Mathematical Modelling in Fish Pond

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Ecological M odelling, 21 (1983/1984) 315-337 315

Elsevier Science Publishers B.V., Amste rdam - Printed in The Ne therlands

M A T H E M A T I C A L M O D E L L I N G O F A F I S H P O N D E C O S Y S T E M

Yu.M. SVIREZHEV, V.P. KRYSANOVA and A.A. VOINOV

Computer Centre of the U.S.S.R. Academy of Sciences, Moscow (U.S.S.R.)

(Accepted for publication 19 October 1983)

ABSTRACT

Svirezhev, Yu.M., Krysanova, V.P. and Voinov, A.A., 1984. Mathematical modelling of a fish

pond ecosystem. Ecol. M odelling, 21: 315-337.

A mathematical model is constructed for a fish breeding pond for carp, silver carp and

bighead. The model is a system of ordinary differential equations describing the material

transformations in the ecosystem. It allows a choice of optimal regimes of the aeration,

feeding and fertilization of a pond for different climatic conditions in order to maximize the

yield.

1. INTRODUCTION

F i s h p o n d s h a v e lo n g b e e n u s e d b y m a n t o m e e t h is a l im e n t a r y a n d - - f i r s t

o f a l l - - p r o t e i n r e q u i re m e n t s . F i s h b r e e d i n g w a s h i g h ly d e v e l o p e d i n a n c i e n t

C h i n a . A h i g h e f f ic i e n c y is a c h i e v e d o n l y w i t h o p t i m u m v a l u e s o f t h e c o n t r o l

p a r a m e t e r s , s u c h a s i n p u t o f f o d d e r a n d f e r t i l i z e r s , a n d r e - a e r a t i o n o f t h e

w a t e r b o d y , a n d w i t h a n o p t i m u m c h o i c e o f t h e s e e d p i e c e c h a r a c t e r i s t i c s .

T h e m a n a g e m e n t a f fe c ts t h e e n t i r e f is h p o n d e c o s y st e m , r e s u lt in g i n u n p r e -

d i c t a b l e a n d , e v e n m o r e , n o t a l w a y s d e s i ra b l e c h a n g e s in t h e e c o d y n a m i c s o f

t h e r es e rv o ir . A m a t h e m a t i c a l m o d e l p e r m i t s a n a s s e s sm e n t t o b e m a d e o ft h e c o n s e q u e n c e s o f d i f fe r e n t c o n t r o l s t ra t e g ie s a n d a n e s t i m a t i o n o f a ll

p o s s i b l e t r a n s f o r m a t i o n s i n t h e w h o l e c o m p l e x o f c a u s e - e f f e c t r e l a t i o n s i n

t h e e c o s y s t e m .

A t t e m p t s t o m o d e l a f is h p o n d h a v e b e e n m a d e b y V i n b e rg a n d A n i s im o v

( 19 6 6) , B o r s h e v (1 9 77 ), a n d o t h e r s . H o w e v e r , t h e e n e r g y a p p r o a c h u s e d b y

V i n b e r g a n d h i s su c c e ss o rs , a l th o u g h p r o v i d i n g a n a d e q u a t e q u a l i t a t iv e

e c o s y s t e m d e s c r i p t io n , g iv e s l it tl e in s i g h t i n t o t h e m a n a g e m e n t o f a n e c o sy s -

t e m . T h e r e f o r e o u r m o d e l w a s b a s e d o n t h e m o d e l s o f la k e e c o s y s te m s

( J o rg en s en , 1 9 8 0 ; V o i n o v e t a l ., 1 9 81 ) .T h i s p a p e r p r e s e n t s a f i s h p o n d s i m u l a t i o n m o d e l . T h e f o o d c h a i n

s t r u c t u r e a n d t h e s e t o f m o d e l p h a s e v a r i a b l e s a r e f i x e d . I t i s a s s u m e d a

0304-3800/84/$03.00 © 1984 Elsevier Science Publishers B.V.



316

p r i o r i t h a t t h e e c o s y s t e m i n q u e s t i o n a l r e a d y i n c l u d e s t h e b i o l o g i c a l s p e c i e s

a n d t h a t a m a n a g e m e n t m o d e is e n vi s ag e d t h a t c a n r e su l t i n t h e o p t i m u m

y i e ld . T h e m o d e l w a s i d e n t i f i e d f o r l it e r a t u r e d a t a a n d g i ve s a q u i t e r e a s o n a -

b l e r e s p o n s e t o c h a n g e s i n c l i m a t i c f a c t o r s a n d c o n t r o l p a r a m e t e r s , t h e r e b ys e r v i n g a s a s i m u l a t o r f o r t h e a n a l y s i s o f a l l k i n d s o f p o s s i b l e d e v e l o p m e n t s

i n a f i s h p o n d e c o s y s t e m . S u i t a b l e o p t i m i z a t i o n t e c h n i q u e s , d e r i v e d o n t h e

b a s i s o f t h i s m o d e l , a l l o w o n e t o d e f i n e o p t i m u m c o n t r o l s t r a t e g i e s f o r a n

e c o s y s t e m .

O n t h e o t h e r h a n d , f i s h p o n d s m a y s e r v e a s u s e f u l m o d e l s f o r t h e a n a l y s i s

o f e c o s y s t e m p r o p e r t i e s i n g e n e r a l , d u e t o t h e i r r e l a t i v e l y s i m p l e t r o p h i c

s t r u c t u r e a n d t h e h i g h i n t e n s i t y o f th e b i o t i c m a t e r i a l a n d e n e r g y t r a n s f o r m a -

t io n s . F r o m t h is p o i n t o f v ie w th e m a t h e m a t i c a l m o d e l l i n g o f f is h p o n d s i s o f

s o m e g e n e r a l e c o l o g i c a l a n d t h e o r e t i c a l s ig n i f ic a n c e .

2. MATERIAL CYCLES IN A FISH PO ND

W h e n a n a l y z i n g a c o n c r e t e p r o b l e m , i t i s n e c e s s a r y t o c h o o s e a d e g r e e o f

g e n e r a l i t y t h a t i s a m p l e f o r m e e t i n g t h e p u r p o s e s o f m o d e l l i n g . I n o u r c a s e ,

t o m o d e l a n o p t i m u m f i s h p o n d , w e h a v e t o c h o o s e v a r i a b l e s t h a t w o u l d

fu l ly r e f l ec t the spec i f i c i ty o f f i sh ponds g iv ing s t ab le h igh y ie ld s ove r long

p e r i o d s o f t i m e .

F r o m e x p e r i e n c e , t h e j o i n t b r e e d i n g o f c a r p a n d h e r b i v o r o u s f i s h ( B ig -h e a d , S i l v er C a r p , W h i t e A m u r , e t c .) is v e r y e f fe c t iv e . T h e y c o m p l e m e n t e a c h

o t h e r w e l l e n o u g h , a s t h e y o c c u p y a l m o s t n o n - o v e r l a p p i n g e c o l o g i ca l n i ch e s .

A l t h o u g h t h e s e s p e c i e s m a y c o m p e t e f o r f o o d , t h e y p r e f e r d i f f e r e n t n a t u r a l

f e ed s : b e n t h o s f o r C a r p , p h y t o p l a n k t o n f o r S il ve r C a r p , z o o p l a n k t o n f o r

B i g h e a d , a n d m a c r o p h y t e s f o r W h i t e A m u r . T h e l a s t s p e c i e s h a s n o t b e e n

i n c l u d e d i n t o t h e m o d e l d u e t o it s re l a ti v e i n d e p e n d e n c e f r o m t h e o t h e r

e c o s y s t e m c o m p o n e n t s .

