The assumptions
of phytoplankton
and rationales of a computer
population
dynamic?
model
John T. Lehman2
Haskins
Laboratories,
Daniel
B. B&kin
School of Forestry
Department
and Environmental
of Biology,
Yale University,
New Haven,
Connecticut
Studies, Yale University
Gene E. Likens
Section of Ecology
and Systematics,
Cornell
University,
Ithaca,
New York
Abstract
Predictions
of phytoplankton
growth dynamics and nutrient assimilation
by a computer
simulation
model are consistent with studies of field and laboratory
populations.
The
model simulates population
dynamics and gross physiology
of phytoplankton
species in
the epilimnion
of a lake where algal growth is subject to temperature,
light, and nutrient
constraints and includes luxury consumption,
end-product
inhibition
of both carbon fixation and nutrient uptake, and species-specific
differential
efficiencies
of nutrient assimilation. C : P, C : N, and N : P ratios of the algal cells respond to changes in external nutrient conditions,
and nutrient storage by the cells permits biological
effects of nutrient
pulses to be evident long after assimilation
of dissolved nutrients
forces the pulses to
decline. Species succession results when abundances of specific taxa decline due to such
factors as sinking or grazing, which assume overriding
importance
when cell division rates
arc slowed by chemical or physical limitations.
The physiological
tenets and limiting
assumptions of the model have been used to formulate
patterns of competition
among speties for light and nutrients in natural systems and to predict temporal changes in plankton
biomass and spccics composition.
Patten in 1968 reviewed the major plankton productivity
models that had been proposed and concluded that they justify
Levins’ ( 1966) axiomatic denial of simultaneous generality,
precision, and reality.
The pace of limnological investigations has
nonetheless prompted continuing attempts
to survey and synthesize our knowledge of
lacustrine ccosys terns. Models should be
realistic and precise enough to deal with urgencics of watershed management and cultural eutrophication but must maintain the
aspect of generality valued by theoreticians.
They must serve in twin roles as repositories
for hypotheses already supported by empirical observations and as testing grounds
for new hypotheses. The models themselves
IA contribution
to the II&bard
Brook Ecosystcm Study.
Financial
support was provided
by
grants from the National
Science Foundation
to
G. E. Likens and F. H. Bormann.
2 Present address:
‘Department
of Zoology NJ15, University
of Washington,
Seattle
98195.
LIMNOLOGY
AND
OCEANOGRAPHY
must be completely tentative, composed of
connected hypotheses each of which is subjcct to change in light of fresh data. The
models must interact with experimental results from the field and laboratory, and if
their theoretical framework is sound, they
should help us to cvaluatc each new datum
as it is produced.
The early models of Riley (1946) and
Riley ct al. ( 1949)) by incorporating physiological factors that contribute to marinc
plankton
productivity,
permitted
phytoplankton distribution and abundance to be
predicted over depth and time. Subsequent
attempts to describe pelagic dynamics more
accurately have rcfincd and extended the
formulative equations of these early papers.
Chen and Orlob (1972) and Male (1973),
among many others, have used computer
simulations to predict plankton population
dynamics and patterns of productivity.
These models usually treat the phytoplankton as a single unit, without regard for dif-
343
MAY
1975, V. 20(3)
344
Lehrmm et al.
ferences among species in the relation between maximum photosynthesis and light
intensity or for species-specific differences
in nutrient uptake efficiencies. There has
been little effort to simulate the seasonal
periodicity of individual species and little
attempt to model changes in species composition during cultural eutrophication.
Believing that such efforts would serve both
theoretical and practical aims, we prepared
a simulation of plankton growth dynamics
at the spccics level.
In this model, we examine the consequences, in terms of community dynamics
and species interactions, of a series of established relations in phytoplankton
physiology. Although the simulator treats growth
primarily as cell division, cell size is not
necessarily held constant, because cell
quotas of carbon and other elements are allowed to vary. Some of the general assumptions explicit in this model are outlined
below.
Each cell possesses a maximum division
rate possible during optimal conditions;
Growth rate can be reduced from the
maximum due to suboptimal temperature,
light, or nutrient conditions;
Each species can be characterized by a
light intensity that saturates photosynthesis.
Greater intensities
progressively
reduce
carbon fixation due to inhibition or photooxidation effects;
Each species can be characterized by an
optimal temperature and by upper and
lower thermal limits;
Each species possesses a growth limiting,
nutrient-specific minimum cell nutrient content and a finite nutrient-specific
storage
capacity beyond this minimum;
Nutrient uptake of each species can be
characterized by Michaelis-Menten
equations with both maximum rates of nutrient
uptake and nutrient-specific
half-saturation
constants;
Actual uptake velocities are dependent
on both internal and external nutrient concentrations;
Cells divide at rates determined by temperature and cellular nutrient contents;
Phvsiological
death is insignificant
ex-
cept under very suboptimal growth conditions;
Cells are characterized with species-specific sinking velocities and with speciesspecific light extinction coefficients.
The algal growth simulator is written in
APL, a language that is very compact:
operations requiring involved subroutines
in FORTRAN or ALGOL are reduced to
single characters in APL. Alternate hypotheses or expansions of existing hypotheses
can often be tested within a matter of minutcs. Such ease of formulation and functional versatility help to focus emphasis on
the essence of the model, rather than on the
technical aspects of its implementation.
Numerical results are achieved by timestep solutions of difference equations. Because important events in nutrient dynamics occur over periods much shorter than a
single day, calculations are performed on an
iterative basis over intervals not exceeding
4 h. On an IBM 370/I%, average execution
time is 0.18 s species-l day-l with a calculation interval of 3 h. During each interval,
cell nutrient quotas are increased by uptake
and diluted by cell division. The principal
environmental variables that need be specified for this model are initial concentrations
and daily ( or hourly) rates of supply of nutrients per liter of epilimnion. Some simple
mass-balance equations have not been
spelled out here in equation form but will
be evident to the interested modeler.
We thank L. Provasoli and G. E. Hutchinson for helpful discussion and criticism.
The moclel
The model simulates the growth of algae
in the epilimnion of a lake during a 360day year. Growth, tabulated as increases
in numbers of cells per liter, is computed
for each species with regard to light, temperature, and nutrient relations. Cell loss
through sinkagc and physiological death is
also computed.
Temperature-The
epilimnetic temperature is assumed constant over 24 h and is
computed once per day to determine species-specific relative growth rates. The annual temperature variation is represented
,
Phytoplankton
Table
plankton
1. Half-saturation
algae.
Cyclotella
nana
NO3
Dunaliella
tertiolectaNH4
blonochrysis
lutheri
Carpenter
MacIsaac
0.35
Caperon
0.5
Eppley
NH4
0.4
NO3
0.21
Caperon
& Guillard
4 Dugdalc
C Meyer
et al.
1971 Leptocylindricus
danicus
1972
1969
G Flcyer
1972
1.4
NH4
0.6
NO3
0.7-1.3
Eppley
& Thomas 1969
1.0
Eppley
et al.
NH4
1.0
NO3
0.42
NH4
0.29
Fragilaria
pinnata
NO3
sp. NO3
Coccochloris
stagnina
NO3
PhaeodactylumN03
tricornutum
N03
N02
Eppley
Caperon
0.6
Eppley
et al.
Fc Meyer
et al.
1969
1969
1972
1969
0.4
0.6-l
.6
Carpenter
4 Guillard
0.1-0.9
Rhizosolenia
stolterfothii
NO3
1.7
Nil4
0.5
Rhizosolcnia
robusta
NO3
3.0
NH4
7.5
Ditylum
brightwellii
NO3
0.6
NH4
1.1
Coscinodiscus
1 ineatus
NO3
2.6
NH4
2.0
Coscinodiscus
wailesii
NO3
3.6
NH4
4 .9
Euglena
gracilis
PO4
16.
