A Dynamic Model of the Phytoplankton Population in the Sacramento

5 A Dynamic Model of the Phytoplankton Population in the

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Sacramento-San Joaquin Delta D O M I N I C M . D I TORO, D O N A L D J . O'CONNOR, and ROBERT V. T H O M A N N Environmental Engineering and Science Program, Manhattan College, Bronx, Ν. Y. 10471 The by

quality the

of natural

growth

population

and

waters

development

of nutrients

resulting

esses. The resulting inorganic

nutrients

is commonly

kinetic

plankton with

two

Joaquin

of phytoplankton

This

determined

data. The resulting

problems.

and

by an analysis equations portion

This

are

as a The

is based

The growth

phytoplankton

the tidal

proc ample

investiga

populations

of mass.

of the

years of data from River,

growth.

model of this phenomenon

of conservation

are empirically

ing experimental

or natural

of eutrophication

formulations

addition

more than

to as eutrophication.

in the solution

on the principle

provides

whose

by the

for excessive phytoplankton

referred

influenced

phytoplankton,

from man's activities fertilization

model of the dynamics death

of

can be accelerated

tion presents a mathematical component

can be markedly

distribution

and zoo-

of exist compared

of the

San

California.

' " I ^ h e q u a l i t y of n a t u r a l waters c a n b e m a r k e d l y i n f l u e n c e d b y the g r o w t h A

a n d d i s t r i b u t i o n of p h y t o p l a n k t o n . U t i l i z i n g r a d i a n t energy,

these

m i c r o s c o p i c p l a n t s a s s i m i l a t e i n o r g a n i c c h e m i c a l s a n d c o n v e r t t h e m to c e l l m a t e r i a l w h i c h , i n t u r n , is c o n s u m e d b y the v a r i o u s a n i m a l species i n the next t r o p i c l e v e l . T h e p h y t o p l a n k t o n , therefore, are the base of the f o o d c h a i n i n n a t u r a l w a t e r s , a n d t h e i r existence is essential to a l l a q u a t i c life. T h e q u a l i t y of a b o d y of w a t e r c a n be a d v e r s e l y affected i f the p o p u l a t i o n of p h y t o p l a n k t o n b e c o m e s so large as to i n t e r f e r e w i t h either w a t e r 131 Hem; Nonequilibrium Systems in Natural Water Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

132

NONEQUILIBRIUM

SYSTEMS

IN

NATURAL

WATERS

use o r the h i g h e r forms of a q u a t i c life. I n p a r t i c u l a r , h i g h concentrations of a l g a l biomass cause large d i u r n a l v a r i a t i o n s i n d i s s o l v e d o x y g e n w h i c h c a n b e f a t a l to fish life. A l s o , the g r o w t h s c a n b e nuisances i n themselves, e s p e c i a l l y w h e n t h e y d e c a y a n d e i t h e r settle to the b o t t o m or a c c u m u l a t e i n w i n d r o w s o n the shoreline. P h y t o p l a n k t o n c a n cause taste a n d o d o r p r o b l e m s i n w a t e r s u p p l i e s a n d , i n a d d i t i o n , c o n t r i b u t e to filter c l o g g i n g i n the w a t e r treatment p l a n t . T h e d e v e l o p m e n t of l a r g e p o p u l a t i o n s of p h y t o p l a n k t o n a n d , i n some cases, l a r g e r a q u a t i c p l a n t s c a n b e a c c e l e r a t e d b y the a d d i t i o n of n u t r i e n t s


w h i c h result f r o m man's activities or n a t u r a l processes.

T h e resulting

f e r t i l i z a t i o n p r o v i d e s m o r e t h a n a m p l e i n o r g a n i c n u t r i e n t s , w i t h the r e s u l t i n g d e v e l o p m e n t of excessive p h y t o p l a n k t o n . T h i s sequence of events is c o m m o n l y r e f e r r e d to as e u t r o p h i c a t i o n . G e n e r a l l y , the m a n a g e m e n t of w a t e r systems s u b j e c t e d to a c c e l e r a t e d e u t r o p h i c a t i o n because of w a s t e discharges has b e e n l a r g e l y subjective. E x t e n s i v e p r o g r a m s of n u t r i e n t r e m o v a l h a v e b e e n c a l l e d for, w i t h l i t t l e or no q u a n t i t a t i v e p r e d i c t i o n of the effects of s u c h treatment p r o g r a m s . A q u a n t i t a t i v e m e t h o d o l o g y is r e q u i r e d to estimate the effect of p r o p o s e d treatment p r o g r a m s that are p l a n n e d to restore w a t e r q u a l i t y or to p r e d i c t t h e effects of e x p e c t e d

f u t u r e n u t r i e n t discharges.

This

methodology

s h o u l d i n c l u d e a m o d e l of the p h y t o p l a n k t o n p o p u l a t i o n w h i c h a p p r o x i mates the b e h a v i o r of the p h y t o p l a n k t o n i n the w a t e r b o d y of interest a n d , therefore, c a n b e u s e d to test the effects of t h e v a r i o u s c o n t r o l p r o cedures a v a i l a b l e . I n this w a y , r a t i o n a l p l a n n i n g a n d w a t e r q u a l i t y m a n agement c a n b e i n s t i t u t e d w i t h at least some degree of confidence that the p l a n n e d results a c t u a l l y w i l l b e a c h i e v e d . T h i s p a p e r presents a p h y t o p l a n k t o n p o p u l a t i o n m o d e l i n n a t u r a l w a t e r s , c o n s t r u c t e d o n the basis of t h e p r i n c i p l e of c o n s e r v a t i o n of mass. T h i s is a n e l e m e n t a r y p h y s i c a l l a w w h i c h is satisfied b y n a t u r a l systems.

macroscopic

T h e use of this p r i n c i p l e is d i c t a t e d p r i m a r i l y b y the

l a c k of a n y m o r e specific p h y s i c a l l a w s w h i c h c a n b e a p p l i e d to these b i o l o g i c a l systems.