I n d e s c r i b i n g t h e n a t u r a l e n r i c h m e n t o f t h e f o d d e r s u p p l ie s , it w i l l b e

l o g i c a l t o t a k e i n t o a c c o u n t t h e c o n c e n t r a t i o n s o f t h e t w o m o s t u s u a l l yl i m i t i n g n u t r i e n t s , i . e . n i t r o g e n a n d p h o s p h o r u s . T h e y a r e s u p p l e m e n t e d b y

t h e b a c t e r i a l d e s t r u c t i o n o f d e a d o r g a n i c s - - d e t r i t u s - - a n d a l s o f r o m t h e

i n p u t o f a r t i f i c i a l f e r t i l i z e r s . F i n a l l y , t h e d i s s o l v e d o x y g e n c o n c e n t r a t i o n i s

q u i t e a n - i m p o r t a n t , a n d s o m e t i m e s d e t e r m i n i n g f a c t o r o f t h e f i s h p o n d

e c o s y s t e m .

H e n c e , t h e m o d e l i n c l u d e s t h e f o ll o w i n g p h a s e v a r ia b le s : p h y t o p l a n k t o n

( F ) , z o o p l a n k t o n ( Z ) , b e n t h o s (B ) , C a r p ( C ) , B i g h e a d ( H ) , S ilv er C a r p

( S ) , d i s s o l v e d m i n e r a l p h o s p h o r u s ( P ) , d i s s o l v e d i n o r g a n i c n i t r o g e n ( N ) ,

d i s s o l v e d o x y g e n ( O ) , a r t i f i c i a l f o d d e r (A) , d e t r i t u s c o m b i n e d w i t hb a c t e r i a ( D ) . I t i s a s s u m e d t h a t a c o n c r e t e e c o s y s t e m c a n r o u g h l y b e d e -

s c r ib e d b y s u b s t it u t in g t h e c o m p l e x m u l t i- s p e ci e s c o m m u n i t y s t r u c tu r e w i t h



317

a s i m p l i fi e d b l o c k p a t t e r n . I n t h is c a se a s e p a r a t e b l o c k ( f o r i n s t a n c e , F o r Z )

m a y c o n t a i n d o z e n s o f s p e c i e s . S u c h a s u b s t i t u t i o n m a y b e c o n s i d e r e d

c o r r e c t i f a l l t h e s p e c i e s w i t h i n o n e b l o c k h a v e c l o s e v a l u e s o f t h e i r p r i m e

e c o l o g ic a l p a r a m e t e r s ( m a x i m u m g r o w t h r a te s , r e s p i r a t i o n c o e f f i c ie n t s , e tc .) .L a t e r , c e r t a i n v a r i a b l e s c a n b e d i s a g g r e g a t e d ( e. g. F , Z , B , D ) ; n e w v a r i a b l e s

c a n b e a d d e d i n t o t h e m o d e l (e .g . m a c r o p h y t e s , o t h e r f is h s pe c ie s) .

T h e i n t e r a c t i o n b e t w e e n t h e p h a s e v a r i a b l e s i s d e s c r i b e d a c c o r d i n g t o t h e

s c h e m e o f t h e m a t e r i a l c y c le p r e s e n t e d i n F i g . 1 . I t is a s s u m e d t h a t s u c h a

s c h e m e c o m p r e h e n s i v e l y r e fl e c ts t h e m a t e r i a l t r a n s f o r m a t i o n p r o c e s s e s i n t h e

p o n d . S i n c e t h e d i s s o l v e d o x y g e n (DO) h a s a c o n t r o l l i n g , r e g u l a t i n g e f f e c t

u p o n d i f f e r e n t c h e m i c a l a n d e c o l o g ic a l p r o c e s se s , it is r e g a r d e d a s a s p e c ia l

/ • BIOGENIC ~ M INERALELEMENTS P , N ) FER TILIZERS

~ I SILVER ARP

• .

BIGHEAD ~ PHYTOPLANKTON

FORAGE I~

CARP

I I

I IZOOPLANKTON BOTTOM AUNA

DETRITUS + BACTERIA

Fig . 1 . Mate r ia l cyc le in the f i sh pond .



318

reaeration

p h o t o s y n t h e s i ~

O X Y G E N

r . ? .~

n

D E T R I T U S

Fig. 2. Inflows and outflows of oxygen n the fish pond.

variable. Figure 2 shows the consumption and replenishment of DO in theecosystem. It should be noted that the specific times inherent in the main

DO transformations are shorter than the times of other ecosystem processes

and, unlike the latter, should be measured in hours rather than in days. For

instance, many scientists stress that fish kills in ponds are most common in

the morning, i.e., the DO concentration definitely depends upon the hour of

the day. This is quite natural, taking into account that the intensity of

photosynthesis, the main source of DO in the ecosystem, is determined by

the intensity of the solar radiation. Therefore, the DO concentration is

analysed separately in the model with its own time step.When modelling a shallow pond with depths of about 1 m and a small

area (less than 1 ha) we may neglect the effects of the spatial distribution of



319

o r g a n i s m s a n d m a t e r i a l , a s w e a r e c o n s t r u c t i n g a lo c a l , i .e . a p o i n t m o d e l . A l l

t h e v a r ia b l e s a r e r e g a rd e d a s c o n c e n t r a t i o n s a n d t h e u n i t o f m e a s u r e m e n t is

m g / 1 . B y t h e c o n c e n t r a t i o n o f l i v i n g o r g a n i s m s w e m e a n t h e r a t i o o f t h e i r

t o t a l b i o m a s s t o t h e v o l u m e o f t h e w h o l e r e s e r v o i r . F u r t h e r , t h e s q u a r eb r a c k e t s [ ] w i l l s t a n d f o r t h e c o n c e n t r a t i o n o f t h e r e s p e c t i v e e c o s y s t e m

v a r ia b l e s . T h e f i s h - b r e e d i n g p o n d i s m o d e l l e d f o r f iv e m o n t h s ( f r o m A p r i l 1 5

t o S e p t e m b e r 1 5 ) .

T h e e x t e rn a l f o r c in g f u n c t io n s i n t h e m o d e l a r e t h e c l i m a t i c f a c t o r s - - w a t e r

t e m p e r a t u r e s a n d t h e t o t a l s o l a r r a d i a t i o n - - a s w e l l a s t h e c o n t r o l e l e m e n t s ,

s u c h a s t h e i n p u t o f a r t i f i c i a l f e e d , m i n e r a l f e r t i l i z e r s a n d t h e i n t e n s i t y o f

a r t i f i c i a l wa te r ae r a t ion .

3. BASIC M OD EL EQUATIONS

(a) N utrients uptake by phytoplankton

T h e p h y t o p l a n k t o n g r o w t h is a n i m p o r t a n t p r o c e ss , w h i c h d e p e n d s o n t h e

p r e s e n c e o f n u t r i e n t s i n t h e w a t e r a n d a l s o o n t h e e x t e r n a l f a c t o r s s u c h a s

t e m p e r a t u r e ( T ) a n d i l l u m i n a t i o n ( L ) . T h e n u t r i e n ts l im i t i n g t h e p h y t o -

p l a n k t o n g r o w t h i n a fi sh p o n d m a y b e n i t ro g e n o r p h o s p h o r u s . T h e c y c l e s

o f t h e s e e l e m e n t s a r e c lo s e l y r e l a t e d i n t h e e c o s y s t e m . T h e u p t a k e r a t e s o f

t h e t w o e l e m e n t s m a y b e v i e w e d a s s y n c h r o n i s e d a c c o r d i n g t o t h e s t o ic h i o-m e t r i c ra t i o , i .e . t h e N / P r a t i o in t h e li v i n g o r g a n i c m a t t e r . F o r d i f f e r e n t

e s ti m a te s t hi s r a ti o c a n b e t a k en f r o m N / P = 1 0 / 1 t o N / P = 5 / 1 . T h e

p h y t o p l a n k t o n g r o w t h is l i m i te d a c c o r d i n g t o t h e s t o i c h io m e t r i c r a ti o . T h u s ,

a c c o u n t i n g f o r t h e e f f e c t o f e x t e r n a l f a c t o r s , w e c a n r e p r e s e n t t h e p h y t o -

p l a n k t o n g r o w t h r a t e a s f o ll ow s :

= t~T"x × FT (1 ) x F F (L , [ F ] , [ D ] )

x m i n ( [ P ] " 1 [ N ] " ) X [ F ]

K~FT[-P]" ' m K~VF+[N]nw h e r e g ~ ' = p h y t o p la n k t o n m a x i m u m g r o w t h r ate , K p F = h a l f - s a t u r a t i o n

c o n s t a n t f o r p h o s p h o r u s u p t a k e , K N F h a l f - s a tu r a t i o n c o n s t a n t f o r n i t ro g e n

u p t a k e , a n d m = s t o i c h i o m e t r i c r a ti o .