Mum 1966
0.58
Fuhs et al,
Cyclotella
nana
po4
Thalassiosira
fluviatilis
po4
1.72
Chlorella
pyrenoidosa
PO4
4.-s.
Jeanjean
Nitzschia
actinastreoides
PO4
1.0
Miiller
1969
1972
Ketchum
1939b
Scenedesmus
PO4
0.6
Rhee 1973
PO4
1.1
Lehman unpublished
lfattori
1962
Pediastrum
duplex
Dinobryon
cylindricum
PO4
0.8
1965
D. sociale
var.
americanum
PO4
0.5
Nitzschia
actinastreoides
si
3.5
Miiller
Thalassiosira
pseudonana
Si
Si
1.4-2.9
Paasche
1973a
1.39
Paasche
1973b
Thalassiosira
decipiens
Si
3.37
Skeletonema
costatum
Si
0.80
Si
2.58
Si
2.96
40.
Knudsen
Chaetoceros
gracilis
NO3
0.2
Eppley
et al.
1969
NH4
0.4
NO3
9.5
NI I4
5.5
Gymnodinium
splendens
NO3
3.8
NH4
1.1
Coccolithus
huxleyi
NO3
0.1
N1f4
0.1
Skeletonema
costatum
NO3
0.45
Licmophora
0.8
Ditylum
NO3
1972
2.6
25,
Isochrysis
galbana
1969
C Meyer
NO2
NH4
et al.
0.7
Caperon
Chlorella
pyrenoidosa
Gonyaulax
polyedra
Eppley
1.25
“If;
0.31
70.
1972
1971
for marine and freshwater
NO,
1969
NO3
No3
NI I4
Anabaena
cylindrica
.9
1.8
345
for N, P, and S4 ,uptake (PM) reported
0.17
Asterionella
j aponica
Bellochia
0.4-l
constants
dynamics
0.1
by a cosine curve during ice-free months
and by a constant function during ice cover:
T = Tmin
+ 0.5T,,,,[lcos 24D - Dm)/(Dc - Dm)]
for D, < D < D,,
(1)
T = Tlllill otherwise, where D = day of the
year, D,, = day that ice cover melts, D, =
day that ice cover is established, TIllin =
sp.
sp.
1972
brightwellii
minimum yearly epilimnetic temperature,
and Tmin + Tnlax = maximum yearly cpilimnetic temperature. When the model is integrated with data from a particular lake,
this temperature curve can be replaced by
a Fourier scrics expansion conforming to
actual temperature measurements (Birge
ct al. 1927).
Temperature dependent relative growth
346
Lehman et al.
Table 2. Maximum
rates of nutrient
uvtake
(Fmoles cell-’ h-‘) reported for ‘marine and freshwater plankton algae or calculated from available
data.
Cyclotella
nana
N03
Biddulphia
aurita
NH4
Dunaliclla
tertiolecta
Monochrysis
luthcri
N03
NH4
NO_
4.-9.
4.-g.
Cyclotclla
nana
NO3
0.3-l.G
Ditylum
brightwellii
N03
Euglena
gracilis
Phaeodactylum
tricornutum
Astcrionella
formosa
Scenedesmus
po4
PO4
po4
Carpenter
Guillard
FI
1971
Lui & Roels 1972
2.1 x1o-8
0.9 x1o-8
2.G-10.6 x10a8 Caperon 8
Meyer 197 2
2.2-3.7 x~O-~
2.1-3.8 x~O-~
-8
1.4 x10
NO,
NO:
Coccochloris
stagnina
x1o-g
x10-lo
x1o-8
1.2 xlo-6
Epplcy E
Coatsworth
2,4 x~O-~
Blum 1966
0.7-l
.4 x10 -’
8. x10-’
Kuenzler &
Kctchum 1962
Mackercth 1953
PO4
1.2 x1o-8
4. x10 -8
Rhee 1973
Pediastrum
duplex
Navicula
pelliculosa
Si
1. x1o-8
Busby E
Lewin 1967
Nitzschia
alba
Si
2.5-5.7
Skclctoncma
costatum
Si
3.4 x1o’g
Thalassiosira
Si
oscudonana
Licmophora sp. Si
2.6 x10-’
PO4
Ditylum
brightwellii
Si
Thalassiosira
dccipiens
Si
1968
Lehman
unpubl ishcd
x1o-8
Lcwin &
Chcn 1968
Paaschc 1973b
7.7 x1o-8
1.46 x10-’
rates ( TD ) are reprcsentcd by skewed normal distributions:
TD =
exp{-2.X (T - To&( L - T,,t) I”>
for T > Topt,
TD=e~~~-~~(~-To~t)/(T~~-T,t)l”)
’ opt*
(2)
Growth rate is highest at the optimal tempcrature ( Topt) and declines as temperature
deviates from the optimum toward either
higher (T,J or lower (Tn) limits. The limits arc set arbitrarily at some fraction (here
10%) of the maximal growth rates under
used
optimal conditions. The distribution
is a somewhat inexact approach to the
Arrhenius equation of enzyme activity that
Johnson et al. ( 1954) suggested may fit the
growth patterns of microorganism populations undergoing exponential increase. Data
on temperature limits for algae are from
laboratory studies of growth rates of individual species (e.g. Thomas 1966; Smayda
1969; Fogg 1965) and field data (Klotter
1955). Hutchinson ( 1967) compiled ranges
and optima for many common species of
phytoplankton.
Nutrient
uptaJce---Nutrient
uptake by
plankton algae generally follows MichaelisMcnten kinetics (e.g. Caperon 1967; Dugdale 1967; Droop 1968) ; half-saturation
constants and maximal velocities of nutrient
uptake for a variety of species have been
determined (Tables 1 and 2). The range in
values indicates that algae differ markedly
in their efficiencies of nutrient uptake. Organisms from pelagic, nutrient-poor localitics are generally characterized by low halfsaturation
constants and thus by very
efficient nutrient uptake. MacIsaac and
Dugdale ( 1969) found that half-saturation
constants for inorganic nitrogen uptake
were an order of magnitude lower among
phytoplankton
assemblages from “oligotrophic” than from “eutrophic” regions of
the ocean, Eppley et al. ( 1969) have shown
how such differences in uptake efficiencies,
when translated to specific growth rates,
can control the outcome of competition between potentially
sympatric species, and
how the patterns of competition can be altered by physical conditions or by the type
of nutrient supplied.
In formulation of our model, we assume
that nutrient uptake depends on external
substrate concentration
in standard Michaclis fashion, i.e. the plot of uptake velocity u ( pmoles cell-i h-i) versus external
nutrient concentration S ( PM ) yields a rectangular hyperbola:
u = [ (Qm - Q)/(Qm - kQ)]u,,,S/(k,
+ S), (3)
where urnal = maximum uptake velocity,
k, = half-saturation constant of nutrient uptake, Q = quantity or quota of nutrient contained per cell, kg = minimum cell nutrient
content (below which division cannot pro-
Phytoplankton
Table 3. Minimum cell nutrient quotas (pmoles
cell-‘) of N, P, and Si for some marine and freshwater phytoplankton.
347
dynamics
90
t
80 Phosphorus :
Asterionclla
f ormo sa
Asterionclla
j aponica
2 x10
-9
blackereth
1.5-3. x10
1.5 x1o-g
-9
-9
i&ller
Fuhs 1969
Cyclotella
nana
0.9 x10
Nitzschia
actinastreoidcs
3 x1o-g
4 x1o-g
Phaeodactylum
tricornutum
2 x1o-g
Chlorella
pyrenoidosa
3 x10-9
Scencdcsmus
quadricauda
4.5 x10-9
Scencdcsmus sp.