A n alternate c o n s e r v a t i o n l a w , that of c o n s e r v a t i o n

of energy, c a n also be used. H o w e v e r , the details of h o w mass is t r a n s f e r r e d f r o m species to species are better u n d e r s t o o d t h a n the c o r r e s p o n d i n g e n e r g y transformations.

T h e mass i n t e r a c t i o n s are r e l a t e d , a m o n g

other factors, to the k i n e t i c s of the p o p u l a t i o n s , a n d i t is this t h a t the b u l k of the p a p e r is d e v o t e d to e x p l o r i n g . C o n s e r v a t i o n of mass has b e e n successfully a p p l i e d to the m o d e l i n g of the d i s s o l v e d o x y g e n d i s t r i b u t i o n i n n a t u r a l waters as w e l l as t h e d i s t r i b u t i o n of s a l i n i t y a n d other d i s s o l v e d substances. T h e r e s u l t i n g m o d e l s have proved useful i n guiding engineering and management

decisions

c o n c e r n e d w i t h the efficient u t i l i z a t i o n of w a t e r resources a n d the p r o t e c t i o n of t h e i r q u a l i t y . It is felt t h a t the p h y t o p l a n k t o n m o d e l p r e s e n t e d

Hem; Nonequilibrium Systems in Natural Water Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

5.

Di TORO E T A L .

Phytoplankton

133

IPopulation

h e r e i n c a n serve a s i m i l a r p u r p o s e b y p r o v i d i n g a basis for p r e d i c t i n g the effects of n u t r i e n t c o n t r o l p r o g r a m s o n the e u t r o p h i c a t i o n of n a t u r a l waters. T h u s , the p r i m a r y p u r p o s e of this p a p e r is to i n t r o d u c e a q u a n t i t a t i v e m o d e l of p h y t o p l a n k t o n p o p u l a t i o n d y n a m i c s i n n a t u r a l waters.

I t is

w i t h i n this p r o b l e m context that the s i m p l i f i c a t i o n s , a s s u m p t i o n s , a n d g e n e r a l l y the structure of the m o d e l is f o r m u l a t e d . A n a t t e m p t is m a d e to m a k e the equations representative of t h e b i o l o g i c a l m e c h a n i s m s w h i l e s t i l l r e t a i n i n g a sufficient s i m p l i c i t y so that the result is t r a c t a b l e a n d


useful. Review of Previous Models T h e i n i t i a l attempts to m o d e l the d y n a m i c s of a p h y t o p l a n k t o n p o p u l a t i o n w e r e b a s e d o n a v e r s i o n of the l a w of c o n s e r v a t i o n of mass i n w h i c h the h y d r o d y n a m i c transport of mass is a s s u m e d to b e insignificant. L e t P(t)

b e the c o n c e n t r a t i o n of p h y t o p l a n k t o n mass at t i m e t i n a s u i t a b l y

chosen r e g i o n of w a t e r .

T h e p r i n c i p l e of c o n s e r v a t i o n of mass c a n

be

expressed as a d i f f e r e n t i a l e q u a t i o n dP

ς

w h e r e S is the net source or sink of p h y t o p l a n k t o n mass w i t h i n t h e r e g i o n . I f h y d r o d y n a m i c t r a n s p o r t is not i n c l u d e d , t h e n the rate at w h i c h Ρ i n creases or decreases d e p e n d s o n l y o n the i n t e r n a l sources a n d sinks of p h y t o p l a n k t o n i n the r e g i o n of interest. T h e f o r m of the i n t e r n a l sources a n d sinks of p h y t o p l a n k t o n is d i c t a t e d b y t h e m e c h a n i s m s w h i c h are a s s u m e d to g o v e r n the g r o w t h a n d d e a t h of p h y t o p l a n k t o n .

F l e m i n g ( 1 9 3 9 ) , as d e s c r i b e d b y R i l e y

p o s t u l a t e d that s p r i n g d i a t o m

flowering

(I),

i n the E n g l i s h C h a n n e l is d e

s c r i b e d b y the e q u a t i o n ^

= [a -

(b +

ct))P

w h e r e Ρ is the p h y t o p l a n k t o n c o n c e n t r a t i o n , α is a constant g r o w t h rate, a n d (b +

ct)

is a d e a t h rate r e s u l t i n g f r o m the g r a z i n g of z o o p l a n k t o n .

T h e z o o p l a n k t o n p o p u l a t i o n , w h i c h is i n c r e a s i n g o w i n g to its g r a z i n g , results i n a n i n c r e a s i n g d e a t h r a t e w h i c h is a p p r o x i m a t e d b y t h e l i n e a r increase of the d e a t h rate as a f u n c t i o n of t i m e . A less e m p i r i c a l m o d e l has b e e n p r o p o s e d b y R i l e y ( 1946) ( I ) b a s e d o n the e q u a t i o n f

=

[P -R-G]P k


134

NONEQUILIBRIUM SYSTEMS IN N A T U R A L WATERS

w h e r e P is the p h o t o s y n t h e t i c g r o w t h rate, R is t h e e n d o g e n o u s r e s p i r a h

t i o n rate of the p h y t o p l a n k t o n , a n d G is the d e a t h rate o w i n g to z o o p l a n k t o n g r a z i n g . A m a j o r i m p r o v e m e n t i n R i l e y ' s e q u a t i o n is t h e a t t e m p t to r e l a t e the g r o w t h rate, t h e r e s p i r a t i o n rate, a n d t h e g r a z i n g to

more

f u n d a m e n t a l e n v i r o n m e n t a l v a r i a b l e s s u c h as i n c i d e n t solar r a d i a t i o n , t e m p e r a t u r e , e x t i n c t i o n coefficient, a n d o b s e r v e d n u t r i e n t a n d z o o p l a n k ton concentration.

