C s + C 6 ) W : ) (9)d t r

d W ~ 1- = - ( M : , + G W , + C 2 w~ + C 5 ~ - ( C 7 + G + C , ) W ;d t r (10)H e r e M ~ / r a n d M ~ / r a r e t h e r a t e s a t w h i c h p l a n t b i o m a s s a n d f e e d( n o n - li v in g b u t r e t ri e v a b l e o r g a n i c b i o m a s s ) a re i n t r o d u c e d i n to t h es y s t e m v i a p h o t o s y n t h e s i s a n d f e e d a p p l i c a t i o n s , r e s p e c t i v e ly . T h e C ia r e d e f i ne d i n T ab l e 2 , an d W , W_~,a n d I ~ a r e a s p r e v i o u s l y d e f i n e d .N o t e f r o m T a b l e 2 t h a t t h e r e is n o c o n s t r a i n t o n t h e C , e x c e p t t h a t t h e yb e p o s it iv e n u m b e r s . N o w l et

K i = C i /( C , + C , + C 3 )m , = M ~ / ( G + C~ + C3)m 3 = M 3/ (C I + C , + (73)

T = r / ( C I + C , + C 3 )

( 1 ! )(12)( 13 )(14)

E q u a t i o n s ( 8 )- (1 0 ) n o w b e c o m ed Wt 1

= 7..(rn, - ( K , + K z + K: ,) W I)d t 1 (15)


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168 E. ,4. Law sTABLE 2Expressions which Define the C~

Expression 31eaningC t / rC2/rC~/rCJ rC~/ rC . / rC7/rc , / rC ~ r

Fraction of plant biomass converted to heterotroph biomassper unit timeFraction of plant biomass converted to detritus per unit timeFraction of plant biomass converted to irretrievable form perunit timeFraction of heterotroph biomass converted to heterotroph bio-mass per unit timeFraction of heterotroph biomass converted to detritus per unittimeFraction of heterotroph biomass converted to irretrievableform per unit timeFraction of detritus converted to heterotroph biomass per unittimeFraction of detritus converted to detritus per unit timeFraction of detritus converted to irretrievable form per unittime

d W , 1d t T ( K ~ W ' + K ~ W ~ + K v W ' - ( K ~ + K S + K ~ ) W ) (16)d W 3 _d t ( t n 3 + K s W 3 + K 2 W ~ + K ~ W ' - ( K T + K ~ + K g ) W3) (17)

It is obvi ous from the def ini tion of the K~ tha t K~ + K, + K 3 -- 1, andthat m ~ is the am ou nt of new plant biomass intr oduc ed into the systemin time T. Hence rnt has the same meaning as Isaacs' M , , and K~, K2,and K 3 have m eaning s identical to the same symbols as defined byLange and Hurley (1975). However, there is no reason to assume thatK 4 + K s + K 6 = K 7 + K s + K 9 = 1 unless on e beli eves tha t the bio-masses of heterotrophs and of detritus are turning over at the samerate as the plant biomass. According to the model the turnover timesof the plants (rp), het er ot ro ph s (rh) , and detritus (rd) are given by thefollowing expressions rp = T (18)


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Unstructured fo od webs: a m ode l for aquacul ture 169rh = T/( K~ + K5 + K~ ) (1 9)r d = T / ( K 7 + K s + K s ) (20)

N o t e t h a t th e p a r a m e t e r K~ m a y b e c o n s i d e r e d t o i n c l u d e th e e ff ec tso f b o t h r e s p i r a t i o n a n d h a r v e s t i n g c a s h c r o p s , s i n c e b o t h p r o c e s s e st r a n s f e r li v in g b i o m a s s t o a n i r r e t r i e v a b l e f o r m . T h e s t e a d y - s t a t e s o lu -t io n s t o e q n s ( 1 5 ) - ( 1 7 ) a r e a s f o ll o w s :

W I = r n l ( 2 1 )W . = K I r t l ' ( K v + K o ) + K v ( K ' - t n ' + m3 ) (2 2 )

KsK,~ + K6(K7 + K,,)W3 = (Ks + K6)(K2ml + m3)+ K t K s m ~ (23)

K s K ~ + K6(K7 + Kg)T h e p o t e n t i a l b i o m a s s o f v a r i o u s t r o p h i c l ev e ls c a n n o w b e c a l c u l a t edi n a m a n n e r s i m i l a r t o th a t e m p l o y e d b y I sa a cs ( 1 9 7 2 ) a n d L a n g e a n dH u r l e y ( 1 9 7 5 ) . F o r e x a m p l e , t h e p o t e n t i a l b i o m a s s o f s tr ic t h e r b i v o r e s(mh) iS

rn h = K ~ r n , / ( K 4 + K 5 + K ~ ) (24)T h e p o t e n t i a l b i o m a s s o f o m n i v o r e s (m y), w h i c h a r e a s s u m e d t o f e e do n p l a n t s, l i v in g o r g a n i s m s , a n d d e t r i tu s , i s e q u a l t o t h e r i g h t - h a n d s i d eo f eqn ( 22 ) ; t he po t en t i a l b i omas s o f s t r i c t p r eda t o r s ( r np ) i s equa l t oK 4 W z /( K ~ + K 5 + K 6), w he r e W , is g i ven by eq n ( 22 ); t he po t en t i a l b i o -m a s s o f d e t r i t a l f e e d e r s i s e q u a l t o K 7 W 3 / ( K 4 + K s + K c ,) , w h e r e W3 isg i v e n b y e q n ( 2 3 ); a n d s o f o r th .