T h e r e l a t i o n s h i p b e t w e e n t h e p h y t o p l a n k t o n g r o w t h r a t e a n d t e m p e r a t u r e

i s d e s c r i b e d b y t h e m o d i f i e d L e h m a n f u n c t i o n ( J o r g e n s e n , 1 9 8 0 ) ( F i g . 3 ) :

e x p (

F T ( 1 ) =

exp (

- 4 . 6 x

- 4 . 6 x

( T O ( i ) - r ) ' )Fff , r < t O ( l )

( T - T O ( l) ) 4)0 2 ( 1 ) , T>~ t O ( l )



32 0

F T

1 .

o \0 1 0 2 0 3 0 4 b

T E M P

F ig . 3 . T e m p e r a tu r e l im i t a ti o n f u n c t io n f o r g r o wth .

w h e r e T O ( l ) i s t h e o p t i m u m t e m p e r a t u r e f o r t h e p h y t o p l a n k t o n d e v e l o p -

m e n t . Q I ( 1 ) = To~ - T 1 . i s t h e d i f fe r e n c e b e t w e e n t h e o p t i m u m a n d m i n i-

m u m t e m p e r at u re s , Q 2 ( 1 ) = 1m'~x - Tolpt i s the d i f fe ren ce b etw ee n m ax im um

a n d o p t i m u m t e m p e ra t u re s .

Fo l low ing S tee le (1962) , w e r ep resen t the l igh t l imi ta t ion fu nc t io n fo r the

F F

1

o ' 2ob o' 4o'oo' ~'oo ~5 oo 'I o6 ooL I G H T

F ig . 4 . L ig h t lim i t a t i o n f u n c t io n f o r p h y to p l a n k to n g r o wth .



32 1

p h y t o p l a n k t o n g r o w t h a s ( F ig . 4 ):

FF(L,[F] [ D I ) = _ L e xp (1 - L )' L o p ,

w h e r e L = L 0 e x p ( - k × h ) i s th e i l l u m i n a t i o n a t a d e p t h o f h c a l c u l a t e d b y

t h e B u r - L a m b e r t e m p i r ic a l f o r m u l a i n t e r m s o f t h e t o t a l s o la r r a d i a t io n L 0

a n d t h e e x t i n c t io n c o e f fi c ie n t , k . T h e l a t t e r d e p e n d s o n t h e c o n c e n t r a t i o n s o f

p h y t o p l a n k t o n a n d d e t r it u s in t h e w a t er :

k = K W + K F X [ F I + K D X [ D I X K P D

w h e r e KW = t h e l i g h t e x t i n c t i o n c o e f f i c i e n t f o r t h e w a t e r , KF = t h e p h y t o -

p l a n k t o n s e lf - sh a d i ng p a r a m e t e r , KD = t h e s h a d i n g p a r a m e t e r f o r s u s p e n d e d

d e t r i tu s , a n d KPD = a f r a c t io n o f d e t r i t u s s u s p e n d e d i n w a t e r . T h e u p t a k e o fn u t r i e n t s i s p r e s e n t e d b y s - s h a p e d t r o p h i c f u n c t i o n s :

V ( [ X ] ) = [X]"K " + [ X I "

w h e r e [ X ] = is t h e s u b s tr a t e c o n c e n t r a t i o n , n = a d i m e n s i o n l e s s q u a n t i t y

c h a r a c t e r i z i n g t h e s t e e p n e s s o f t h e f u n c t i o n ( s e e F i g . 5 ) . I n o u r m o d e l n = 2 .

I n t h i s c a s e t h e f u n c t i o n v a n i s h e s w i t h z e r o d e r i v a t i v e , w h i c h i s v e r y

i m p o r t a n t f o r t h e s ta b il it y o f t h e c o m p u t e r r e a l is a ti o n o f t h e m o d e l . N o t e

t h a t t h e e x p e r i m e n t a l d a t a c a n n o t p r o v i d e a n o b j e c t iv e c r i t e ri o n f o r t h ec h o i c e o f n: n e a r z e r o t h e s - s h a p e d f u n c t i o n s w i t h n = 1 ,2 o r 3 a p p r o x i m a t e

t h e e x p e r i m e n t a l r e s u lt s w i t h p r a c t ic a l ly t h e s a m e a c c u r a c y .

V2~

J n - - - - 3

n = 2

I - - - -

0 1 0 2 0 3 0 40 5bX

Fig. 5. Trophic functions.



322

(b) F eeding

L e t u s d e s c r i b e t h e f e e d i n g o f h i g h e r tr o p h i c l e v e l o r g a n i s m s o n o r g a n i s m s

o f l o w e r l e v e l s w i t h a n e x a m p l e o f z o o p l a n k t o n g r a z i n g o n p h y t o p l a n k t o n :

qF z = F T ( 2 ) X F O ( 2 ) X V (~ t~ .~ ' , K F Z, [ F ] ) X [ Z ] .

T h e t e m p e r a t u r e f u n c t i o n F T ( 2 ) i s s i m i l a r t o F T ( 1 ) . T h e F O ( 2 ) f u n c t i o n o f

t h e l o g i st ic t y p e ( F i g. 6 ) a c c o u n t s f o r t h e r e l a t i o n s h i p b e t w e e n t h e z o o p l a n k -

t o n g r o w t h a n d t h e p r e s e n c e o f D O i n t h e w a t e r :

1F O ( 2 ) - -

1 + e x p ( - X ( 2 ) × ( [ 0 1 - m ( 2 ) ) )

w h e r e m ( 2 ) i s t h e o x y g e n h a l f - m a i n t e n a n c e c o e f f i c i e n t , i . e . t h e [ 0 ] v a l u e i nw h i c h F O ( 2 ) = 1 / 2 . X (2 ) is t h e p a r a m e t e r c h a r a c t e r i z i n g t h e s te e p n e s s o f t h e

c u r v e . T h e t r o p h i c f u n c t i o n i s d e t e r m i n e d b y a n s - s h a p e d c u r v e :

/zrm~ - [ F ] "

V(ZF , g v z , [ F ] ) - +

where /~vmz' i s t h e z o o p l a n k t o n m a x i m u m g r o w t h r a t e w h e n g r a z i n g p h y t o -

plankton, Kvz i s t h e h a l f - s a t u r a t i o n c o n s t a n t f o r t h e p h y t o p l a n k t o n u p t a k e

b y z o o p l a n k t o n , a n d n - - 2 .

(e) Feeding with switching

I t f o l l o w s f r o m t h e l i t e r a t u r e t h a t s o m e f i s h a r e c h a r a c t e r i z e d b y t h e

se lec t iv i ty o f f eed ing , i .e ., f eed ing wi th swi tch ing . Carp , f o r in s t anc e , p r e f e r s

FO1 " '

o ; ~ ~ ~ ~ 6 ~ & ~ ~ b02

F i g . 6 . O x y g e n l i m i t a t i o n f u n c t i o n f o r g r o w t h .