Thalassiosira
fluviatilis
1.7 x10-9
Rhcc
12.5 x10-’
Fuhs et al.
A.
formosa
3 x10
(35~)
1.1 x10
Dinobryon
0.5 x10
Anabaena
2.S x10
Milller
1972
rso-59)
-8
Gymnodinium
1953
1972
1973
1972
OV
0
I
I
RELATIVE
I
I
I
1
2
3
4
5
NUTRIENT
CONCENTRATION
(k 8” 1.0)
Grim 1939*
-8
-9
l?ig. 1. Relative
in response to both
regimes. Q and ko
as cxplaincd in the
-9
velocities
of nutrient
uptake
external and internal nutrient
rcfcr to ccl1 nutrient contents,
text.
Silicon:
Navicula
pclliculosa
Nitzschia
alba
0.5 x10
3 x10
-7
-7
Busby 6 Lcwin
1967
Lcwin 4 Chcn 1968
-G
2 x10
-6
1.8 x10
4 x19-6
llughcs G Lund 1962
Grim 1939*
2 x1o-8
Paaschc 1973a
Isochrysis
galbana
3 x1rl-8
Droop 1973
Astcrionella
formosa
6 x~O-~
Grim 1939”
Gpnodinium
Dinobryon
-7
3.9 x10
1.8 x~O-~
-7
1 x10
are not necessarily
Astcrionclla
formosa
Fragilaria
crotonensis
Thalassiosira
pseudonana
Nitrogen:
Anabaena
*These quantities
the minimum.
teed) (Table 3), and Qfn = maximum nutrient storage capacity of a cell. The function includes a feedback factor to account
for end-product inhibition
of uptake by
nutrients that accumulate in the cells. Luxury consumption of nutrients
(Ketchum
1939a) is a natural consequence of this
model whenever high external nutrient
levels product uptake rates in excess of
the dilution caused by ccl1 division. Kinetic evidcncc ( Rhec 1973) that internal
phosphate stores act as noncompetitive inhibitors of phosphate uptake provides the
mechanism for limiting luxury consumption
that Droop (1973) sought on an intuitive
basis. Bccausc empirical data for inhibition
constants are limited to Rhee’s measurements for Scenedesmus and these constants
are difficult to mcasurc, we use a simplification in the model to ensure that uptake
rates decline as ccl1 nutrient stores increase.
Each population is characterized by a limiting nutrient store capacity ( Qlrt ) ; maximal
rates of nutrient uptake diminish as internal
nutrient stores ( Q ) approach the maximal
limiting quantity. Uptake rates are greatest
when cell nutrient contents equal a minimum value ( k, ) below which cell division
cannot proceed. Figure 1 shows predicted
relative nutrient uptake rates based on
equation 3. Each curve corresponds to a
diffcrcnt cellular nutrient content ( Q), and
uptake velocities decline as cells become
progressively less starved. Q,n can be estimated as the maximum Q approached in
nutrient excess, and it can bc obtained from
348
Lehman
published data for some species. Lineweaver-Burk transformation of the curves
in Fig, 1 shows that the simplification used
is superficially indistinguishable
from genuine noncompetitive inhibition.
Data by Fuhs (1969), Droop (1973), and
Paasche ( 1973a), among others, show that
ccl1 division rates, plotted as functions of
internal nutrient contents, empirically
resemble rectangular hyperbolas; they are
treated as such in the model, using Droop’s
( 1973) derivation of the kinetic relations.
The nutrient dependence of cell division
( ND ) is given by
ND=
(Q-W/Q>
(4)
with Q constrained to the range [k@, Qm].
ND thus has a lower limit of 0 and approachcs 1 asymptotically, although in practice the function is truncated at Q = Qm,
The model thus recognizes uptake rates as
functions of both internal and external nutrient levels, but cell division as dependent
solely on internal concentrations. This permits formulation of situations in which dissolved phosphate and nitrate disappear
within hours following artificial enrichment
of a lake, but produce long term growth effects among the phytoplankton
that absorb them (e.g. Schindler et al. 1971). The
distinction
bctwecn uptake and division
rates enables changes in cell nutrient quotas
like those observed by Droop ( 1968) to be
reproduced by this model and also predicts
variable intracellular
nutrient ratios, depending on external nutrient
conditions
and the growth regime.
Three nutrients are considered in the
model, treated simply as dissolved inorganic
P, N, and Si. Additional
nutrients, trace
metals, and vitamins could be immediately
included without programing
changes if
the appropriate species-specific kinetic parameters were known. For purposes of illustration, however, and for reproducing
much nutrient-related population dynamics,
the present arrangement is sufficient.
Photosynthesis-The
treatment of photosynthesis is similar to that of Talling (1957)
or Vollenweider
( 1965)) but the photosynthetic function used is that of Steele ( 1962).
et al.
Light intensity (1) and its attenuation with
depth (2) are determined from Beer-Lambert predictions based on a vertical light
path:
I = IOexp(-EZ),
(5)
where I0 = surface intensity in langleys per
minute ( g-cal cm-2 min-l).
The light extinction coefficient (E) is a sum of contributions from the lake water itself and from
plankton, the latter term being density dependent and species-specific, The parameters used to rclatc photosynthesis to light
intensity for different taxa follow observations (e.g. Ryther 1956; Brown and Richardson 1968) of species-specific intensity dependent saturation and inhibition:
P(1) = %,,,( I&t) exp( 1 - W,,t ) , ( 6)
where P,,,,, = maximum photosynthetic rate,
occurring at Iopt, and Iopt = light intensity at
which photosynthetic rate is saturated. Both
the appearance of surface inhibition
and
the correlation
dcmonstratcd
by Rodhe
( 1965) between light attenuation and photosynthesis are predicted by the model.
Though Rodhe stressed the use of the intensity of the most penetrating spectral componcnt in each particular water mass as a
correlate with photosynthesis, our model
ignores spectral modification
with depth
and treats light intensity in terms of photosynthetically active radiation ( PHAR : 380720 nm: Strickland 1958). The success of
the treatment is partly due to the factors inducing Hutchinson’s (1957, p. 392) remark
that spectral modification
below the top
mctcr of penetration is often sufficiently
small for a mean vertical extinction coefficient for white light to be useful.
Incident light intensity at full noon sun
is represented by a cosine function of period
1 year:
I ,nnx=I,+1,[1-cos~(D+8)/180].
(7)
This function predicts a minimum insolation of I, at winter solstice (22 December)
and a maximum of I, + 21, at summer solstice (22 June). WC have used I,, = 0.10
ly min-’ (PHAR) and I, = 0.25, corresponding approximately to expectations for north
Phytoplankton
d
,
temperate regions (ca. 42”N).
A similar
curve for photopcriod predicts daylengths
varying from 9 to 15 h over the year. Incident light intensity from sunrise to sunset
is likewise rcprcsented by a cosine function.
Phytoplankton populations are assumed to
be uniformly
distributed
throughout
the
epilimnion.
Photosynthetic
rates, determined from
the average light intensity within
each
meter interval below the surface, are integrated over the calculation interval of several hours, The model includes allowances
for end-product inhibition by carbohydrate
or lipid reserves stored during previous periods of intense fixation.
Each species is
characterized
with a minimum,
growth
limiting carbon content (Co) and a maximum carbon store capacity (C,).