A s a c o n s e q u e n c e , t h e coefficients of the e q u a t i o n s

are t i m e - v a r i a b l e since t h e e n v i r o n m e n t a l p a r a m e t e r s v a r y t h r o u g h o u t t h e year. T h i s p r e c l u d e s a n a n a l y t i c a l s o l u t i o n to the e q u a t i o n , a n d n u -


m e r i c a l i n t e g r a t i o n m e t h o d s m u s t b e u s e d . T h r e e separate a p p l i c a t i o n s ( 2 , 3, 4)

of these e q u a t i o n s to t h e near-shore o c e a n e n v i r o n m e n t h a v e

b e e n m a d e , a n d the r e s u l t i n g a g r e e m e n t w i t h o b s e r v e d d a t a is q u i t e encouraging. A c o m p l e x set of e q u a t i o n s , p r o p o s e d b y R i l e y , S t o m m e l , a n d B u m p u s (1949) (5)

first

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

w i t h respect to d e p t h i n t o t h e c o n s e r v a t i o n of mass e q u a t i o n . I n a d d i t i o n , a c o n s e r v a t i o n of mass e q u a t i o n for a n u t r i e n t ( p h o s p h a t e ) w a s also i n t r o d u c e d , as w e l l as s i m p l i f i e d e q u a t i o n s for t h e h e r b i v o r o u s a n d c a r n i v o r o u s z o o p l a n k t o n concentrations.

T h e phytoplankton and nutrient

e q u a t i o n s w e r e a p p l i e d to 20 v o l u m e elements w h i c h e x t e n d e d f r o m the surface to w e l l b e l o w the e u p h o t i c zone. I n o r d e r to s i m p l i f y the c a l c u lations, a t e m p o r a l steady-state w a s a s s u m e d to exist i n e a c h v o l u m e element.

T h u s , the equations a p p l y to those p e r i o d s of the y e a r d u r i n g

w h i c h the d e p e n d e n t v a r i a b l e s are not c h a n g i n g s i g n i f i c a n t l y i n t i m e . S u c h c o n d i t i o n s u s u a l l y p r e v a i l d u r i n g the s u m m e r m o n t h s . T h e results of these c a l c u l a t i o n s w e r e c o m p a r e d w i t h o b s e r v e d d a t a , a n d a g a i n the results w e r e e n c o u r a g i n g . Steele ( 1 9 5 6 ) (6)

f o u n d t h a t the steady-state a s s u m p t i o n d i d not

a p p l y to t h e seasonal v a r i a t i o n of the p h y t o p l a n k t o n p o p u l a t i o n . I n s t e a d , h e u s e d t w o v o l u m e segments to represent the u p p e r a n d l o w e r w a t e r levels a n d k e p t the t i m e d e r i v a t i v e s i n the equations. T h u s , b o t h t e m p o r a l a n d spatial variations were considered. I n addition, the differential equations for p h y t o p l a n k t o n a n d z o o p l a n k t o n c o n c e n t r a t i o n w e r e c o u p l e d so t h a t t h e i n t e r a c t i o n s of the p o p u l a t i o n s c o u l d b e s t u d i e d , as w e l l as the nutrient-phytoplankton dependence.

T h e coefficients

of the equations

w e r e not f u n c t i o n s of t i m e , h o w e v e r , so that the effects of t i m e - v a r y i n g solar r a d i a t i o n i n t e n s i t y a n d t e m p e r a t u r e w e r e not i n c l u d e d . T h e e q u a tions w e r e n u m e r i c a l l y i n t e g r a t e d a n d the results c o m p a r e d w i t h t h e o b s e r v e d d i s t r i b u t i o n . Steele

a p p l i e d similar equations

d i s t r i b u t i o n of c h l o r o p h y l l i n the G u l f of M e x i c o

to

the vertical

(7).

T h e m o d e l s p r o p o s e d b y R i l e y et al. a n d Steele are b a s i c a l l y s i m i l a r . E a c h c o n s i d e r the p r i m a r y d e p e n d e n t v a r i a b l e s to b e the 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 n u t r i e n t c o n c e n t r a t i o n . A c o n s e r v a t i o n of mass e q u a -


5.

Phytoplankton

DI TORO E T A L .

135

Population

t i o n is w r i t t e n f o r e a c h species, a n d the s p a t i a l v a r i a t i o n is i n c o r p o r a t e d b y c o n s i d e r i n g finite v o l u m e elements w h i c h i n t e r a c t b e c a u s e o f v e r t i c a l eddy diffusion a n d d o w n w a r d advective transport of the phytoplankton. T h e i r e q u a t i o n s differ i n some details ( f o r e x a m p l e , the g r o w t h coefficients that w e r e u s e d a n d t h e assumptions o f s t e a d y state) b u t t h e p r i n c i p l e is t h e same. I n a d d i t i o n , these e q u a t i o n s w e r e a p p l i e d b y t h e authors t o a c t u a l m a r i n e situations a n d t h e i r solutions c o m p a r e d w i t h o b s e r v e d d a t a . T h i s is a c r u c i a l p a r t o f a n y i n v e s t i g a t i o n d i s c u s s i o n w h e r e i n t h e a s s u m p tions t h a t a r e m a d e a n d t h e a p p r o x i m a t i o n s t h a t a r e u s e d a r e difficult


to justify a priori. T h e m o d e l s o f b o t h R i l e y a n d Steele h a v e b e e n r e v i e w e d i n greater detail b y R i l e y ( I ) i n a discussion of their a p p l i c a b i l i t y a n d possible future development.

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

FLOW

ZOOPLANKTON

PREY

GRAZING

TEMPERATURE PHYTOPLANKTON

-ι

M. NUTRIENT LIMITATION

SOLAR RADIATION

NUTRIENTS

MAN —'

Figure

1.

NUTRIENT USE

MADE

INPUTS

Interactions of environmental variables and the phyto plankton, zooplankton, and nutrient systems


136

NONEQUILIBRIUM

SYSTEMS

IN

NATURAL

WATERS

t h e o r e t i c a l m o d e l s of p h y t o p l a n k t o n - z o o p l a n k t o n p o p u l a t i o n models w e r e discussed b y Steele

(8).

