Journal of Plankton Research Vol.18 no.l pp.63-85,1996
Modeling phytoplankton growth rates
Jeffrey D.Haney and George A Jackson1
Department of Oceanography, Texas A&M University, College Station, TX
77843, USA
'To whom correspondence should be addressed
Abstract. Mathematical models of planktonic ecosystems use a variety of different formulations to
relate phytoplankton growth rates to environmental conditions. Does the formulation influence the
model result? We have modified the model of Fasham, Ducklow and McKelvie (/. Mar. Res., 34,591639, 1990) to test how its results would respond to changes in algal growth rate formulations. The
original model uses a Monod relationship between nutrient concentration and relative growth rate, and
a multiplicative rule to combine light and nutrient effects. Use of a Droop formulation for algal growth
rate or a threshold (Blackman's law) mechanism to combine light and nutrient limitation produced
significant changes in simulation results. One important effect was to increase zooplankton population
and, as a result, the regenerated production. While there are aesthetic reasons to prefer these alternate
formulations, a more accurate formulation will require more laboratory work on algal physiology. Such
laboratory work should be encouraged as an adjunct to modeling work.
Introduction
Models of planktonic processes become ever more important as oceanographers
work to understand how the different parts of planktonic ecosystems function
together. Such models are being used to study global carbon balances (e.g. Najjar
etal., 1992; Fasham et al., 1993; Sarmiento et al., 1993), distribution and movement
of copepods (e.g. Hofmann, 1988; Hofmann and Ambler, 1988), and deep chlorophyll layer dynamics (Varela et al., 1992). Each new model seems to use different
formulations to describe interactions between biological variables. The large differences in model formulations and the paucity of discussion about the implications of the formulations are surprising.
Steele and Henderson (1992) have studied model formulations in a simple
nitrogen-phytoplankton-zooplankton (NPZ) system that focused on zooplankton interactions, primarily grazing. They examined how grazing of zooplankton on
phytoplankton affects nutrient, phytoplankton and zooplankton populations.
They noted that the results are very sensitive to the choice of zooplankton grazing
formulation and of parameters used in it, and need to be studied further.
The most commonly used relationship between phytoplankton growth concentration originates with the works of Caperon (1967) and of Dugdale (1967), which
used Michaelis-Menten kinetics to describe nitrogen uptake rate by algal cells as a
function of ambient nutrient concentration. This hyperbolic relationship has been
used to justify a Monod relationship between specific growth rate and nutrient
concentration (e.g. Dugdale, 1967). In an alternative approach, Droop (1973) has
argued that growth actually responds to the size of an internal nitrogen pool (cell
quota) rather than directly to the external nutrient concentration. The cell quota
is, in turn, determined by nutrient uptake rates (given by Michaelis-Menten
kinetics) and cell division. Burmaster (1979) has shown that these two formulations
© Oxford University Press
63
J.D.Haney and G.AJactson
are equivalent under steady-state conditions. Both formulations have been used in
recent phytoplankton growth models.
Nitrogen is the nutrient most often considered to limit phytoplankton growth
rate and, hence, used as model currency. Its use to control phytoplankton nutrient
uptake/growth in models is complicated by the fact that seawater has at least two
important chemical forms of nitrogen, nitrate and ammonia, which act differently.
Nitrate is usually considered to be added to the euphotic zone by physical movement of subeuphotic zone water up into the light; ammonia is usually considered to
be released by animals within the euphotic zone (e.g. Dugdale and Goering, 1967).
The two forms are referred to as new and regenerated nitrogen. Ammonia is frequently assumed to be taken up preferentially by algal cells, both by being more
rapidly taken up at equal concentrations of nitrate and ammonia, and by suppressing nitrate uptake. Computer simulations use a variety of formulations and constants to express the suppression of nitrate uptake. Walsh (1975) used a
multiplicative factor to suppress nitrate uptake in the presence of ammonia.
Wroblewski (1977) used the data of Walsh and Dugdale (1972) to develop a different inhibition expression.
Light limitation of algal growth has been an extensively studied physiological
process (e.g. Platt etal., 1977). Although irradiance and resulting photosynthetic
rates vary through the water column, photosynthesis in models is usually averaged
through the mixed layer.
The problem of joining expressions for light- and nutrient-limited algal growth
rates has resulted in quite different solutions. The difficulty arises when describing
algal growth rate under conditions where either light irradiance or nutrient concentration could limit it. Traditional approaches start with the calculation of specific growth rates relative to maximum rates for light limitation and for nutrient
limitation. One approach then uses the smaller of the fractional growth rates as the
appropriate one for the model. This is Blackman's 'law of the minimum' (threshold) approach. A second approach multiplies the two fractional growth rates to
generate a combined fractional rate. In either case, there is a single combined value
for the relative specific growth rate.
The different choices that have been made to express the relationships between
algal growth rates and environmental conditions may or may not have consequences for simulation results. A problem with interpreting published modeling
studies is that it is difficult to know the extent to which the environmental nature of
a given situation determines the results and the extent to which the details of the
model implementations do.