3 2 3

t o f e e d o n b e n t h i c o r g a n i s m s , b u t , i n t h e c a s e o f a d e f i c it i n b e n t h o s , i t c a n

s w i t c h to f e e d i n g o n z o o p l a n k t o n . T o d e s c r ib e f e e d i n g w i t h s w i t c h i n g w e

u s e d t h e f i n d i n g s o f W . S . T a n w h o a n a l y z e d t h e e x p e r i m e n t a l d a t a o f I v l e v

(1 9 55 ) o n t h e r e l a ti o n s h i p b e t w e e n f e e d i n g a n d c o n c e n t r a t i o n o f f e e d i t e m sf o r c a r p . T h e i n t e r p r e t a t i o n o f t h e e x p e r i m e n t a l d a t a i s i m p e d e d b y t h e f a c t

t h a t m o d e l s t u d i e s u s u a ll y i n c o r p o r a t e n o t i o n s t h a t g i ve in s i g h t in t o t h e

e s s e n c e o f t h e p r o c e s s. T h e i r c h o i c e is a r b i t ra r y , t o a g r e a t e x t e n t ; w h e r e a s a n

e x p e r i m e n t o r u s e s o n l y t h e e m p i r i c a l l y a v a i l a b l e v a l u e s . N e v e r t h e l e s s , T a n

h a s f o u n d a r e l a t io n s h i p b e t w e e n t h e p r o b a b l i l i t y x i o f t h e u p t a k e o f t h e i t h

t y p e f e e d i t e m a n d t h e t o t a l c o n c e n t r a t i o n o f f e e d Q - i= a qi, w h e r e qi i s t h e

c o n c e n t r a t i o n o f t h e i t h t y p e p re y . T h e f e e d i te m s a r e li st e d a c c o r d i n g t o t h e

p r e f e r e n c e ; qa is t h e f a v o u r i t e f e e d . T h e n t h e p r o b a b i l i t y x a = 1 f o r a n y s e t

q = ( q a , q 2 , - . . , q n ) , w h e r e a s t h e c u r v e s x i (Q) f o r i = 2 ,3 . . . . . h a v e a n i n v e r t e ds - s h a p e d f o r m ( F ig . 7 ), w i t h x j < x i f o r j > i a n d l im x~(Q)= 1, l imx~ (Q ) = 0 . Q-~0 Q- - ,~

T h i s m e a n s t h a t t h e t r a n s i e n t r e g im e i s n o t c l e a rl y d e f i n e d b u t c o v e r s a n

i n t e rv a l o f Q - v a lu e s. H o w e v e r , th e d e p e n d e n c e o f x i o n Q w i t h l a r g e Q

v a l u e s s e e m s d o u b t f u l . A p p a r e n t l y , t h e f u n c t i o n s x 2 ( q l ) , x 3 ( q l , q 2 ), e tc . w i ll

b e m o r e a p p r o p r i a t e . T h u s t h e i n v e r t e d s - s h a p e d f u n c t i o n in t h e m o d e l is

u s e d t o c h a r a c t e r i z e th e f e e d i n g o f c a r p w i t h s w i tc h in g :

~ / ( [B ] , )~ , , rn B) = e x p ( - X , ( [ B ] - m , ) )

1 + e x p ( - ) ~ , ( I B ] - m s ) ) "

H e r e t h e Xs a n d m s p a r a m e t e r s h a v e t h e s a m e m e a n i n g a s f o r t h e s - s h a p e d

sw

oo Q 10 2b

F i g . 7 . U p t a k e p r o b a b i l i t i e s , x i , o f t h e i t h t y p e o f f ee d . Q = t o t a l c o n c e n t r a t i o n o f f e ed .



324

f u n c t i o n f o r o x y ge n . T h e r e f o r e, t h e u p t a k e o f z o o p l a n k t o n b y c a r p u n d e r a

d e f i c i t o f b e n t h o s c a n b e w r i t t e n i n t h e f o r m :

q z c = F T ( 4 ) × F O ( 4 ) × r a in ( [ K B c , B cR ) - - x , K s c , [ B ] ) ] ,

[ Z ] ) X n ( [ B ] , X . , m B ) ] ) X [ C ] .

H e r e F T ( 4 ) i s t h e c a r p g r o w t h r a te a s a f u n c t i o n o f te m p e r a t u r e , F O ( 4 ) is t h e

c a r p g r o w t h r a te a s a f u n c t i o n o f D O c o n c e n t r a t i o n .

T h e F T ( 4 ) a n d F O ( 4 ) f u n c t i o n s h a v e f o r m s s i m il a r t o t h o s e f o r p h y to - a n d

z o o p l a n k t o n , B cR is t h e c r it ic a l v a lu e o f t h e b e n t h o s c o n c e n t r a t i o n a t w h i c h

c a r p s w i tc h e s t o f e e d i n g o n z o o p l a n k t o n .

T h e d i f fe r e n c e b e t w e e n t h e t w o t r o p h i c f u n c t io n s u n d e r t h e m i n i m u m s ig n

e n s u r e s t h a t t h e g r o w t h r a t e o f c a r p i s n o m o r e t h a n t h e o n e a t t a i n e d f o r

B c R , w h e n s w i tc h i n g t o a n e w t y p e o f f e ed . T h e c o n s u m p t i o n o f b e n t h o s b y

c a r p i s r e p r e s e n t e d a s f o l l o w s :

qBC = F T ( 4 ) × F O ( 4 ) × v ( ~ r ~ ' , K B C , [ B ] ) × [ C ] .

B y a n a l o g y , w e d e s c r i b e t h e f e e d i n g w i t h s w i t c h i n g f o r b i g h e a d , w h o s e

f a v o u r i t e f e e d a c c o r d i n g t o t h e li t e ra t u r e is z o o p l a n k t o n . P h y t o p l a n k t o n is

t h e s u b s t i t u t i n g f e e d , a n d d e t r i t u s i s t h e c o n s t r a i n e d f e e d . I n t h i s c a s e w e

h a v e a t w o - s t e p s w i t c h i n g .

( d ) M e t a b o l i s m

T h e e x c r e t i o n o f t h e p r o d u c t s o f m e t a b o l i s m b y t h e l i v i n g o r g a n i s m s o f

t h e e c o s y t e m m a y b e c o n s i d e r e d to b e a p p r o x i m a t e l y i n p r o p o r t i o n t o t h e

t o ta l u p t a k e o f f o o d . T h u s , t h e e x c r et io n o f t h e p r o d u c t s o f m e t a b o l i s m a n d

t h e i r t r a n s f o r m a t i o n i n t o d e t r i t u s is r e p r e s e n t e d i n t h e f o l l o w i n g w a y ( f o r

z o o p l a n k t o n ) :

q ( 1 ) = M B z × ( q F Z + q DZ )D

w h e r e M B z is t h e m e t a b o l i s m p a r a m e t e r f o r z o o p l a n k t o n , qF Z i s t h e u p t a k e

f u n c t i o n o f t h e p h y t o p l a n k t o n b y z o o p l a n k t o n , qD z is t h e f u n c t i o n o f

d e t r i t u s u p t a k e b y z o o p l a n k t o n , ( q F z + q D Z ) is t h e r a t i o n o f z o o p l a n k t o n .

E n e r g y l o s s e s a r e t a k e n i n t o a c c o u n t b y t h e o u t f l o w

q zE = M B O z × [ Z ]

w h e r e M B O z is t h e z o o p l a n k t o n r e s p i r a t i o n c o e f f i c ie n t .

M o r e o v e r , t h e r e l a t i o n s h i p b e t w e e n f o o d a s s i m i l a t i o n a n d t h e r a t i o n

v a l u e s is t a k e n i n t o a c c o u n t f o r fi sh . F o r i n s t a n c e , i n t h e a b u n d a n t f e e d i n g o fs il v er c a r p , t h e f o o d i s c o n t i n u o u s l y s w a l l o w e d a n d p a s s e s t h r o u g h t h e

i n t e s t in e s o q u i c k l y t h a t o n l y 3 0 - 4 0 % o f it c a n b e a s s i m i la t e d . W i t h



325

m o d e r a t e f e e di n g, a lm o s t t w ic e a s m u c h f o o d c a n b e a s s im i l a te d ( P u s h c h i n a ,

1975) .