The
carbon stores ( C ) of active cells will lie
somewhere between these extremes. Photosynthetic carbon fixation, P(I,C), reduced
by end-product inhibition, is given by:
P(W) = [(Cnt-C>/(C,,a-Co)lP(I), (f-3)
where P( 1) is the function of light intensity
described in equation 6. This function is
analogous to that for nutrient uptake (cquation 3), since carbon must be treated not
simply as a source of energy but also as a
nutrient clement. Further reduction of carbon fixation rates, resulting from CO2 limitation caused by rapid removal of inorganic
carbon from solution during periods of intense photosynthesis, is critical in soft-water
lakes at high cell densities; it is treated by
Lehman et al. (in press).
Respiration-Cellular
carbon stores are
increased by photosynthetic gains and decreased by respiratory losses. Respiration
rate (R) is assumed to vary with both carbon stores and water temperature:
I-l = Rrl,x( c/c,)“.“7
- expW3(T- Topt)/(Tll- ~o,t)l,
(9)
where R,,,, = the maximum respiration rate
obtained at T = Tol,t. This treatment of respiration, a modification
of that used by
Riley (1946) based on his light and dark
bottle studies (Riley 1943)) is suggested by
Strickland’s (1960) note that respiration
349
dunamics
.‘j
remb,:
I
rates generally increase in cells entering the
stationary growth phase as nutrient limitapronounced.
tion becomes increasingly
Fogg ( 1965-j, Fulls ct al. ( 19r12), and others
have shown that mean cell carbon contents
generally increase over the same period,
mainly as lipid or carbohydrate stores. Carbon quotas are raised to the 0.67 power in
accord with the empirical “surface law” of
respiration, which claims that most of a
cell’s cncrgy is expcndcd at its surface to
maintain its integrity vis a vis the environment, A possible alternative to this equation is that suggested by Bannister (1974)
for nutrient saturated phytoplankton.
The carbon dependence of cell division
rate ( CD) is given by:
CD = (C - C,.),‘C,
(10)
a quantity analogous to that used for other
nutrients.
CeZZ division-The
temperature, carbon,
and nutrient dcpendenccs (TD, CD, and
ND) described above were defined as normalized functions which take on values
only bctwcen 0 and 1. Actual calculated
cell division rates (p) for each species are
given as
,u=p,,TD
CD r NDi,
(11)
where p,n = the species-specific maximum
division rate, approached asymptotically
under optimal growth conditions, and TD,
CD, and NDi are the calculated relative
growth dcpcndences for temperature, carbon, and P, N, and Si. A multiplicative
nutrient dependence ( Vcrduin 1964; Droop
1973) is used in our simulations, rather
than Liebig’s classic law of the minimum,
Both alternatives were readily examined
with the model, since the opposing hypotheses involve changing only a single character
in the program. They give similar results
when only one factor is limiting, but equation 11 is more versatile and permits simultaneous limitation by more than one elemcnt, although it perhaps predicts it to be
more severe than is realistic. The true relation may lie s0mewher.e between equation
11 and Liebig, as Bloomfield et al. (1974)
have suggested.
350
Lehman et al.
Mortality-Physiological
death is assumed to be insignificant under optimal cell
growth conditions. Cushing ( 1955, 1959)
calculated natural mortality of actively dividing marinc diatom populations to be
less than 1% of either daily production or
grazing rates. Under extreme suboptimal
conditions when growth rate falls below
some small fraction of maximum, set by us
arbitrarily
at 5%, physiological
death becomes significant. The longer a population
remains at these conditions, the greater the
mortality effects become, until some maximal fraction (M,,,)
of the population dies
each day.
M = M,,,[
1 - exp(-kSG)]
(12)
where SG = number of days at suboptimal
conditions (p/pm < 0.05)) and k = (In 2)
divided by the number of days at suboptima1 conditions until M increases to half of
M rnaxSince the quantitative aspects of physiological death are so poorly understood, we
have simply lumped them together in terms
of their gross effects. Treatment of mortality in this fashion avoids the difficulty
of determining
separate contributions
of
temperature, photo-oxidation,
or nutrient
starvation, We believe that a suitably flcxiblc mortality function of this sort is preferable to constant mortality rates, but it
still does not explain such nonobvious and
poorly studied aspects of cell mortality as,
for instance, time-lagged effects of temperatures above the Grenzoptimum
like those
demonstrated for Melosira by Rodhe (1948).
The treatment of physiological death must
therefore be regarded as a working simplification introduced to explain certain gross
similarities of mortality resultant from a
variety of possible causes and to simulate
phenomena about which very little is
known.
Dead cells that do not sink release their
nutrient contents to an epilimnion particulate nutrient pool; we assume that decomposition rates back to the dissolved pool
are proportional
to particulate pool size
because of the swift reproductive potential
of bacterial populations. The nutrient stores
of cells that do sink are assumed to be lost:
some fraction is lost to deep sediments, the
remainder is redistributed
to the mixed
layer at vernal and autumnal overturns.
Mean epilimnion depth, or the depth of
uniform surface mixing, can vary over the
year according to any predetermined pattern specified by the user. Stewart (1965) illustrated the variety of annual temperature
profiles possible in dimictic
temperate
lakes, depending on weather conditions.
Empirical
depths were originally
fit by
Fourier series, but comparative runs showed
that results are almost identical if the points
are simply connected by a series of straight
lines, the method we now use.
Treatment of ccl1 sinkage from the epilimnion is simplified by the assumption of
uniform cell distribution. Hutchinson (1967)
tabulated maximal sinking rates for some
freshwater species; Smayda (1970) prescnted linear regressions of sinking, rates on
mean diameters of living and senescent individual cells and maximum diameters of
palmelloid colonies, albeit for marine species. If the vertical distribution
of each
species is assumed to be uniform, changes
in population numbers can be given by
dN/dt=[pln2-(V/D)-M]N,(13)
where N is cell concentration, V is sinking
rate, D is the mean epilimnetic depth, and
/A and M arc the previously defined growth
and death rates. The assumption of thorough mixing is obviously an utter simplification. Rilcy ct al. ( 1949) proposed a more
realistic method based on eddy turbulence,
and some workers have attempted discrete
simulations based on their approach (Male
1973). The adoption of such methods was,
however, unnecessary for the types of biological questions posed here.
Results
The implications
of the assumptions
stated explicitly
at the outset encompass
enough empirical characteristics of phytoplankton communities to provide the model
with realism and precision, but not sacri-
Phytoplankton
Table 4. Principal parameters used in describing
in the text. Published values are used when possible.
sons et al. 1961) are used to estimate parameters.
Green
Iopt
(ly
P
k, (MO
s
(~10~ poles
cell-1
kQ (x109 poles
cell-l)
h-l)
(X108 poles
0
s"
0
P
N
Sl
10
30
0
10
0
P
N
4.5
90
0
1
20
Pmax (1~10~ poles
C cell-l
Rmax (z&O7 pmoles C cell-l
kC (~10~ jxnoles C cell -1 )
Cm (x106 poles
pm (divisions
C cell-l)
day")
V (m day-l)
Ed (x107 liter
N
si
cell -1 m-1)
7e2
72
0
Diatom
0.1
Dinoflagellate
0
1.6
16
0
l-5
7
3
7'o"o
4:
2000
3-2
32
Chrysophyte
0.2
0.06
5"
0
0.6
0.75
0
1
1
2
si
cell-l)
0.03
N
si
P
s
model species according to the scheme described
Otherwise published ratios (e.g. Gkln 1939; Par-
Blue-green
0.045
min-1)
351
dynamics
10
100
0
1
10
0
10
200
0
10
0
16
160
0*5
0.8
8
1000
0
0
15
1.0
h-l)
7
1.5
3
hY1)
1.4
0.3
0.6
7
l-5
7
3
0.36
3
15
1.8
1.5
3
15
1.8
3
2.5
3
1.5
1.5
0.8
0.2
2.5
0.5
0.5
1.2
1
3
fice its generality in the process. The entities treated here are composed of values
gleaned and synthesized from the literature
( Table 4) ; they arc necessarily composite
due to the lack of sufficient data to characterize any single real species with all the
parameters necessary. To this extent they
are hypothetical
and their characteristics
are subject to the needs and improved data
of each user; we hope to, show that efforts
to characterize ecologically important phytoplankton with regard to certain defined
parameters may be rewarded by enhanced
predictability
for natural communities. Figure 2 shows the predicted photosynthetic
profiles of four hypothetical species in water of moderate clarity ( E = 0.4 m-l) at a
surface light intensity of 0.36 ly rnml. The
different patterns of surface inhibition and
attenuation with depth are ascribable to the
different Iopt values assigned to each. Taxa
30
0.3
displaying photosynthetic
maxima lowest
in the water column are those which tolerate or require subdued light intensities.