Fasham et al. (1990) published a model (hereafter referred to as FDM) that they
developed to describe the euphotic zone ecosystem in the Sargasso Sea, near
Bermuda. The model is a fairly simple description of a planktonic ecosystem in
which temporal changes in the thermocline depth affect the ecosystem. It is a welldocumented and well-thought-out model. The authors compared its results with
field observations. This model forms the basis for recent global models of carbon
cycling (e.g. Fasham et al., 1993; Sarmiento et al., 1993).
We have used the FDM model modified with different algal growth formulations to test their effects on simulation results. Our goal has been to determine
64
Modeling pbytoplankton growth rates
which formulations have minimal effect on the results and which have profound
effects.
Model description
Base model
The FDM model follows the evolution of average mixed-layer concentrations of
seven nitrogen compartments, including phytoplankton (P), zooplankton (Z),
bacteria (B), nitrate (Na), ammonia (Nr), dissolved organic nitrogen (jVd) and
detritus (D). Unless otherwise noted, we have used the same expressions used in
FDM for the non-phytoplankton compartments and will not discuss them further.
The readers should consult their paper for further details.
Changes for the vertically averaged concentrations of phytoplankton and
nitrate in the mixed layer in FDM are given by:
where crn is the phytoplankton specific growth rate for nitrate uptake, G, is the rate
of phytoplankton loss to grazing, u,, is the phytoplankton specific mortality rate, No
is the subsurface nitrate concentration, •>, is the fraction of phytoplankton growth
excreted as DON, M is the mixed layer depth, h* is the rate of increase of M when
the mixed layer is getting deeper, 0 when it is not, and m is a mixing velocity. Equation parameters and values are listed in Table I. Much of the uncertainty that we
will address concerns the model formulation used to calculate the algal specific
growth rate a.
Nutrient uptake
Almost all formulations of nutrient uptake rate by cells as a function of nutrient
concentration N use a hyperbolic relationship of the form N/(K + N), where K is
the half-saturation constant.
Ammonia uptake inhibition of nitrate uptake has been expressed by Wroblewski (1977) as an exponential reduction factor e"**r multiplying the nitrate uptake
term. He used a value ofty= 1.5 (J.M"1. Hofmann and Ambler (1988) used the factor
with t|i = 5.59 JJLM-'. Walsh (1975) developed a different inhibition factor of nitrate
uptake that is essentially a two-term Taylor series approximation to the other,
(1 - tyN,) - e*"; with a much smaller v|/ (= 0.02 u,M"').
FDM used Michaelis-Menten kinetics to describe phytoplankton uptake of
nitrate and ammonia. Nitrate and ammonia uptake rates were summed to calculate the total nutrient uptake rate. Ammonia inhibition of nitrate uptake was
expressed using Wroblewski's (1977) inhibition factor:
65
J.D.Hanej and G.A Jackson
Table I. Notation. The first value shown is the standard FDM value. Note that u.M is equivalent to mmol
Symbol
Meaning
Units
Values
B
C
(C/N)
U.M-N
mol
mol/mol
(iM-N d a y '
21
P
Bactenal concentration
Carbon content of algal cell
Maximum C/N composition of algae
Phytoplankton loss to zooplankton grazing
Rate of mixed-layer depth increase
Relative light-controlled specific growth rate
Half-saturation concentration for nitrate
uptake and growth
Half-saturation concentration for ammonia
uptake and growth
Half-saturation concentration for nitrate
uptake in Droop formulation
Half-saturation concentration for ammonia
uptake in Droop formulation
Minimum cell quota
Eddy diffusion mixing rate
Mixed-layer depth
Dissolved organic N concentration
Nitrate (new N) concentration
Subsurface nitrate concentration
Ammonia (regenerated N) concentration
Phytoplankton concentration
Q. + Q.
Relative nitrate uptake rate
Relative ammonia uptake rate
Nutrient content of cell (cell quota)
Zooplankton concentration
Fraction of phytoplankton growth excreted as
DON
Specific growth rate
Maximum specific growth rate
Droop model specific growth rate
Droop model maximum specific growth rate
Phytoplankton specific mortality rate
Cell concentration
Nutrient uptake rate
P.
Maximum nutrient uptake rate
a
Phytoplankton average daily specific growth
rate
Phytoplankton average daily specific growth
rate for nitrate uptake
Nitrate inhibition constant
c.
hJ
Kn
K
KD
Ko
m
M
jVd
N
No
Nr
P
Q
Qo
Q,
Qat
z7.
M-I
a.
N.
a-r^
•
e~*Nr
m day'
U.M-N
0.5
JJLM-N
0.5,0.05
M.M-N
2.625
u,M-N
U.M-N cell'
m day-'
m
u.M-N
M.M-N
M.M-N
^M-N
H.M-N
u.mol-N cell"'
U.M-N
2.625
0.24 x 10^
0.1
2,5
0.05
day-'
day'
2.9
day-'
day-'
3.58
day '
0.045,0.54
cell 1'
u.mol-N cell '
day'
jimol-N cell 1
day'
3.6 x 10-*
day-'
day'
HM-'
1.5,5.59,0.02
(3)
(4)
(5)
66
Modeling phytoplankton growth rates
FDM used the same values for half-saturation constants for nitrate and
ammonia uptake (0.5 u.M). Ammonia uptake half-saturation constants can be
much lower than those for nitrate (e.g. McCarthy and Goldman, 1979).