(e ) Mor ta l i t y

A s w e k n o w f r o m t h e l it e ra t u r e , t h e m o r t a l i t y o f l iv i n g o r g a n i s m s d e p e n d s

o n t h e D O c o n c e n t r a t i o n i n t h e w a t e r . I f t h is e x t e r n a l f a c t o r i s c o n s t a n t ,

m o r t a l i t y i s i n t h e f ir s t a p p r o x i m a t i o n p r o p o r t i o n a l t o t h e b i o m a s s o r

c o n c e n t r a t i o n o f l i v i n g o r g a n i s m s . T h i s m o d e l a c c o u n t s f o r m o r t a l i t y o n l y

f o r p h y to - a n d z o o p l a n k t o n , a n d b e n t h o s . T h e r e s h o u l d b e n o n a t u r a l

m o r t a l i t y o f f is h in t h e o p t i m u m c o n d i t i o n s o f a f i sh - b r e ed i n g p o n d . P r o b a -

b l e f i s h k i l l s f r o m o x y g e n d e f i c i t a r e d e s c r i b e d b y a n o t h e r m e c h a n i s m ,

d e p e n d i n g o n t h e D O c o n c e n t r a t i o n i n t h e w a t e r. A r t if i ci a l a e r a t i o n i sp r o v i d e d w h e n o x y g e n c o n c e n t r a t i o n p a s s e s t h e t h r e s h o l d a n a e r o b i c v a l u e .

T h u s , t h e z o o p l a n k t o n m o r t a li t y c a n b e d e s c r ib e d i n t h e f o l lo w i n g w a y :

q(2) = F O X ( [ O ] ) X M z X [ Z ]Z D

H e r e F O X is t h e f u n c t i o n o f t h e m o r t a l it y d e p e n d e n c e o n t h e D O c o n c e n t r a -

t i o n .

F O X ( [0 ] ) = 1 + K A / [ 0 ] ,

M z i s t h e m o r t a l i t y c o e f f i c i e n t , K A i s t h e c o e f f i c i e n t o f m o r t a l i t y i n c r e a s e

u n d e r o x y g e n d e f i c i t .

(d ) Des t ruc t ion

T h e d e s t r u c t io n p r o c e ss o f t h e d e a d o r g a n ic m a t er ia l , w h i c h p r o d u c e s t h e

b a si c n u t r i e n t s - - p h o s p h o r u s a n d n i t r o g e n , - - d e p e n d s o n t h e th e r m a l c o n d i-

t i o n s a n d o n t h e p r e s e n c e o f D O i n t h e w a t e r . T h e r e f o r e , t h e f o r m a t i o n o f ,

s a y, m i n e r a l p h o s p h o r u s a s a r e s u lt o f d e t r i tu s d e s t r u c t i o n a n d d i s s o l u t i o n

c a n b e d e s c r i b e d i n t h e f o l l o w i n g w a y :

q o p = U D P X E l ( T ) X E 2 ( [ 0 ] ) X [ D ] .

H e r e t h e d e p e n d e n c e o f th e d e s t r u c t i o n r a te o n t e m p e r a t u r e is g iv e n b y

t h e V a n t - H o f f f u n c t io n

E 1 (T ) = 2 ( r -2 ° )/ a °

T h e o x y g e n f u n c t i o n f o r t h i s p r o c e s s is ( F ig . 8 ):

E 2 ( [ 0 ] ) = e x p [ C O P ( M - [ 0 ] ) ] / ( 1 + e x p [ C O P ( M - [ 0 ] ) 1 ) ,

w h e r e C O P i s t h e p a r a m e t e r o f t h e s t e e p n e s s o f th e o x y g e n c u rv e , M is t h e

t h r e s h o l d b e t w e e n t h e a e r o b i c a n d a n a e r o b i c c o n d i t io n s .



326

E2

1

oo ~ ~ b

O2

F i g . 8. O x y g e n f u n c t i o n f o r d e t r i t u s d e s t r u c t i o n p r o c e s s .

T h u s , u n d e r a n o x i c c o n d i t i o n s , m i n e ra l p h o s p h o ru s b e g i n s t o f l o w i n t e n -

s i v e l y f ro m s e d i me n t s i n t o t h e w a t e r . U n d e r a e ro b i c c o n d i t i o n s , o n l y t h e

i n f lo w o f p h o s p h a t e s f r o m d e tr it u s, d e c o m p o s e d in t h e w a t e r b o d y , i s w o r t h

c o n s i d e r i n g ; in t h e ma i n , t h e p ro c e s s t a k e s t h e o p p o s i t e t e n d e n c y : d i s s o l v e dp h o s p h a t e s t u rn t o i n s o l u b l e f o rms a n d s i n k t o t h e b o t t o m.

qpD = SEDP X E 3 ( [0 ] ) X [ P ] ,

w h e r e

/ 0 , [ 0 ] [ 0 ] < M

E 3 ( [0 1 ) = - M [ 0 ] > ~ M

[ [ 0 1 - COD'

H e r e COD = M- CK, w h e r e C K i s t he s t eepness o f t he oxygen func t ion ,a n d S E D P is t h e m a x i m u m s i n k in g ra t e o f p h o s p h o r u s .

(h) Oxyge n f lows

T h e o x y g e n c o n t e n t i n t h e f is h p o n d w a t e r d e p e n d s o n t h e e n r i c h m e n t o f

w a t e r w i t h o x y g e n a n d t h e r a te o f i ts c o n s u m p t i o n . T h e i n f l o w o f o x y g e n d u e

t o p h o t o s y n t h e s i s is p r o p o r t i o n a l t o t h e p h y t o p l a n k t o n p r o d u c t i o n a n d c a n

b e d e s c r i b e d b y t h e fu n c t i o n .

qFo = P H OT x It,

w h e r e P H O T i s t he a ss im i l a t ion coe f fi c i en t.



327

T h e e x c h a n g e p r o c e s s e s w i t h a t m o s p h e r i c o x y g e n a r e c h a r a c t e r i z ed b y t h e

f o r m u l a

R=RE×(Os--[OI),w h e r e R E i s t h e r e a e r a t i o n c o e f f i c i e n t t h a t i s d e p e n d e n t o n t h e w i n d v e l o c i t y

i n t h e g e n e r a l c a s e . O s i s t h e o x y g e n s a t u r a t io n c o n c e n t r a t i o n . F o l l o w i n g

W a n g e t a l . ( 1 9 7 8 ) ,

O s - - 1 4 . 6 1 9 9 6 - 0 . 4 0 4 2 0 × T + 0 . 0 0 8 4 2 × T 2 - 0 . 0 0 0 0 9 × T 3.

I n f l o w o f o x y g e n w i t h a r t i f i c i a l a e r a t i o n i s a l s o t a k e n i n t o a c c o u n t .

