mathematical modelling in fish pond
TRANSCRIPT
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 1/23
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.
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 2/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 3/23
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 .
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 4/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 5/23
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 )
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 6/23
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 .
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 7/23
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.
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 8/23
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 .
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 9/23
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 .
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 10/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 11/23
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 .
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 12/23
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.
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 13/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 14/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 15/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 16/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 17/23
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
Maximum uptake rate of zooplankton
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)
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 18/23
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
Maximum uptake rate of zooplankton
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 19/23
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
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 20/23
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.
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 21/23
'° °
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 ,
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 22/23
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.
8/3/2019 Mathematical Modelling in Fish Pond
http://slidepdf.com/reader/full/mathematical-modelling-in-fish-pond 23/23
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) .