O t h e r m o d e l s have b e e n p r o p o s e d w h i c h f o l l o w the outlines of the equations a l r e a d y discussed.

E q u a t i o n s w i t h parameters that v a r y as a

f u n c t i o n of t e m p e r a t u r e , s u n l i g h t , a n d n u t r i e n t c o n c e n t r a t i o n h a v e b e e n presented b y D a v i d s o n a n d C l y m e r (9) a n d simulated b y C o l e (10).

A

set of equations w h i c h m o d e l the p o p u l a t i o n of 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 a species of fish i n a large l a k e h a v e b e e n p r e s e n t e d b y P a r k e r (11).

T h e a p p l i c a t i o n of the t e c h n i q u e s of p h y t o p l a n k t o n m o d e l i n g

to


the p r o b l e m of e u t r o p h i c a t i o n i n rivers a n d estuaries has b e e n p r o p o s e d b y C h e n (12).

T h e interrelations b e t w e e n

the n i t r o g e n c y c l e a n d the

p h y t o p l a n k t o n p o p u l a t i o n i n the P o t o m a c E s t u a r y has b e e n i n v e s t i g a t e d using a feed-forward-feed-back

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

interact l i n e a r l y f o l l o w i n g first o r d e r k i n e t i c s

(13).

T h e f o r m u l a t i o n s a n d equations p r e s e n t e d i n the subsequent

sections

are modifications a n d extensions of p r e v i o u s l y presented equations w h i c h i n c o r p o r a t e some a d d i t i o n a l p h y s i o l o g i c a l i n f o r m a t i o n o n the of p h y t o p l a n k t o n a n d z o o p l a n k t o n p o p u l a t i o n s .

behavior

I n contrast to the m a -

j o r i t y of the a p p l i c a t i o n s of p h y t o p l a n k t o n m o d e l s w h i c h have b e e n m a d e p r e v i o u s l y , the equations presented i n the subsequent sections are a p p l i e d to a r e l a t i v e l y s h a l l o w r e a c h of the S a n J o a q u i n R i v e r a n d the estuary further downstream.

T h e m o t i v a t i o n for this a p p l i c a t i o n is a n i n v e s t i g a -

t i o n of the p o s s i b i l i t y of excessive p h y t o p l a n k t o n g r o w t h s as e n v i r o n m e n t a l c o n d i t i o n s a n d n u t r i e n t l o a d i n g s are c h a n g e d i n this area. T h u s , t h e p r i m a r y thrust of this i n v e s t i g a t i o n is to p r o d u c e a n e n g i n e e r i n g t o o l w h i c h c a n be u s e d i n the s o l u t i o n of e n g i n e e r i n g p r o b l e m s to protect the w a t e r q u a l i t y of the r e g i o n of interest.

Phytoplankton

System Interactions

T h e major obstacle to a rigorous q u a n t i t a t i v e t h e o r y of p h y t o p l a n k t o n p o p u l a t i o n d y n a m i c s is the enormous

c o m p l e x i t y of the b i o l o g i c a l a n d

p h y s i c a l p h e n o m e n a w h i c h influence the p o p u l a t i o n . It is necessary, therefore, to i d e a l i z e a n d s i m p l i f y the c o n c e p t u a l m o d e l so that the result is a m a n a g e a b l e set of d e p e n d e n t systems or v a r i a b l e s a n d t h e i r interrelations. T h e m o d e l c o n s i d e r e d i n the f o l l o w i n g sections is f o r m u l a t e d o n the basis of three p r i m a r y d e p e n d e n t systems: the p h y t o p l a n k t o n p o p u l a t i o n , w h o s e b e h a v i o r is the object of c o n c e r n ; the h e r b i v o r o u s z o o p l a n k t o n

popula-

t i o n , w h i c h are the predators of the p h y t o p l a n k t o n , u t i l i z i n g the a v a i l a b l e p h y t o p l a n k t o n as a f o o d s u p p l y ; a n d the n u t r i e n t system, w h i c h r e p r e sents the n u t r i e n t s , p r i m a r i l y i n o r g a n i c substances, that are r e q u i r e d b y the phytoplankton d u r i n g growth.

T h e s e three systems are affected

not

o n l y b y t h e i r interactions, b u t also b y e x t e r n a l e n v i r o n m e n t a l v a r i a b l e s .


5.

Di

TORO

E T

AL.

Phytoplankton

137

Population

T h e three p r i n c i p a l v a r i a b l e s c o n s i d e r e d i n this analysis are t e m p e r a t u r e , w h i c h influences a l l b i o l o g i c a l a n d c h e m i c a l reactions, d i s p e r s i o n a n d a d v e c t i v e flow, w h i c h are the p r i m a r y mass t r a n s p o r t m e c h a n i s m s i n a n a t u r a l b o d y of w a t e r , a n d solar r a d i a t i o n , the energy source for the p h o t o s y n t h e t i c g r o w t h of the p h y t o p l a n k t o n . I n a d d i t i o n to these e x t e r n a l v a r i a b l e s , the effect of man's a c t i v i t i e s o n the system is felt p r e d o m i n a t e l y i n the n u t r i e n t system. Sources of the necessary n u t r i e n t s m a y b e the r e s u l t of, for e x a m p l e , i n p u t s of w a s t e w a t e r f r o m m u n i c i p a l a n d i n d u s t r i a l discharges or a g r i c u l t u r a l runoff.


T h e m a n - m a d e w a s t e loads are i n most cases the p r i m a r y c o n t r o l v a r i a b l e s w h i c h are a v a i l a b l e t o affect changes i n the p h y t o p l a n k t o n a n d z o o p l a n k t o n systems. A s c h e m a t i c r e p r e s e n t a t i o n of these systems a n d t h e i r i n t e r relations is p r e s e n t e d i n F i g u r e 1. I n a d d i t i o n to the c o n c e p t u a l m o d e l w h i c h isolates the m a j o r i n t e r a c t i n g systems, a f u r t h e r i d e a l i z a t i o n is r e q u i r e d w h i c h sets the l o w e r a n d u p p e r l i m i t s of t h e t e m p o r a l a n d s p a t i a l scales b e i n g

considered.