T h e c o n s u m p t i o n o f o x y g en f or th e r e sp i r a ti o n o f a q u e o u s o r g a n i s m s a n d

p l a n t s i s p r o p o r t i o n a l t o th e ir b i o m a s s , f o r e x a m p l e :

qoz = RESPz × [ Z ] f or z o o p l a n k t o n ,

T A B L E I

D i f f e r e n t i a l e q u a t i o n s , l e v e l 1

d F

d - ~ = q a F - - q F Z - - q F B - - q F $ - - q F H - - q F D - - q F E

d Z

d t = q F z + q o z - - q z B - - q z c - - q z n - - q z D - - q Z E

d B

d -- 7 = q ~ m + q z B + q D B - - q n c - - q B D - - q B E

d C

d- -- 7 = q A c + q B c + q z c - q c D - q c e

d S

d ~ = q ~ + q o s - q s D - q s E

d H

d t = q z H + q F n + q D H - - q n o - - q H E

d P

d- -- ~ = q o e + P U ( t ) - q e F - q e D

d N- ~ = q o N + N U ( t ) - q N F

d[O]d t = q F o + R E > ( R E A - [0])+ O U ( t ) - q o F - - q o z - - q o B

- qoc - - qos - qOH - - q o D

d A

d ~ = A U ( t ) - q A c - q , ~ D

d D

d ~ - t = q F D + q Z D + q B D + q C D + q S D + q n D + q A D

- q o e - q D ~ - q D z - q D B - - q D s - - q D u - - S E D × D



328

TABLE II

Flows, level 2

p2 1 N 2

qGF = FT(1) × FF (L , F, D) × #~ x × min( 2Kp F + p2 ' 5 K ~ r + N2 ) X F

__ 1

qPr -- Trig )< qGF

__ 5

qNF -- T6g X qGr

qFZ = FT(2)X FO(2) × V( p,~x, Krz , F)× Z

qoz = FT(2)× FO(2)× V( #~ , Koz, O )× Z

qpn = FT (3) × V(Fn~, KFB F ) × B

qzn = FT(3)X V(/,t~x, KzB, Z)X Bqos = FT(3)× V (# ~ , Kn s , D)× B

qAc = FT (4)X FO(4)X V(#'~.~, K a o a ) x C

qBc = FT(4) x FO(4) × V(I.t~", Knc, B)X C

qrs = FT(5) X FO(5) × V(#~-s x, K FS, F) × S

qos = FT(5) X FO(5) X V(#n~, Kos, D)X S

qzn = FT(6) X FO(6)X V ( # ~ , K z n , Z)X H

q z c = F T ( 4 ) × FO(4)×min([V(#~,K n c, B c R ) - - V ( t* ~ , K a o B ) ] ,

[ V ( # ~ x , K z o Z ) × 7q(B, An, r a n ) l) × C

Y1 = V( It ~, Kzn , ZcR ) - V(Ix'~, KZH, Z)

Y2 = V( #~ , KFr, F)X ~(Z, )~z, mz)

Y3 = V ( # ~ ~, KFr, FcR)× ~I( Z, )~z , mz)

qeH = FT(6 )× FO(6)×min(Y1, Y2)× H

qDn = FT (6)× FO ( 6) ×rrfin([min( Y1, Y3 )-nf in ( Y1, Y2 )],

[ V(# ~, g o n , D)X r/(Z, Xz, mz)X rt(F, XF, m F ) ] )

MB F X qcl:, F < Fo

qro = ( MB r x qcF + MF X F, F > Fo

qan = MBB X(qFB + qzB + qoB)

qzo = MBz X(qvz + qDz) + FOX([O])X M z X Z

q c D = ( M B c + M B B c X R C ) X R CRCm~,

RS

qsD = ( M Bs + MBBs × ~--~ff---)× RSetOmax



TABLE II (continued)

329

R Hqno = (M Bn + MB Bn × ~ )X RH

RC = qac + qBc + qzc

RS = qrs + qDs

RH = qZH + qFn + qon

qFE = MBOv X F

qZE = MB Oz × Z

qnE = MBOo × B

qce = MBOc X C

qse = MBOs X S

qHE = MBOn × H

qDP = UDP × E l( T) × E2([0])× D

qPD = SEDP × E3([0])× P

qDN = UDN X EI(T)X E2([0])× D

qFO = P H O T X qCF

qOF = RESPF X F

qoz = RES Pz X Z

qoB = RESPn X B

qoc = RESPc X C

qos = RESPs × S

qon = RESPH x H

qAD = AL PH A X A

w h e r e R E S P z i s t h e [ 0 ] c o n s u m p t i o n c o e f f i c i e n t f o r t h e z o o p l a n k t o n r e s p i r a -

t i o n .

T h e o x y g e n c o n s u m p t i o n f o r o x i d a t i o n o f d i ss o lv e d a n d s u s p e n d e d o r g a n i c

m a t e r i a l in w a t e r i s p r o p o r t i o n a l t o t h e d e t r i tu s a m o u n t i n v o l v e d i n t h e c y c l e

q o o = O K × [ D ] ,

w h e r e O K i s t h e o x i d a t i o n c o e f f i c i e n t .

T h u s , w e h a v e p r e s e n t e d t h e m a i n f l o w s o f t h e m a t e r i a l n e c e s s a r y f o r t h ed e s c r i p t i o n o f th e f is h p o n d e c o s y s te m . I t s h o u l d j u s t b e a d d e d t h a t a n i n p u t

o f f e e d f o r f i s h a n d a n i n p u t o f m i n e r a l fe r t il iz e r s a r e g i v e n b y e x t e r n a l



330

TABLE III

Functions, level 3

[ e x o ( 6 4 1Q I ( I ) ] 1'

F T ( I) = F T ( T , T O ( I) ,Q I ( I) ,Q 2 ( I )) = ( 4 " 6X ( T - T O ( I ) ] 4 ]

e x p 1 1 ,

L LF F ( L , F, D) = ~'opt exp(- k x h)xexp[1 - ~optmexp(- k × h)]

k = K W + K F x F + K D x D x K P D

FO (1 ) = 1/(1 + exp( - X(I)([0]- m(I))))• (X, )~, m) = e-X(x-'n)/(1 + e -x(x-m))

# m a x × X 2V(/.tmax, K, X)

K 1 + X 2

FOX([O])= 1 + KA/[0l

E l ( T ) = 2(r-z°)/l°

E2([0]) = exp[C O P (M - [0])]/(1 + exp[C O P ( M - [0])l)

O, [01 < ME3([0]) = [0] - M

~ [O]-COD' [O]>/M

T < T O (I )

T>~ T O ( I )

inflows, while the sedimentation process of detritus and the settling of

phosphorus (losses from the material cycle) are characterized by outflows.

All the model equations are given in Table I. The flows they include and the

functional relationships are listed in Tables II and III respectively.

4. SIMULATION EXPERIMENTS

The comp uter runs of the Fort ran pro gram for the model were carried out

on BESM-6. The system of 11 ordinary differential equations is solved by

the Runge-Kutta technique with an automatic choice of the step in the

interval (0,150). To solve the system we had to specify 112 parameters, the

arrays of temperature and illumination, the inflows of nitrogen and phos-

phorus fertilizers, of feed, and the regime of artificial aeration.As with all simulation models of this kind, paramet er estimation is quite a

problem. Some of the parameters were determined from previous works



331

( J o r g e n s e n e t al ., 1 9 7 8; W a n g e t a l. , 19 7 8) , s o m e c a n b e e s t i m a t e d o n t h e

b a s i s o f t h e l i t e r a t u r e d a t a , a n d o t h e r s s h o u l d b e i d e n t i f i e d a c c o r d i n g t o t h e

v a r i a b l e d y n a m i c s . M o r e d i f f i c u l t i e s a r i s e b e c a u s e c e r t a i n p a r a m e t e r s c h a r -

a c t e r i z e e x t r e m e l y a g g r e g a t e d p r o c e s s e s ; i t is r a t h e r d i f f i c u l t to f i n d u n i q u eq u a n t i t a t i v e v a l u e s f o r t h e m .