T a b l e 3 is a s i m p l e a p p l i c a t i o n o f e q n s ( 2 1 ) - ( 2 3 ) to a p r a w n p o n d i nH a w a i i . C a r b o n l o a d i n g r a t e s a s s o c i a t e d w i t h p h o t o s y n t h e s i s a n da p p l i e d f e e d s a s w e l l a s t h e t u r n o v e r t i m e o f t h e p h y t o p l a n k t o n a r ek n o w n o r c o u l d b e c a l c u l a t e d f r o m d a t a i n C o s t a - P i e r c e ( 1 9 8 4 ) . T h et u r n o v e r t im e s o f th e h e t e r o t r o p h s a n d d e t ri tu s w e r e s i m p l y a s s u m e d .T h e v a l u e s o f K j, K2 , a n d K 3 a p p e a r t o b e r e a s o n a b l e b a s e d o n I sa a c s'( 1 9 7 2 ) c a l c u l a t i o n s . T h e v a l u e s o f K 4 - K 9 w e r e c h o s e n t o h a v e t h es a m e r e l a t i v e m a g n i t u d e a s K j , K 2 , and K ~ w h i l e s a t i s f y i ng eqns( 1 9 ) -( 2 0 ) . E q u a t i o n s ( 2 2 ) a n d ( 2 3 ) th e n r e d u c e t o t w o s i m p l e li n e a re q u a t i o n s i n v o l v i n g rn~ a n d rn 3. T h e c a l c u l a t e d b i o m a s s o f h e t e r o -t r o p h s i n te r m s o f c a r b o n i s 1 3 8 k g h a - ~ c o m p a r e d t o th e k n o w n


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170 E. A . La wsT A B L E 3Appl ica t ion of Uns t ruc tured Food Web Mode l to P rawn Ponds ; Biomass Expressed

i n T e r m s o f Or g a n ic C a r b o nm~ = 22.5 kg ha- ~ Co sta-Pierce , 1984, p . 58)rp = 3 da ysm 3 = 12 kg ha - t ( 40 kg ha - ~da y- l x 10% ca rbon x 3 days ; Co s ta P ie r ce , 1984 ,p. 90)r h = 30 daysr d = 60 daysK l = 0 . 2Ka = 0-02Kr =0-01

K, = 0 - 3 K3 = 0 . 5K.~ = 0-03 K a= 0 .0 5K s = 0"015 K , = 0"025W , =4 rn ~ + 4 m 3 = 13 8 k g h a -~,"~= 1 2 m I + 3 2 m 3 = 6 5 4 k g h a - 1S tanding c rop of p r a w ns= 112 kg ha -~ (900 kg ha -~ x 25% organic con ten t x5 0 % c )Harves t ing ra te = 0.69 kg ha - ~d a y - ~= 1 .9% of p r awn s tand ing c rop every th r ee days

s t a n d i n g c r o p o f p r a w n c a r b o n o f 1 1 2 k g h a -~ . N o t e t h a t th e a c t u a lp r a w n h a r v e s t i n g r a t e o f a b o u t 1 . 9 % e v e r y t h r e e d a y s is a b o u t 4 0 % o ft h e a s s u m e d v a l u e o f K 6. I f t h e p r a w n h a r v e s t i n g r a t e w e r e i n c r e a s e d ,K 6 w o u l d a l so i n c r e a s e a n d t h e t u r n o v e r t i m e o f t h e p r a w n s a t s t e a d ys ta te w o u l d d e c r e a s e . T h e m o d e l o b v i o u s l y p r o v i d e s a m e c h a n i s m f o rp r e d i c t in g t h e e f f e c t s o f m a n a g e r i a l m a n i p u l a t i o n s o n t h e p r o d u c t i v i t yo f t h e s y s t e m . F o r e x a m p l e , r e d u c i n g t h e r a te o f f e e d a p p l i c a t i o n s b y af a c t o r o f t w o i s p r e d i c t e d t o r e d u c e t h e h e t e r o t r o p h s t a n d i n g c r o p b yo n l y 1 7 % .

R E F E R E N C E SC o s t a - P i e r c e , B . ( 1 9 8 4 ) . E c o l o g i c a l s t u d i e s o f se m i - i n t e n s i v e p r a w n a q u a -c u l t u r e p o n d s . P h D d i s s e r t a t i o n , U n i v e r s i t y o f H a w a i i, 2 0 8 p p .I sa a cs , J . D . ( 1 9 7 2 ) . U n s t r u c t u r e d m a r i n e f o o d w e b s a n d ' p o l l u t a n ta n a l o g u e s ' . Fish . Bull . , 7 0 , 1 0 5 3 - 9 .I sa a c s, J . D . ( 1 9 7 3 ) . P o t e n t ia l tr o p h i c b i o m a s s e s a n d t r a c e - s u b s t a n c e c o n c e n -t r at i o n in u n s t r u c t u r e d m a r i n e f o o d w e b s. Mar. Bio l . , 2 2 , 9 7 - 1 0 4 .L a n g e , G . D . & H u r l e y , A . C . ( 1 9 7 5 ) . A t h e o r e t i c a l t re a t m e n t o f u n s t r u c t u r e df o o d w e b s. Fish . Bull . , 7 3 , 3 7 8 - 8 1 .

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