W i t h i n the context of the p r o b l e m of e u t r o p h i c a t i o n a n d its c o n t r o l , the seasonal d i s t r i b u t i o n of the p h y t o p l a n k t o n is of m a j o r i m p o r t a n c e , so that the l o w e r l i m i t of the t e m p o r a l scale is o n the o r d e r of days. T h e s p a t i a l scale is set b y the h y d r o d y n a m i c s of the w a t e r b o d y b e i n g c o n s i d e r e d . F o r e x a m p l e , i n a t i d a l estuary, the s p a t i a l scale is o n the o r d e r of m i l e s whereas i n a s m a l l l a k e i t is l i k e l y a g o o d d e a l s m a l l e r . T h e u p p e r l i m i t s for the t e m p o r a l a n d s p a t i a l extent of the m o d e l are d i c t a t e d p r i m a r i l y b y p r a c t i c a l considerations s u c h as the l e n g t h o f t i m e for w h i c h a d e q u a t e i n f o r m a t i o n is a v a i l a b l e a n d the size of the c o m p u t e r b e i n g u s e d for t h e calculations. T h e s e s i m p l i f y i n g assumptions are m a d e p r i m a r i l y o n the basis of a n i n t u i t i v e assessment of the i m p o r t a n t features of the systems b e i n g c o n s i d e r e d a n d the experience g a i n e d b y p r e v i o u s a t t e m p t s to address these a n d r e l a t e d p r o b l e m s i n n a t u r a l b o d i e s of w a t e r . T h e b a s i c p r i n c i p l e to b e a p p l i e d to this c o n c e p t u a l m o d e l , w h i c h c a n t h e n b e t r a n s l a t e d i n t o m a t h e m a t i c a l terms, is t h a t of c o n s e r v a t i o n of mass. Conservation of Mass T h e p r i n c i p l e of c o n s e r v a t i o n of mass is the basis u p o n w h i c h the mathematical development

is s t r u c t u r e d . A l t e r n a t e f o r m u l a t i o n s , s u c h

as those b a s e d o n the c o n s e r v a t i o n of energy, h a v e b e e n p r o p o s e d .

How-

ever, c o n s e r v a t i o n of mass has p r o v e d a u s e f u l s t a r t i n g p o i n t for m a n y m o d e l s of the n a t u r a l e n v i r o n m e n t . T h e p r i n c i p l e of c o n s e r v a t i o n of mass s i m p l y states that the mass of the substances b e i n g c o n s i d e r e d w i t h i n a n a r b i t r a r i l y selected

volume

m u s t b e a c c o u n t e d for b y e i t h e r mass t r a n s p o r t i n t o a n d out of the v o l u m e


138

NONEQUILIBRIUM SYSTEMS IN N A T U R A L WATERS

o r as mass p r o d u c e d or r e m o v e d w i t h i n the v o l u m e .

T h e t r a n s p o r t of

mass i n a n a t u r a l w a t e r system arises p r i m a r i l y f r o m t w o

phenomena:

d i s p e r s i o n , w h i c h is c a u s e d b y t i d a l a c t i o n , d e n s i t y differences, t u r b u l e n t d i f f u s i o n , w i n d a c t i o n , etc.; a n d a d v e c t i o n o w i n g to a u n i d i r e c t i o n a l flow — f o r e x a m p l e , the fresh w a t e r flow i n a r i v e r or estuary or the p r e v a i l i n g c u r r e n t s i n a b a y or a near-shore e n v i r o n m e n t . T h e d i s t i n c t i o n b e t w e e n t h e t w o p h e n o m e n a is that, over the t i m e scale o f interest, d i s p e r s i v e mass t r a n s p o r t mixes adjacent v o l u m e s of w a t e r so t h a t a p o r t i o n of the w a t e r i n adjacent v o l u m e elements is i n t e r c h a n g e d , a n d the mass t r a n s


p o r t is p r o p o r t i o n a l to the difference i n concentrations of mass i n adjacent volumes.

A d v e c t i v e t r a n s p o r t , h o w e v e r , is transport i n t h e d i r e c t i o n of

t h e a d v e c t i v e flow o n l y .

I n a d d i t i o n to the mass t r a n s p o r t p h e n o m e n a ,

mass i n the v o l u m e c a n increase r e s u l t i n g f r o m sources w i t h i n the v o l u m e . T h e s e sources represent t h e rate of a d d i t i o n or r e m o v a l of mass p e r u n i t t i m e p e r u n i t v o l u m e b y c h e m i c a l a n d b i o l o g i c a l processes. A m a t h e m a t i c a l expression of c o n s e r v a t i o n of mass w h i c h i n c l u d e s t h e terms to d e s c r i b e the mass t r a n s p o r t p h e n o m e n a a n d t h e source t e r m is a p a r t i a l d i f f e r e n t i a l e q u a t i o n of the f o l l o w i n g f o r m =

V

· EVP

-

V

· QP + S

w h e r e Ρ (x, y, ζ, t) is the c o n c e n t r a t i o n of the substance of phytoplankton biomass—as

(1)

P

interest—e.g.,

a f u n c t i o n of p o s i t i o n a n d t i m e ; Ε is the

d i a g o n a l m a t r i x of d i s p e r s i o n coefficients; Q is t h e a d v e c t i v e

flow

rate

v e c t o r ; S ρ is t h e v e c t o r w h o s e terms are the rate of mass a d d i t i o n b y the sources a n d s i n k s ; a n d V is t h e g r a d i e n t operator. t i a l e q u a t i o n is too

T h i s partial differen

g e n e r a l to b e s o l v e d a n a l y t i c a l l y , a n d n u m e r i c a l

t e c h n i q u e s are u s e d i n its s o l u t i o n . A n effective a p p r o x i m a t i o n to E q u a t i o n 1 is o b t a i n e d b y s e g m e n t i n g t h e w a t e r b o d y of interest i n t o η v o l u m e elements of v o l u m e V , a n d r e p r e s e n t i n g the d e r i v a t i v e s i n E q u a t i o n 1 b y differences. η χ