F o r m o d e l c a l i b r a t i o n w e u s e d t h e d a t a o f t h e P o l i s h s c i e n t i s t s ( O p u s z y n -

s k i, 1 9 7 8; W a s i l e w s k a , 1 9 7 8 ; G r y g i e r e k , 1 9 78 , et c .) , w h o c a r r i e d o u t c o m p l e x

s t u d i e s o n a g r o u p o f te s t p o n d s w i t h c a r p a s a m o n o c u l t u r e (4 ,0 0 0 f i s h / h a . )

a n d o n t h r e e g r o u p s o f p o n d s w i t h t h e a d d i t i o n o f s i l v e r c a r p ( 4 , 0 0 0 , 8 , 0 0 0

a n d 1 , 2 0 0 f i s h / h a . ) . T h e f i s h p o n d s w e r e r e g u l a r l y f e r t i l i z e d ( b y u r e a a n d

s u p e r p h o s p h a t e ) , a n d b a r l e y w a s u s e d a s t h e f o d d e r f o r c a r p . T h e p a r a m e t e r s

r e s u l ti n g f r o m t h e m o d e l i d e n t i f i c a t io n a r e p r e s e n t e d i n T a b l e I V .

TABLE IV

Ecological parameters

Notation Ecological meaning Units

~,~axma x

~ F Z

KFz

ma x

IXDZ

KD~

ma xI~ FB

KFB

I~ZB

KZB

ma xI~DB

KDB

I~ZC

K z cm a x

l~ BC

KBCm a x

I~ Ac

K ~

K FS

Maximum growth rate of phytoplankton 3.0 1/ day

Maximum uptake rate of phytoplankton

by zooplankton 1.4 1/day

The corresponding half-saturation

parameter 15.0 mg/lMaximum uptake rate of detritus by

zooplankton 0.5 1/day

The corresponding half-saturationparameter 60.0 mg/lMaximum uptake rate of phytoplankton

by benthos 0.2 1/dayThe corresponding half saturation constant 15.0 mg/1

Maximum uptake rate of zooplankton

by benthos 0.4 1/day

The corresponding half-saturation constant 1.0 mg/1

Maximum uptake rate of detritus bybenthos 0.2 1/day

The corresponding half saturation constant 60.0 mg/1


by carp 0.02 1/day

The corresponding half saturation constant 1.0 mg /l

Maximum uptake rate of benthos by carp 0.06 1/ dayThe corresponding half-saturation constant 5.0 mg/lMaximum uptake rate of artificial

feed by carp 0.03 1/dayThe corresponding half saturation constant 0.2 mg/1Maximum uptake rate of phytoplanktonby silver carp 0.1 1/day

The corresponding half-saturation constant 20.0 mg /l

(continued)



332

TABLE IV (continued)

Notation Ecological meaning Units

ma x

tXns Maximum uptake rate of detritusby silver carp 0.07 1/day

The corresponding half saturation constant 60.0 mg/1

Maximum uptake rate of phytoplankton

by bighead 0.1 1/da y

The corresponding half-saturation constant 20.0 mg/1


by bighead 0.15 1/da yThe corresponding half-saturation constant 1.0 mg/1

Maximum uptake rate of detritus by

bighead 0.1

The corresponding half-saturation constant 60.0Metabolism parameter for phytoplankton 0.3Metabolism parameter for zooplankton 0.3

Metabolism parameter for benthos 0.3

Minimum metabolism parameter for carp 0.3

for silver carp 0.3

for bighead 0.3

Additional metabolism parameter for carp 0.4

for silver carp 0.4

for bighead 0.4

Maximum ration for carp 13.0for silver carp 10.0

for bighead 10.0Respiration coefficient for phytoplankton 0.001

for zooplankton 0.001

for benthos 0.001

for carp 0.001for silver carp 0.001

for bighead 0.001Mortaility coefficient for phytoplankton 0.09

for zooplankton 0.005

for benthos 0.05DO consumption parameter for F respira-tion 0.001 1/day

for Z 0.11 1/dayfor B 0.01 1/dayfor carp 0.01 1/dayfor silver carp 0.01 1/ day

for bighead 0.01 1/ dayPhosphorus destruction parameter 0.00004Nitrogen destruction parameter 0.002

Assimilation coefficient 1.0

Transformation coefficient of fodder intodetritus 0.2Reaeration coefficient 0.3Detritus oxidation parameter 0.085Sedimentation parameter of detritus 0.05

KDSmax

~FH

KFHmax

I~zH

KZHmax

~DH

KDH

M B r

M B z

M B s

M B c

M Bs

MBI~

M B B c

M B B s

M B B ~

RCmaxRSm~

RHm~,

MBOF

M B O z

M B O B

M B O c

M B O s

M B O H

MF

M z

MeR E S P F

R E S P z

R E S P B

R E S P c

R E S P s

R E S P H

UDP

UD N

P H O T

A LP HA

R E

O K

S ED

1/day

mg/ldimensionlessdimensionless

dimensionless

dimensionlessdimensionless

dimensionless

dimensionless

dimensionless

dimensionless

rag/1rag/1

rag/1

dimensionless

dimensionless

dimensionless

dimensionless

dimensionless

dimensionless

1/day1/day

1/day



TABLE IV (continued)

Nota tion Ecological meaning Units

333

S E D P Sedimentation parameter of phosphorus 0.1TO (l ) Optim um temperature for F growth 24.0

TO(2) for Z 24.0

TO(3) for B 24.0

TO(4) for C 26.0

TO(5) for S 26.0

TO(6) for H 26.0

T l i . Minimum temperature for F growth 9.0

T E i . for Z 9.0

Tm3in for B 9.0

T4 . for C 10.0

TmSi. for S 13.0

T r . for H 13.0

T~ax Maximum temperature for F growth 34.0

T2ax for Z 34.0

Tm3ax for B 34.0

T4~x for C 35.0

TmS~ for S 35.0

Trax for H 35.0

Lop Opt imu m illumination for photosynthesis 3000.0

K W Light extinc tion coeffic ient in water 0.2

K F Self-shading parameter for phytoplankton 0.03K D Shading parameter of detritus 0.4

K P D Fractio n of detritus suspended in water 0.5

h Mean photosynthesis depth 0.1

m(2) Oxygen half-maintainance parameter for Z 3.0

m(4) for C 3.0

m(5) Oxygen half-maintainance parameter for S 3.0

rn (6) for H 3.0

X(2) Steepness of the oxygen curve for Z 1.0

X(4) for C 1.0

A(5) for S 1.0

A (6) for H 1.0B c R Critical value of benthos concentrati on 20.0

Z c R Critical value of zooplankton concentrati on 5.0

F cR Critical value of phytoplankto n concentrat ion 30.0

rn B Parameter of the switch function for B 10.0

rn z for Z 3.0

m F for F 15.0

As Steepness of the switch function for B 1.0

A Z for Z 1.0

A F for F 1.0

KA Parameter of mortality increase at D O deficit 2.0

m Stoichiometr ic ratio 5.0

C O P Parameter of the oxygen function 2.0

M Parameter of the oxygen function 1.0

C O D Threshold between oxic and anoxic condi-

tions 2.0

oC

oC

oC

oC

oC

oC

oC

oC

oC

oC

oC

*CoC

oC

°CoC

oC

oC



334

5. RESULT S, CONC LUSIONS AND DISCUSSION

T h e f o l l o w i n g g r a p h s s h o w d i f f e r e n t v a r i a n t s o f t h e e c o s y s t e m d e v e l o p -

m e n t . W h e n t h e n u t r i e n t c o n c e n t r a t i o n is m a i n t a i n e d a t a l e ve l o f P = 0 .0 6- 0 . 6 m g / 1 , N = 1 - 2 m g / 1 ( m o s t l y d u e t o f e r ti li z er s ), it t u r n s o u t t h a t t h e

p h y t o p l a n k t o n g r o w t h is l im i t e d b y t h e a m o u n t o f p h o s p h o r u s , a n d t h e

p h y t o p l a n k t o n d y n a m i c s c le a rl y fo ll ow s t h e d y n a m i c s o f p h o s p h a t e s ( in F i g.

1 0 t h e a r r o w s i n d i c a t e t h e m o m e n t s o f i n t r o d u c i n g t h e f e r ti li ze rs ).