Patterns of photosynthesis by a single species change as light intensity varies from
sunrise to noon ( Fig. 3). Light limited in
the early morning hours, the population
progressively shows saturation and surface
inhibition effects as incident light intensity
increases. Daily photosynthesis
by each
species has therefore both a time and a
depth component, and comparative computcr runs have shown that estimates of
species productivity
based on mean daily
insolation intensities (e.g. Male 1973; Hydroscience 1973) rather than on actual
daily integrals may overestimate photosynthesis by 59% or more when severe midday
inhibition is present. For instance, in a 9-m
water column, with e = 0.3 m-l, noon intensity = 0.4 ly min-l, and daylength = 12
352
Lehman
I,=
0.36
et al.
ly mln-’
Em0.4m’1
OOJ;
I
2
3
DEPTH
4
( m 1
5
6
7
ool
*0
I
2
3
DEPTH
4
(ml
a
6
7
Fig. 2. Photosynthesis-depth
profiles for four
hypothetical
species with Iopt values listed in Table
4. The profiles
illustrate
differential
extents of
surface inhibition
and depth attenuation
between
species.
Fig. 3. Photosynthesis-depth
profiles for a single species with Iopt = 0.1 ly min? PHAR.
Profiles are computed at successive hours after sunrise, corresponding
to increasing
incident
light
intensities
( IO).
11, the two methods give results for daily
photosynthesis differing by 60% for the diatom species used here (IoPt = 0.1 ly min-I).
Just as incident light intensities can control species-specific fixation rates, water
transparency can profoundly influence rcalized rates of photosynthetic carbon fixation
by algal cells and can completely alter competitive relationships between species in a
uniformly
mixed water column of finite
depth. At moderately high light intensities,
and in water of high transparency (low E),
the dinoflagellate
and the diatom display
the highest normalized daily fixation rates;
at successively higher values of E, the green
and the blue-green increase relatively and
eventually surpass them. The results imply
that regardless of the incident light level,
if large plankton biomass generates extensive self-shading, species which saturate at
low light intensities become competitively
In fertilized
Canadian shield
superior.
lakes the species dominating at times of
peak biomass are saturated at 0.03 ly mind1
or less ( Schindler and Fee 1973).
fixation
ratesDepressed afternoon
When end-product inhibition of carbon fix-
ation is added to these light intensity effects, photosynthesis becomes a function of
the physiological state of the cells as well
as of their external physical environment.
Figure 4 shows changes in net and gross
carbon fixation and cellular carbon contents over three 24-h cycles for a hypothetical green alga in near-optimal light and
Carbon fixation
is
nutrient
conditions.
maximal in the morning of each day and
declines in the afternoon, even though light
intensities are identical in each period.
Comparison of the gross carbon fixation
curve with that of cell carbon, and reference to equation 8, rcvcals that the afternoon depression is due to end-product inhibition
by accumulated
reserves. Cell
carbon reaches a maximum value just before sunset and a minimum just before
sunrise of each day. Afternoon depression
of net carbon fixation rates is further cnhanced due to the correction for respiration,
which increases in the model in proportion
to the cell carbon quota. Carbon reserves
that accumulate within cells in daylight
product increased rates of respiration and
cell division which themselves act to reduce
Phytoplankton
I
I
IO
20
I
I
HO&
dynamics
&&I
I
MI~&GHT
I
I
60
70
I
80
Fig. 4. Net ( 0) and gross ( 0 ) photosynthesis
by a hypothetical
green alga during three 24-h cycles, and corresponding
cell carbon contents ( A ). Depressed afternoon rates are due partly to end-product inhibition
(gross ) and partly to respiratory
losses (net ).
cell quotas. The resulting pattern is one of
maximal photosynthesis
in the morning,
preceding the period of maximal respiration
and growth. A ncccssary consequence of
the assumptions used here is that cells taken
from the light and placed in the dark in
the presence of adequate nutrients continue
to divide, at an ever-decreasing rate, until
they exhaust their carbon reserves. The
photosynthetic
capacity (P,,,,) of Phaeodactylurn
tricornutum
progressively
declines during, sustained exposure to optimal
light intensities ( Griffiths 1973), even without any significant decrcasc of cell chlorophyll; when the cells are incubated in the
dark for increasing periods, the measurable
Pmax increases. Our model predicts those
results precisely, principally on the basis of
end-product
inhibition
of photosynthesis
by accumulated carbon reserves.
The model is not constrained to produce
depressed afternoon fixation rates, but these
arc a consequence of near-optimal light
regimes. If light intensities are especially
low for instance, a low P : R ratio may keep
carbon stores from accumulating to inhibitory levels and no pronounced afternoon depression will occur. The ability of the
model to account for cell shading at high
cell densities is shown in Fig,. 5. Afternoon
depression due to accumulated
carbon
stores is reduced as carbon fixed per cell
declines during a simulated population expansion.
Steady state growth relations-To
determine whether cell division rates predicted
by the model could be related to the extcrnal nutrient
concentration
directly
and
thereby avoid the necessity of treating nutrient storage, we carried out simulations
Lehman et al.
phyte. These values are not an explicit part
of the model framework, i.e. no K, values
are programed, but they ca.repredicted by
the model for cells growing in steady state
nutrient regimes, The values are all about
one order of magnitude lower than the corresponding
half-saturation
constants of
phosphate uptake (k,) for each species.
The K, values for growth are valid, however, only for cells in equilibrium
with a
constant environment,
e.g. those in a
chemostat. Attempts to relate cell division
rates directly to external nutrient levels
without considering the moderating influences of luxury consumption and cellular
nutrient reserves prove inadequate when
the cells are exposed to continually fluctuating chemical environments, as is the case
1
I
I
I
I
in natural ecosystems.