η d i a g o n a l m a t r i x of v o l u m e s V ; A, the η χ ;

L e t V be

the

η m a t r i x of d i s p e r s i v e

a n d a d v e c t i v e t r a n s p o r t t e r m s ; S , t h e η v e c t o r of source terms S , P

Pj

aver

a g e d over the v o l u m e V,·; a n d P, the η v e c t o r of concentrations P , w h i c h ;

are the concentrations i n the v o l u m e s . T h e n the finite difference e q u a t i o n s c a n b e expressed as a v e c t o r d i f f e r e n t i a l e q u a t i o n VP

= AP

+

(2)

VS

P

w h e r e the d o t denotes a t i m e d e r i v a t i v e . T h e details of the a p p l i c a t i o n of this v e r s i o n of the d i s p e r s i o n a d v e c t i o n e q u a t i o n to n a t u r a l b o d i e s of w a t e r has b e e n p r e s e n t e d b y T h o m a n n (14) etal.

and reviewed by O'Connor

(15).


5.

DI

TORO

E T

Phytoplankton

A L .

139

Population

T h e m a i n interest i n this r e p o r t is c e n t e r e d o n the source terms

S

Pj

f o r the p a r t i c u l a r a p p l i c a t i o n of these equations to t h e p h y t o p l a n k t o n p o p u l a t i o n i n n a t u r a l w a t e r bodies. t e r m of p h y t o p l a n k t o n , S

It is c o n v e n i e n t to express the source

as a difference b e t w e e n the g r o w t h rate,

Pj>

of p h y t o p l a n k t o n a n d t h e i r d e a t h rate, D , S

Pj

where G

Pj

and D

Pj

=

-

(G

Pj

h a v e units [ d a y ] . 1

G, Pj

i n the v o l u m e V ; . T h a t is

Pj

(3)

D )Pj Pj

T h e s u b s c r i p t Ρ identifies the

q u a n t i t i e s as r e f e r r i n g to p h y t o p l a n k t o n ; the s u b s c r i p t / refers to the v o l Downloaded by UNIV OF PITTSBURGH on May 15, 2016 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch005

ume element being considered.

T h e b a l a n c e b e t w e e n the m a g n i t u d e of

the g r o w t h rate a n d d e a t h rate d e t e r m i n e s the rate at w h i c h p h y t o p l a n k t o n mass is c r e a t e d or d e s t r o y e d i n the v o l u m e element V ; . T h u s , the f o r m of the g r o w t h a n d d e a t h rates as f u n c t i o n s of e n v i r o n m e n t a l p a r a m eters a n d d e p e n d e n t

v a r i a b l e s is a n i m p o r t a n t element i n a successful

phytoplankton population model. Phytoplankton

Growth

Rate

T h e g r o w t h rate of a p o p u l a t i o n of p h y t o p l a n k t o n i n a n a t u r a l e n v i r o n m e n t is a c o m p l i c a t e d

f u n c t i o n of the species of

phytoplankton

present a n d t h e i r d i f f e r i n g reactions to solar r a d i a t i o n , t e m p e r a t u r e , a n d the b a l a n c e b e t w e e n

nutrient availability and phytoplankton require

ments. T h e c o m p l e x a n d often c o n f l i c t i n g d a t a p e r t i n e n t to this p r o b l e m have been reviewed recently b y H u t c h i n s o n (1967)

(16),

Strickland

(1965) (17), L u n d (1965) ( I S ) , and Raymont (1963) (19).

The avail

a b l e i n f o r m a t i o n is not sufficiently d e t a i l e d to s p e c i f y the g r o w t h k i n e t i c s for i n d i v i d u a l p h y t o p l a n k t o n species i n n a t u r a l e n v i r o n m e n t s .

Hence,

i n o r d e r to a c c o m p l i s h the task of c o n s t r u c t i n g a g r o w t h rate f u n c t i o n , a s i m p l i f i e d a p p r o a c h is f o l l o w e d .

T h e p r o b l e m of different species a n d

t h e i r associated n u t r i e n t a n d e n v i r o n m e n t a l r e q u i r e m e n t s is not addressed. I n s t e a d , the p o p u l a t i o n is c h a r a c t e r i z e d as a w h o l e b y a m e a s u r e m e n t of the biomass of p h y t o p l a n k t o n present.

T y p i c a l q u a n t i t i e s u s e d are the

c h l o r o p h y l l c o n c e n t r a t i o n of the p o p u l a t i o n , the n u m b e r of organisms p e r u n i t v o l u m e , or the d r y w e i g h t of the p h y t o p l a n k t o n p e r u n i t v o l u m e (20).

W i t h a c h o i c e of biomass u n i t s e s t a b l i s h e d , the g r o w t h rate ex

presses the rate of p r o d u c t i o n of biomass as a f u n c t i o n of the i m p o r t a n t e n v i r o n m e n t a l v a r i a b l e s . T h e e n v i r o n m e n t a l v a r i a b l e s to be

considered

b e l o w are l i g h t , t e m p e r a t u r e , a n d the v a r i o u s nutrients w h i c h are neces sary for p h y t o p l a n k t o n g r o w t h . Light and Temperature.

C o n s i d e r a p o p u l a t i o n of p h y t o p l a n k t o n ,

either a n a t u r a l association or a single species c u l t u r e , a n d assume that the o p t i m u m or s a t u r a t i n g l i g h t i n t e n s i t y for m a x i m u m g r o w t h rate of biomass is present a n d i l l u m i n a t e s a l l the cells, a n d f u r t h e r that a l l the


140

NONEQUILIBRIUM

SYSTEMS

IN NATURAL

WATERS


4.0

0

5

10

15

20

25

30

TEMPERATURE °C

Figure

2.