U n d e r o t h e r c o n d i t io n s , w h e n t h e p h y t o p l a n k t o n g r o w t h is n o t l im i t e d b y

n u t r i e n t s , t h e n a t u r a l f o d d e r r e se r v e s a r e b e t t e r d e v e l o p e d . F i g . 11 s h o w s t h e

s u c ce s si o n o f t h e m a x i m a o f p h y t o p l a n k t o n , z o o p l a n k t o n , a n d b e n t h o s

c o n c e n t r a t i o n s . H o w e v e r , i n t h i s c a s e t h e z o o p l a n k t o n a n d b e n t h o s c o n -

c e n t r a t i o n s o n c e a g a i n d r o p s h a r p l y i n t h e m i d d l e o f t h e s e a s o n , t h i s b e i n gq u i t e n a t u r a l f o r s u c h a d e n s e f i s h p o p u l a t i o n .

T h e n e x t v a r i a n t s h o w s t h e e c o s y s t e m d e v e l o p m e n t i n t h e c o n d i t i o n s o f a

w a r m e r c l i m a t e ( F i g . 1 2 ) .

A s a r e s u l t o f t h e s i m u l a t i o n s , w e h a v e d e r i v e d t h e d y n a m i c s o f t h e

v a r ia b l e s t h a t a d e q u a t e l y r e f le c t th e r e a l p i c t u r e o f t h e d e v e l o p m e n t o f t h e

e c o s y s t e m fo r o n e s e as o n . T h e p h y t o p l a n k t o n g r o w t h is l im i t e d b y n u t r i e n t s

( m o s t l y p h o s p h o r u s ) . F i s h c o n s u m e s a lo t o f z o o p l a n k t o n , b u t i ts c o n c e n t r a -

t i o n m a y i n c r e a s e a t t h e b e g i n n i n g o f th e s e a s o n . A l o t o f b e n t h o s i s

c o n s u m e d b y c a r p . C a r p b e g i n s to g a i n w e i g h t a t t h e v e r y b e g i n n i n g o f t h es e a s o n , w h e r e a s b i g h e a d a n d s il ve r c a r p b e g i n t o g r o w o n l y i n J u ly , s i n c e

t h e i r g r o w t h i s t o a g r e a t e r e x t e n t l i m i t e d b y t e m p e r a t u r e . R i g h t a f t e r t h e i r

E3I"

J

J

o ~ lb02

Fig. 9. Oxyg en function for p hosphorus sinking process.



'° °

335

lOC

5 0

O 0 5~)--"~ ~ 1 0 0 ~ 1 5 0

TIME

F i g . 1 0. E c o s y s t e m d e v e l o p m e n t i n p h o s p h o r u s l i m i t e d co n d i t i o n s . + = p h y t o p l a n k t o n ,

× --- z o o p l a n k t o n , ~ = b e t h o s , [ ] = c a r p , t ~ = s i l v er c a r p , t~ = b i g h e a d .

150

5 0

. -- -- -- -- + ~ + ......_...

+ +

0 ~ " - - - - - - ~ - - ~ - - - - - - - % - - ~ - - - - - - - W ¢ ~ - - ' - - ' - - ; ~0 50 100 150

T I M E

F i g . 11 . E c o s y s te m d e v e l o p m e n t i n th e c o n d i t i o n s o f n u t r i e n t a b u n d a n c y . + = p h y t o p l a n k t o n ,



336

200-

150

1 0 0

50

0 5 0 1 0 0 1 5 0

T I M E

Fig. 12. Ecosystem development in the conditions of a warmer climate. (T= To +4°C).+ = phytoplankton, x = zooplankton, ~ = bethos, [] = carp, X = silver carp, N = bighead.

application, ferti l izers are quickly consumed, and the phytoplankton con-

centration increases. The oxygen control unit is so designed that the oxygen

content never drops below 3 mg/1. The artificial aeration is increased when

this threshold is passed. The amount of detritus has increased by the middle

of the season, and then begins to diminish due to its uptake by bighead and

silver carp.

The next stage envisages the search for the optimum regimes of the fish

pond. The control parameters are the inflows of fertilizers and fodder, the

artificial aeration, and the density of implantation. The mathema tical mo del

allows one to find the operating modes that maximize the yield.

REFERENCES

Borshev, V.N., 1977. Mathematical modelling experience in a simple fishpond ecosystem. In:

Fish Breeding in Ponds. VNIIPRH, 18:260-281 (in Russian).Grygierek, E., 1978. The influence of the silver carp on eutrophication of carp ponds. IV.

Zooplankton. Rocz. Nauk rol., H, 99: 81-92.



337

Iv lev , V.S ., 1955 . Exper im enta l eco logy o f f ish nour ishme nt . P ishchep rom izda t , M oscow , 252

pp . ( in Russ ian) .

Janus zko , M . , 1978 . The in f luence o f the si lve r ca rp on eu t ro ph ica t ion o f ca rp pon ds . I I I .

P h y to p l a n k to n . R o c z . Na u k r o l . , H , 9 9 : 5 5 - 8 0 .

Jo rgensen , S .E . , F r i i s , M.B . , Henr iksen , J . , Jo rgensen , L .A. and Meje r , H.F . (Ed i to rs ) , 1978 .H a n d b o o k o f E n v i r o n me n ta l D a ta a n d E c o lo g i c a l P a r a me te r s . I .S .E .M . , V~ eerl~ bse, De n -

m a r k .

Jo rgensen , S .E . , 1980 . Lake Management . Pergamon Press . Oxford , 167 pp .

Op uszyn sk i , B ., 1978 . The in f luence o f the s ilve r ca rp on eu t roph ica t ion o f ca rp pond s . V II .

R e c a p i tu l a t i o n . R o c z . Na u k r o l. , H , 9 9 : 1 2 7 - 1 5 1 .

P io t row ska , W . , 1978. The in f luence o f the s ilve r ca rp on eu t ro ph ic a t ion o f ca rp po nds . I .

P h y s i c o - c h e mic a l c o n d i t i o n s . R o c z . Na u k r o l. , H , 9 9 : 7 - 3 2 .

P u s h c h in a , L . I . , 1 9 7 5 . P h y to p l a n k to n o f f i s h b r e e d in g p o n d s i n Kr a s n o d a r r e g io n u n d e r

in tens ive management . Len ingrad , 23 pp . ( in Russ ian) .

S tee le , J .H. , 1962 . Env i ronm enta l con t ro l o f pho to syn th es is in the sea. L inm ol . Oceanog r . , 7 :

1 3 7 - 1 5 0 .

Vin b e r g , G .G . a n d An i s imo v , S . I . , 1 9 6 6 . M a th e ma t i c a l mo d e l o f a n a q u a t i c e c o s y s t e m. I n :

P h o to s y n th e t i c S y s t e ms o f H ig h P r o d u c t iv i t y , Na u k a , M o s c o w ( in R u s s i a n ) .

Vinberg , G.G . and L iachnov i tch , V.P ., 1965 . Fer t i l iza t ion o f the ponds . P ishche prom izda t ,

Moscow, 271 pp . ( in Russ ian) .

W a n g , L .K . , V ie lk in d , D . a n d W a n g , M .H. , 1 97 8. M a th e ma t i c a l m o d e l s o f d i s s o lv e d o x y g e n

concen t ra t ion in f resh wate r . Eco l . Model l ing , 5 : 115-123 .

W as i lewska , W. , 1978 . Th e in f luence o f the s ilve r ca rp on eu t roph ica t ion o f ca rp pond s . V.

B o t to m f a u n a , R o c z . Na u k r o l . , H , 9 9 : 9 3 - 1 0 8 .

Vo in o v , A .A . , Vo r o n k o v a , O .V ., L u c k y a n o v , N .K . a n d S v i re z h ev , Yu .M . , 1 98 1. L a k e e c o s y s -

t e ms mo d e l l in g . I n : E c o lo g i ca l M o n i to r in g P r o b l e ms , I V . G id r o m e te o i z d a t , L e n in g r a d ( i nRuss ian) .

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