20
40
60
80
100
HOURS AFTER MIDNIGHT
Table 5 lists the C : P and N : P molar
Fig. 5. Net photosynthesis
ratios computed for the cells in the phos(01, cell carbon
( 0 ), and population
density ( A) of a hypothetiphate-limitation
experiment ( Fig. 6). Cells
cal green alga.
under extreme phosphate starvation showed
many-fold increases in C : P and N : P ratios. Most of the increase was due simply
in which small inocula ( 500 cells liter-l)
were incubated for 5 days at each of a to phosphate depletion, but storage capabilities for other nutrients also played an
range of nutrient concentrations. The interimportant
role. Cells whose division rates
val was short enough so that the cells were
exposed to stable light and temperature re- are slowed by low phosphate concentrations
continue to photosynthesize and to absorb
gimes. One nutrient was tested at a time,
nitrogen
salts until end-product inhibition
with all others in great excess, and populaforces
a
balance among carbon fixation,
tion densities were so low that uptake did
nutrient uptake, and the dilution rates atnot significantly
affect nutrient concentrations in the medium. The interval was tending respiration and cell division. For
more than adequate, however, to assure example, if C and N contents of the starved
( 0.01 ,xM P ) cells had been the same as
that the internal nutrient stores of the cells
for the normal cells, then the C : P ratio of
stabilized at a level determined by uptake
kinetics and cell division rates. At the end the green alga would have been 228 and
the N : P ratio 70.
of the 5 days division rates were calculated
Fuhs et al. (1972) found that P-starved
for the P-limited cells, based on their internal phosphate stores, and these rates arc Thalassiosira pseudonana accumulated cxcess carbohydrate reserves and that cell
plotted for three model species against
carbon
quotas increased markedly during
phosphate concentrations in the medium
phosphorus
deplction.
Cell quotas of ni(Fig. 6); curves for nitrogen from analogous
trogen, on the other hand, did not increase
simulations are qualitatively the same. The
in ThaZassiosira and even declined slightly.
points plotted in Fig. 6 can be fit by Michaelis-Menten equations specified in terms This discrepancy between model prediction
and empirical fact can be traced to our
of t.~and S. The appropriate half-saturation
constants, calculated from these data, and conceptual treatment of nutrient interaccalled K, values, are 0.09, 0.32, and 0.03 tions. That is, the equations presented
PM for the green, blue-green, and chrysoabove implicitly assume that uptake mech-
Phytoplankton
355
dynamics
CHRYSOPHYTE
GREEN
c5 0.8
BLUE-GREEN
9
$ 0.7
V
$! 0.8
d
g 0.5
H
1 0.4
0
w
2 0.3
2
g 0.2
0.1
0.0
II
0.0
11111
02
0.4
I
0.8
0.8
PHOSPHATE
Ill
I
1.0
I
1.2
CONCENTRATION
l
1.4
III
III
1.8
C:P
P
(Pa
Green
0.01
0.03
0.1
0.3
1.0
523
320
‘2
34
Blue-green
592
51-5
347
176
76
species
derived
with
from
dissimilar
halfcurves fitted to
little over a wide range of phosphate concentrations, but carbon reserves accumulate as described in relation to Table 5. We
lack precise data on the influence of single
nutrient deplction on the kinetic parametcrs for uptake and storage of other nutrients, but our model could incorporate
new data with a minimum of effort. Ketchum ( 193%) and Rhee ( 1974) emphasized the importance of such nutrient inter-
Table 5. Predicted C : P and N : P atomic ratios of cells experiencing
ent assi,milation mechanisms are assumed mutually
independent.
Dissolved
2.0
IN THE MEDIUM (NM / LITER )
Fig. 6. Steady state relative growth curves for three hypothetical
saturation constants for phosphate uptake. K, (growth)
constants were
points generated by the model as described in the text.
anisms
for different
substances
are completely
independent,
being determined
only by internal and external supplies of
individual nutrients. If nitrogen uptake is
made dependent on cell phosphorus co,ntent as well as ccl1 nitrogen by the same
relation used to determine cell division
(ND), simulation results (Table 6) conform closely to those measured by Fuhs et
al. ( 1972). Nitrogen quotas change very
1.8
phosphate
limitation.
Nutri-
N:P
Chrysophyte
3
Green
94
70
37
18
7
Blue-green
Chrysophyte
g
;:
26
I.4
8
51
10
356
Lehman
et al.
Table 6. Cell nitrogen quotas as multiples of
kQ for cells growing in different phosphate concentrations.
Nitrogen
uptake is assumed dependent
on cell phoSPhorus quotas.
)JMP
Green
0.01
0.1
4.86
5.07
5.19
1.0
lxfltm
5.15
5-30
5.36
Chrysophyte
4.39
4.41
4.42
actions, and their results arc reproduced if
phosphorus uptake is made dependent on
cell nitrogen quotas.
Many workers have observed dramatic
increases in starch or lipids in cells under
nutrient limitation (see Fogg 1965; Strickland 1960). Since increased carbon quotas
generally imply increased cell volumes, the
model similarly predicts the common phenomenon of cell enlargement in nutricntpoor media or stationary phase cultures
(Prakash et al. 1973).
Luxury consumptioni-The
pattern of nutrient assimilation and growth observed by
Ketchum (1939a) and Kuenzler and Ketchum (1962), involving luxury consumption
of dissolved nutrients by algal cells, follows
directly from the precepts of this model.
The predicted consequences of suspending
P-starved cells in a phosphate-containing
medium or, alternatively, of sudden phosphate enrichment of a P-limited lake, are
shown in Fig. 7. High cell concentrations
were used to dramatize the results and nitrogen concentrations were held at great
excess. The initial phosphorus contents of
the cells were adjusted to i&, i.e. to the
growth-rate limiting minimum cell phosphorus content. The cells immediately responded to the external enrichment by absorbing the compound, as shown by a
decrease in the phosphate concentration
in the medium and an increase in the
phosphate contents of the cells. Cell
division, on the other hand, underwent
an apparent lag phase of 8 h before measurable population expansion. Division actually proceeded throughout the
“lag” phase at an ever-increasing rate but
became graphically perceptible only after
8 h. Because cellular nutrient quotas (Q)
,-
I,
HOURS
AFTER
P ADDITION
Fig.
7. Luxury
consumption
of phosphate by
a hypothetical
green alga, shown as an apparent
lag period in cell division
during which cell P
quotas steadily accumulate.
are determined by the balance between uptake velocity and cell division, Q evcntually declines, but only after dissolved
phosphate is reduced to a very low level.
At the same time, uptake rates, reflected in
the rate of phosphate depletion from the
medium, decline markedly.
Cell division
continues, however, fueled by the phosphate r&serves accumulated by luxury consumption, Division rates successively decline during the period of decreasing Q, as
shown by the almost linear shape of the
population growth curve, which deviates
from the exponential shape expected if cell
concentrations had been lower or the initial
P supply larger. With no additional phosphate enrichments,
the population
size
would cvcntually reach a maximum value
when division is halted by P starvation.
This maximum would be only transitory due
to two processes: the rate of population
depletion in natural systems due to sinkage
or grazing would exceed the waning rate
of cell division; and physiological
death
would begin to operate on the senescent
cells. The second factor is beyond doubt
responsible for population declines in laboratory stationary phase cultures, but physi-
Phytoplankton
ological death may not bc of comparable
importance in natural systems. Densities of
most populations decline due to sinkage
although the cells are still perfectly viable.
This point is particularly
applicable to
diatoms, characterized
by high sinking
speeds due to their siliceous frustulcs, as
the following
simulation run will show.
Phosphorus starved populations were adjustcd to low initial cell densities (1,000
cells liter-l), with the exception of the diatom, which was set at 2 x lo6 cells liter-l,
the concentration of Syneclra acus var. angustissima in Linsley Pond on 17 April 1937
( Hutchinson
1944). Initial nutrient concentrations were, as given by Hutchinson,
0.003 mg PQ4-P liter-l and 4 mg SiOz liter-l.
Nitrogen was set at 0.12 mg NOs-N liter-l.