Phytopknkton saturated growth rate (base e) as a function of temperature

necessary n u t r i e n t s are present i n sufficient q u a n t i t y so t h a t n o n u t r i e n t is i n short s u p p l y . F o r this c o n d i t i o n , t h e g r o w t h rate t h a t i s o b s e r v e d is c a l l e d the m a x i m u m o r s a t u r a t e d g r o w t h rate, K'. M e a s u r e m e n t s of K ' ( b a s e e ) as a f u n c t i o n of t e m p e r a t u r e are s h o w n i n F i g u r e 2 a n d l i s t e d i n T a b l e I . T h e e x p e r i m e n t a l c o n d i t i o n s u n d e r w h i c h these d a t a w e r e c o l l e c t e d a p p e a r t o meet the r e q u i r e m e n t s of o p t i m u m l i g h t i n t e n s i t y a n d sufficient n u t r i e n t s u p p l y . T h e d a t a presented a r e selected f r o m l a r g e r g r o u p s o f r e p o r t e d values, a n d t h e y represent t h e m a x i m u m o f these r e p o r t e d g r o w t h rates. T h e p r e s u m p t i o n is that these large values reflect the m a x i m u m g r o w t h rates a c h i e v a b l e . F r o m a n e c o l o g i c a l p o i n t of v i e w , i t is necessary t o c o n s i d e r the species most a b l e t o c o m p e t e , a n d , i n terms of g r o w t h rate, i t is t h e species w i t h t h e largest g r o w t h rate w h i c h w i l l predominate.

A s t r a i g h t - l i n e fit to this d a t a appears t o b e a c r u d e b u t

r e a s o n a b l e a p p r o x i m a t i o n o f t h e d a t a r e l a t i n g s a t u r a t e d g r o w t h rate K' to t e m p e r a t u r e , Τ Κ'

X

w h e r e K has values i n the range 0.10 ± 0.025 d a y " x

(4)

= KT 1

°C

_ 1

. T h i s coefficient

i n d i c a t e s a n a p p r o x i m a t e d o u b l i n g o f t h e s a t u r a t e d g r o w t h rate f o r a t e m p e r a t u r e c h a n g e f r o m 10° t o 20 ° C , i n a c c o r d a n c e w i t h t h e g e n e r a l l y reported temperature-dependence

of b i o l o g i c a l g r o w t h rates. T h e o p t i

m u m temperature for algal growth appears to b e i n the range

between

20° a n d 2 5 ° C , a l t h o u g h t h e r m o p h i l i c strains a r e k n o w n t o exist (27). A t h i g h e r temperatures, there is u s u a l l y a s u p p r e s s i o n o f t h e saturated g r o w t h rate, a n d t h e s t r a i g h t - l i n e a p p r o x i m a t i o n i s n o l o n g e r v a l i d .

It should


5.

Di TORO E T A L .

Phytoplankton

141

Population

also b e n o t e d that t h e scatter i n the d a t a i n F i g u r e 2 is sufficiently large so that t h e l i n e a r d e p e n d e n c e o n t e m p e r a t u r e a n d also the m a g n i t u d e of K' c a n v a r y c o n s i d e r a b l y i n p a r t i c u l a r situations. I n t h e n a t u r a l e n v i r o n m e n t , the l i g h t i n t e n s i t y t o w h i c h the p h y t o p l a n k t o n are exposed is not u n i f o r m l y at the o p t i m u m v a l u e b u t i t varies as a f u n c t i o n of d e p t h because of the n a t u r a l t u r b i d i t y present a n d as a f u n c t i o n of t i m e over the d a y .

T h u s , the p h y t o p l a n k t o n i n t h e l o w e r

layers are exposed to intensities b e l o w the o p t i m u m a n d those at the surface m a y b e exposed to intensities a b o v e the o p t i m u m so that t h e i r


g r o w t h rate w o u l d b e i n h i b i t e d . F i g u r e 3 b , c , d f r o m R y t h e r (28)

are

plots of the photosynthesis rate n o r m a l i z e d b y the photosynthesis rate at the o p t i m u m or s a t u r a t i n g l i g h t i n t e n s i t y vs. t h e l i g h t i n t e n s i t y , I, i n c i d e n t o n the p o p u l a t i o n s . F i g u r e 3a is a p l o t of f u n c t i o n (5) for I

8

=

2000 ft-candles, p r o p o s e d b y Steele (8)

to d e s c r i b e the l i g h t -

d e p e n d e n c e of the g r o w t h rate of p h y t o p l a n k t o n . T h e s i m i l a r i t y b e t w e e n this f u n c t i o n a n d d a t a f r o m R y t h e r is sufficient to w a r r a n t the use of this expression to express the influence of n o n o p t i m u m l i g h t i n t e n s i t y o n the g r o w t h rate of p h y t o p l a n k t o n . w o r k e r s h a v e suggested Table I. Ref. 21 22 23

5 2^ 24 25 25 25 25 26

Other

different forms for this r e l a t i o n s h i p (29,

Maximum G r o w t h Rates as a Function of Organism

Chlorella ellipsoidea (green alga) Nannochloris atomus (marine flagellate) Nitzschia closterium (marine d i a t o m )

N a t u r a l association Chlorella pyrenoidosa Scenedesmus quadricauda Chlorella pyrenoidosa Chlorella vulgaris Scenedesmus obliquus Chlamydomonas reinhardti Chlorella pyrenoidosa (synchronized culture) (high-temperature strain)

Temperature, 25 15 20 10 27 19 15.5 10 4 2.6 25 25 25 25 25 25 10 15 20 25

30).