Thermal data compiled by Riley ( 1940)
indicate that Linsley Pond is generally
stratified by mid-April, and mean cpilimnion depth was accordingly set at 5 m in
the model. Nutrient input rates to the cpilimnion were assumed to be low: 0.01 rug
Pod-P liter-l day-“, 0.5 pg N03-N liter-’
day-l, and 3.5 ,ug SiOa liter-l day-l. All species were assigned the same temperature
relations to reduce the number of variables
and to make interpretation
easier. The
physical parameters of the simulation corresponded to the interval 17 April to 20
June. Hutchinson
( 1944) reported that
Synedra reached peak population size on
30 April, subsequently declined, and was
replaced by Dinobryon
divergent which
reached a maximum on 1 June. This same
replacement of a diatom by a chrysophyte
species occurred in the simulation
run
(Fig. 8).
The model can thus be used to dcvclop
hypotheses for the species composition
change in nature. In both the model and
Linsley
Pond, phosphate concentrations
plummeted to less than 0.001 mg liter-1 as
the diatom population
reached maximal
size. As external concentrations dropped,
uptake rates and cell nutrient quotas declined, causing division rates to’ slow, Once
division rates slowed in the model diatom,
the observable reduction
in population
numbers was due to sinkage. Diatoms need
dynamics
357
Fig. 8. Simulated succession in Linsley Pond,
1937. Actual cell counts of Synedra ( 0 ) and
f)li;wqbyn
( x 10 ) ( 0 ) are after Hutchinson
high turbulence and rapid division rate to
produce population expansions in pelagic
When either prerequisite
cnvironmcnts.
for suspension is curbed, populations tend
to decline. At its population maximum, the
diatom was dividing at almost 0.7 divisions
day-l, a growth rate which just balanced
sinking losses. Division rate then declined
slowly due to continued nutrient deficiency,
and cells could no longer be reproduced as
fast as they were lost.
Because of their more efficient uptake of
dissolved phosphorus and slower sinking
rates, chrysophytes continued to increase
after the dcclinc of the diatom. The other
species included, green, blue-green, and
dinoflagellate,
never achieved substantial
population densities because of the low nutrient levels following the diatom bloom,
The expanding chrysophytes eventually exhausted the remaining phosphate, their division rates declined, and loss rates assumed
overriding importance. With the aid of the
model we can thus understand mechanistically how cell division rate, turbulence,
sinking speed, and, though not treated here,
Lehman
CHRYSOPHYTE
MARCH
APRIL
Simulated succession in Linsley Pond,
1938, using the initial conditions
of Hutchinson
( 1944) and Riley ( 1940 ).
grazing rates, operate in species succession.
When either of the first two quantities declines, loss rates take command. Low nutrient levels can thus cause the decline of
a pelagic diatom population by slowing division even before cells become excessively
starved.
The exact patterns of seasonal succession
predicted by our model naturally depend
on the initial conditions specified. When
biological
conditions
( Hutchinson
1944)
were combined with chemical and physical
conditions ( Riley 19140)for Linsley Pond on
23 February 1938, and the same rates of
nutrient addition assumed as in the previous example, the prcdictcd pattern of succession once again agrees well with that
reported by II-Iutchinson ( 1944) (P’ig. 9,).
The blue-green population ( Oscillatoria in
Linsley Pond) started at 9 X lo6 cells liter-“,
rose to a maximum in early March, and
thereafter declined irregularly
due to a
combination of sinkage and physiological
death. It was succeeded by the bloom of a
green ( Scenedesmus in Linsley Pond),
which subsequently gave way to a chrysophyte ( Dinobryon)
in phosphate depleted
waters, Diatoms, started at the same concentration as the green and chrysophyte
(1,000 cells liter-l), never achieved substan-
et al.
tial numbers in the model: they were similarly depauperate in the lake (Hutchinson
1944 ) .
The pattern of succession predicted by
the model was due to interspecific competition for the waning supply of dissolved
phosphorus; similar competition may have
been at least partly responsible for the actual events in nature. Both simulations
dealt with periods near spring overturn
when nutrient supply was variable and
grazing pressure low. Under similar conditions, this model faithfully reproduces patterns of succession in a variety of lakes, but
corroboration between observed and predicted events declines in later months unless additional sources of competition, and
grazing pressure, are included (Lehman et
al. in press).
Temperature effects-The
previous examples ignored temperature for simplicity’s
sake, but empirical distinction of Kiikeformen from Wiirmeformen (Findenegg 1943)
implies that thermal effects may bcl quite
pronounced
among freshwater plankton.
Apart from their explicit effects on division
rates, the temperature dependency functions used in this model affect the physiological condition and nutritional
composition of cells. Growth of small inocula (500
cells liter-l) was simulated for periods of 3
days in the presence of excess nutrients and
a favorable light regime, with only the epilimnion temperature varied to simulate a
series of suboptimal, optimal, and supraoptimal temperatures. The cxperimen t was
analogous to placing a series of laboratory
cultures with the same medium at different
temperatures but under identical light regimes. The average carbon quotas of the
cells were calculated for the third day of
each run, since by that time carbon quotas
had stabilized into regular rhythmic patterns (see Fig. 4). These are listed in Table 7 together with calculated temperature
dependenccs of cell division ( TD ) . Cells
at low temperatures had larger carbon reserves than cells under more optimal conditions, because carbon fixed at low temperatures is not immediately used in growth but
is stored, At optimal temperatures, how-
Phytoplankton
Table 7. Cell carbon quotas as multiples
of
kc for a hypothetical
chrysophyte
uncler several
temperature
regimes.
Temp
("C)
TD
0
0.1
5
15
0.36
0.775
1.0
20
0.1
25
0.001
10
C f kc
2::
2
3-8
2.3
ever, the fixed carbon is rapidly depleted
by division and respiration, and at supraoptimal
temperatures,
abnormally
high
respiration rates dispose of the carbon almost as fast as it is fixed. Increased reserves
at suboptimal temperatures were also observed for the other nutrients (N, I’, and
Si). Another result that emerged was that
rates of nutrient uptake and carbon fixation
were considerably lower at suboptimal temperatures, even though those rates were not
explicitly specified as temperature dcpendent. The larger nutrient reserves resulting
from slow division rates reduced uptake,
through end-product inhibition.
Cells at
low temperatures are therefore characterized by slow growth, low uptake and fixation rates, and large cell nutrient quotas.
Cells at supraoptimal light intensities photosynthesizc
at very rapid rates, even
though ccl1 division is slowed, because
high respiration
prevents carbon stores
from accumulating.
Discussion
The model described here, based on a
few simple empirically
and conceptually
supported relations concerning the physiology of algal cell growth, can be used reciprocally with experimental data and can
serve as a tool in hypothesis testing. We
propose it as an easily expandable start for
future, more extensive models of lake ccoIts conceptual design
sys tcm dynamics,
and its coding in APL make changes easy
and create a flexible tool for testing new
ideas. By its own existence and operation
it suggests experiments that can refine it
and add to its realism. The model adopts
a mechanistic view of competition
and
dynamics
359
niche space, in that both are immediate
conscqucnces of the physiology of the organisms themselves and of the ambient
physicochemical
environment.
This approach has the advantage of being based on
primary observations, for which empirical
tests can easily bc formulated, rather than
on derived indices whose real analogs may
be vague and difficult
to distinguish.
It
has incorporated, synthesized, and formalized the results of diverse observations of
algal physiology and has produced experimentally testable predictions.
It has also
made obvious those areas where information is particularly lacking, as in the factors
influencing
physiological
death, and the
types and methods of nutrieint interaction.
The incorporation
of nutrient
storage
permits the model to mimic such subtle observations from field and laboratory as luxury consumption and starvation induced
changes in cellular nutrient ratios. The nutrient storage and growth relations used
imply the existence of time lags and differenccs in magnitude between rates of substrate assimilation and of cell division. It
does this in a manner that suggests how
phosphate and nitrate added to lakes can
disappear from solution within hours and
how large algal standing crops can develop
in equilibrium
with low concentrations of
dissolved nutrients,
This simulated rapid
uptake and storage likewise helps predict
the empirical rates of nutrient removal
from the epilimnion of a fertilized lake resulting from cell sinkage. Schindler et al.
(1973) estimated losses as high as 8% of
epilimnion P per day, virtually all of which
is contained in phytoplankton
cells. Similar sinking losses are predicted by this
model in certain instances (Lehman ct al.
in press),
Differences of as much as an order of
magnitude
between half-saturation
constants of cell division rates ( K, ) and those
of phosphate uptake (k,), empirically demonstrated by Rhee (1973), follow from
the assumptions of this model. The ability
of submaximal uptake rates to support Aearmaximal growth is due, as Rhee suggests, to
the cell’s ability to store excessive P ac-
360
Lehman
cumulated during periods of nutrient abundance, to the direct dependency of its
growth rate on internal rather than external
concentrations, and to the dependency of
nutrient uptake on both, Simulation models
that measure growth rates purely as functions of immediate external nutrient concentrations
are thus inaccurate
unless
steady state conditions arc a genuine feature of the region or period considered.
Koonce ( 1972) recognized the importance
oE cellular nutrient quotas in determining
division rates, but his formulative equations
do not seem entirely accurate in light oE
recent physiological
data. Bierman et al.
( lQ74) have similarly proposed a model of
phytoplankton
growth that separates uptake from division; they showed that it
could account for luxury consumption and
time lags, whereas models that considered
external concentrations alone could not.
Carbon is treated in our model much
like other macronutrients, with the notable
exceptions that its uptake is light dependent and its loss involves respiration. This
treatment has facilitated the simulation of
many empirical observations of physiological and ecological interest. For example,
the model predicts reduced net photosynthesis in the afternoon caused by increased
respiration
and end-product
inhibition.
Similar empirical observations made by
Harris ( 1973 ) and Harris and Lott ( 1973)
were ascribed entirely to increased photorespiration during the afternoon.
We have also modeled the excretion of
organic carbon by algal cells. In some
simulations, excretion was assumed proportional to the cell carbon quota and the production of extracellular material followed
under different growth regimes. Cells excreted carbon during exponential growth
in quantities that increased during periods
of active photosynthesis, mimicking results
of empirical studies (Fogg et al. 1965; F’ogg
1966). Excretion increased during periods
of nutrient limitation, as in stationary phase
cultures ( Guillard and Wangersky 1958;
Marker 1965). An implication of the results
generated by the model under the above
assumption is that plankton algae in oligo-
et al.
trophic conditions might be expected to excrete a greater percentage of the carbon
they fix than would species in eutrophic
waters. Fogg ct al. (1965) in fact claimed
that “oligotrophic”
species tended to excrete greater percentages of the carbon
fixed than did “eutrophic”
species. The
moderating process appears to be competition for fixed carbon between growth proccsses and losses by excretion.
The phenomenon of phytoplankton
succession is a direct consequence of the limiting assumptions of this model. Species
abundance increases during periods optimal for cell growth, and ccl1 densities decline due to factors that assume prominence
when division rates are slowed by physical
or chemical limitation.
Simple grazing
functions have been coupled with the
model, and some of the consequences are
mentioned by Lehman et al. (in press),
By the same argument used by O’Brien
( 1974) we conjecture that grazing is as important as sinkage in causing population
crashes in nature. IIis thesis is that speciesspecific removal of cells, whether by grazing, death, or sinking, can have major effects on species dominance.
All three
factors can cause exponential declines in
population
numbers when division rates
fall below critical maintenance levels. In
Fig. 8, this critical level was 0.7 divisions
day-l for the diatom, as slower division
rates could not maintain stable populations
in the epilimnion in the face of sinkagc. If
grazing pressure had been superimposed on
the diatom, the critical division rate might
have been greater yet. Reasoning of this
sort argues against the importance of physiological death in most natural plankton
communities.
In fact, Williams
( 1972)
cited the absence of proof of severely nutrient depleted cells in nature as evidence
that other factors, physical or biological,
reduce most populations before extreme nutrient starvation can occur. We retain the
assumption of physiological
death in the
model however, because of indications that
it could be of widespread significance in
some casts (e.g. Jassby and Goldman 1974).
The conceptual treatment outlined here
Phytoplankton
is applicable to all pelagic communities
where competition for light and nutrients
plays a role in phytoplankton
succession.
It is not restricted to the elements C, N, P,
and Si, but is equally applicable to vitamins and tract metals, which Provasoli
( 1969)) Goldman ( 1972)) and Patrick et
al. (1969) advocated as possible detcrminants of algal succession, The main difficulty with modeling these trace substances
is the lack of data regarding uptake parameters and cell quotas. Droop (1968, 1973)
compiled kinetic data for iron and B12, but
they arc restricted to a very few species.
Furthermore, seasonal variations in the concentrations and fluxes of these trace substances in natural waters are less well
studied than are supply rates of other compounds.
By comparison, data for N, P, and Si arc
widely available, but kinetic parameters
are nonetheless wholly lacking for most
freshwater species of ecological interest.
The simulations of succession in Linsley
Pond (Figs. 8 and 9) were conducted using composite species that may bear only
hypothetical similarity to, those populations
actually in the lake. Predicted cell counts
of the chrysophyte, for instance, are aImost
an order of magnitude higher than those
actually achieved by Dinobryon in both
1937 and 1938, though the predicted peaks
coincide well. Recently, uniform labeling
experiments with 83P (Lehman
unpublishcd) have shown that the Dinobryon cell
phosphorus quotas used in the runs, calculatcd from Grim’s (1939) data, are actually
an order of magnitude too low. Correction
for this error makes the predicted and observed cell counts coincide much more
exactly. The ultimate predictive value of
the model thus cannot be properly gauged
until enough parameters are measured to
replace the hypothetical species used here
with defined entities, a goal attainable only
through concerted empirical studies,
It is sometimes argued that detailed models will prove too expensive to provide predictions of general usefulness for real problems, such as those of eutrophication,
On
the contrary, we find that, correctly formu-
dynamics
361
latcd, a model of considerable detail can
predict general patterns precisely without
extcnsivc or costly simulations. Moreover,
the results presented here imply that simple
models that treat all phytoplankton
as a
single unit without regard for species-spccific differences in nutrient uptake, luxury
consumption, and division rates ignore the
implications that species composition holds
for water quality and secondary production
( Hutchinson 1973).
Coefficients such as P,,, and IoBt are
treated as constants here even though they
arc not strictly so in nature. The same is
equally true of half-saturation
“constants”
and other such quantities used by physiologists. Although this treatment is simplistic
when compared to nature, we have tried to
investigate the consequences of conncctcd
hypotheses, which, though they can be improved as adaptations are better understood, arc nonetheless basic to a predictive
model of phytoplankon
dynamics.
If a
model such as this is used not only as a
working research tool, but as a repository
for experimentally defined relations, it can
incorporate each new dependency as it is
quantitatively
described, and the accuracy
of its predictions should increase in proportion to our understanding of the intricacies
of l$custrine systems,
In the ultimate sense, Levins (1966) is
correct in claiming that no model can simultaneously
satisfy all demands for simultancous reality, generality, and precision. Different models, however, possess
different degrees of intrinsic order, order
attained at the expense of energy used to
provide their conceptual and data bases.
WC hope that the predictions of our model
can stimulate the research needed to, reduce
the entropy of these bases. In this way, the
generality of the model will not be sacrificed, but its precision and reality may improve enormously.
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Submitted:
Accepted:
29 May 1974
2 January 1975