Temperature

Saturated Growth Rate, K'(Base , Day- ) 1

e

3.14 1.2 2.16 1.54 1.75 1.55 1.19 0.67 0.63 0.51 1.96 2.02 2.15 1.8 1.52 2.64 0.2 1.1 2.4 3.9



142

NONEQUILIBRIUM SYSTEMS I N N A T U R A L WATERS

0

1

2

LIGHT

3

4

5

INTENSITY

6

7

8

9

10

(FOOT CANDLES χ Ι Ο ) 3

Limnology and Oceanography

Figure 3. Normalized rate of photosynthesis vs. incident light intensity: (a) Theoretical curve after Steele (8); (b,c,d) Data after Ryther (28) T h e s e v a r i a t i o n s a p p r o x i m a t e l y f o l l o w the shape o f E q u a t i o n 5 f o r l o w l i g h t intensities b u t differ f o r the r e g i o n o f h i g h l i g h t intensities, u s u a l l y b y not d e c r e a s i n g after some o p t i m u m i n t e n s i t y is r e a c h e d . I n p a r t i c u l a r , T a m i y a et al. (21) h a v e i n v e s t i g a t e d the g r o w t h rate o f ChloreUa soidea

t o v a r i o u s l i g h t a n d t e m p e r a t u r e regimes.

ellip-

T h e saturated growth

rates as a f u n c t i o n o f t e m p e r a t u r e are i n c l u d e d i n F i g u r e 2. T h e influence o f v a r y i n g l i g h t i n t e n s i t y fits the f u n c t i o n F(I)

=

(6)

I + K'/a

w h e r e K' is the saturated g r o w t h rate a n d α is a constant ( a = 0.45 d a y "

1

k i l o l u x ' ) . H o w e v e r , since K' i s a f u n c t i o n o f t e m p e r a t u r e , t h e s a t u r a t i n g 1

l i g h t i n t e n s i t y f o r E q u a t i o n 6 is also a f u n c t i o n o f t e m p e r a t u r e .

Similar

d a t a o b t a i n e d b y S o r o k i n et al. (26) u s i n g a h i g h - t e m p e r a t u r e s t r a i n o f Chlorella

pyrenoidosa

s u p p o r t the t e m p e r a t u r e - d e p e n d e n c e

o f the s a t u -


5.

DI

TORO

E T

Phytopfonkton

AL.

143

Population

r a t i n g l i g h t i n t e n s i t y f o r c h l o r e l l a . T h e r e f o r e , i n u s i n g E q u a t i o n 5, a temperature-dependent light saturation intensity m a y be warranted. A t this p o i n t i n the analysis, the effect of the n a t u r a l e n v i r o n m e n t o n the l i g h t a v a i l a b l e to the p h y t o p l a n k t o n m u s t b e i n c l u d e d . E q u a t i o n 5 expresses the r e d u c t i o n i n the g r o w t h rate c a u s e d b y n o n o p t i m u m l i g h t intensity. T h i s expression c a n therefore b e u s e d to c a l c u l a t e the r e d u c t i o n i n g r o w t h rate to b e e x p e c t e d at a n y i n t e n s i t y . H o w e v e r , this is

too

d e t a i l e d a d e s c r i p t i o n for c o n s e r v a t i o n of mass equations w h i c h d e a l w i t h h o m o g e n e o u s v o l u m e elements, Vj, a n d the g r o w t h rate w i t h i n these


elements.

W h a t is r e q u i r e d is averages of the g r o w t h r a t e o v e r the v o l

u m e elements. I n o r d e r to c a l c u l a t e the l i g h t i n t e n s i t y w h i c h is present i n the v o l u m e V , the l i g h t p e n e t r a t i o n at the d e p t h of w a t e r w h e r e Vj is l o c a t e d }

m u s t be e v a l u a t e d . T h e rate at w h i c h l i g h t is a t t e n u a t e d w i t h respect to d e p t h is g i v e n b y the e x t i n c t i o n coefficient, k .

T h a t is, at a d e p t h z, t h e

e

i n t e n s i t y at that d e p t h , I(z),

is r e l a t e d to the surface i n t e n s i t y , I , b y the 0

formula I{z) where ζ =

= I

0

exp ( -

(7)

k z) e

0 is the w a t e r surface a n d ζ is p o s i t i v e d o w n w a r d . T h u s , the

r e d u c t i o n of the s a t u r a t e d g r o w t h rate at a n y d e p t h ζ r e s u l t i n g f r o m the n o n o p t i m u m l i g h t i n t e n s i t y present is g i v e n b y E q u a t i o n 7, s u b s t i t u t e d i n t o E q u a t i o n 5. (8) T o a p p l y this e q u a t i o n to the finite v o l u m e elements, w i t h i n w h i c h i t is a s s u m e d that the p h y t o p l a n k t o n c o n c e n t r a t i o n is u n i f o r m , i t is necessary to average this r e d u c t i o n factor t h r o u g h o u t the d e p t h of the e l e m e n t Vj.

Let Hj

and Η

0

volume

be the depths of the surface a n d b o t t o m ,

υ

r e s p e c t i v e l y , of the v o l u m e e l e m e n t Vj.

F o r e x a m p l e , i f the v o l u m e ele

m e n t Vj extends f r o m the w a t e r surface to t h e b o t t o m of the w a t e r b o d y , then H ; = 0

0 and Η

υ

is the w a t e r d e p t h at the l o c a t i o n of Vj.

sake of s i m p l i c i t y , i t is a s s u m e d t h a t this is the case. If H

oj

F o r the

Φ 0, a s t r a i g h t

f o r w a r d g e n e r a l i z a t i o n of the f o l l o w i n g average is r e q u i r e d . I n a d d i t i o n to a n average o v e r d e p t h , i t is also e x p e d i e n t to average the p h y t o p l a n k t o n g r o w t h rate o v e r a t i m e i n t e r v a l . S i n c e the t i m e scale w i t h i n w h i c h this analysis is a d d r e s s e d is the w e e k - t o - w e e k c h a n g e i n the p o p u l a t i o n o v e r a y e a r , a d a i l y average is a p p r o p r i a t e . F o r s i m p l i c i t y , it is a s s u m e d t h a t the i n c i d e n t solar r a d i a t i o n as a f u n c t i o n of t i m e o v e r a d a y is g i v e n b y the f u n c t i o n

Io(t)

=

I

0

/·(

A Dynamic Model of the Phytoplankton Population in the Sacramento

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