We have tested the effect of using different expressions for the Qn and Q, in the
model calculations. We have tested both the effect of a different ammonia inhibition factor by substituting the factor (1 - \\>Nr) for e-*"' in equation (3) and the
effect of using different \\i values found in the literature.
We also investigated another formulation of ammonia inhibition by using the
Michaelis-Menten inhibition kinetics to combine both the nitrate and ammonia
substrates in a form used to express competitive inhibition of a single enzyme
(Neame and Richards, 1972; O'Neill etal., 1989):
where N'0 = NJKB and N', = NJ/Kr. This was the formulation recommended by
Cullen etal. (1993).
Nutrient-growth model
FDM used a Monod formulation for the algal specific growth as a function of nutrient uptake:
M- = VJQ
(8)
where \i.m is the maximum algal specific growth rate.
Droop (1973) suggested that algal growth rate is related to the internal nutrient
cell content, known as the cell quota Q^:
m
where Ko is the minimum cell quota and (i.nU3 is the maximum specific growth rate
for the Droop formulation, different than |xm. The cell quota, in turn, depends on
the cell division rate and on the nutrient uptake rate per cell p:
where pm is the maximum uptake rate per cell and Q is, as before, the fraction of
maximum nutrient uptake rate (not the cell quota).
The concentration of nitrogen in phytoplankton is the product of algal cell number concentration <}> and cell quota:
67
J.D.Haney and G.A Jackson
P = ^Qat
(11)
Changes in the phytoplankton nitrogen concentration are calculated using
(12)
Changes in nitrate and ammonia concentrations must be similarly altered.
Equations (11) and (12) can be used to calculate changes in Q^.
dP
dt
dt
^ dt
^
dt
dt
n
dt
v
oQ
(14)
This use of the cell-quota approach requires additional parameters: KQ, pm, and
half-saturation constants for nutrient uptake, KDjl and KDj, that can differ from
those for growth. For the comparison of the Droop and FDM models, we required
that Kr = Kn and KDj = KDjl ** KD. We derived the Droop parameters by setting the
Droop and Monod formulations equal at steady state (e.g. Burmaster, 1979):
where C is the carbon content of the cell and (CIN)^ is the maximum value of algal
C:N composition. We set (C/N)m = 21 after Tett and Droop (1988). We calculated
C (mol) as 1.67 x lO^cm)" 8 (Mullin et ai, 1966), assuming r = 5 x l ( H c m ( 5 jim).
The maximum specific growth rate ( f i ^ ) in the Droop equation is the specific
growth rate at an infinite cell quota. The parameter ^ in the Monod equation is
the maximum growth rate at infinite external nutrient concentration and is related
toj
68
Modeling phytoplankton growth rates
PmM-n,
(17)
Pm ~
Light limitation
Evans and Parslow (1985) derived an analytic solution for algal specific growth
rate averaged over the mixed layer for a simplified light profile as a function of
daylength and peak surface irradiance. It was based on the relationship between
photosynthesis and irradiance developed by Smith (1936), and recommended by
Jassby and Platt (1976). Daytime irradiance is described by triangular function and
does not explicitly consider cloudiness (Evans and Parslow, 1985). FDM used the
Evans and Parslow result.
The algal specific growth rate under light limitation can be expressed as:
V = »mJ(t,lMP)
(18)
where J is the relative growth rate as a function of peak daily surface irradiance /,
day of the year t, mixed-layer depth M and phytoplankton concentration P.
Joint nutrient-light limitation
FDM expressed the algal specific growth rate in the presence of both light and
nutrient control as the product of the two relative rates:
a=\K*JQ
(19)
An alternative multiplicative formulation to express joint light-nutrient
limitation using the Droop equation is:
(20)
The interaction of light and nutrient has also been represented as the smaller of
the two relative growth rates (e.g. Walsh, 1975; Tett etal., 1986; Varelae/a/., 1992):
<r = ,xmmin(/,Q)
(21)
This is known as the 'threshold' hypothesis or as the 'law of the minimum' after
Blackman (1905).
Mixed-layer model
Deepening of the mixed layer in FDM decreases phytoplankton concentrations by
diluting mixed-layer phytoplankton with phytoplankton-free water, but it does not
decrease the total, vertically integrated phytoplankton population. Shoaling in
mixed-layer depth has the opposite effect of decreasing the total, integrated
phytoplankton population, but not directly affecting the concentrations.
69
J.D.Haney and G.AJackson
Mathematically, this is calculated using h* = max(—— ,0) (Evans and Parslow,
d
1985).
'
Unlike the phytoplankton, the total zooplankton population, summed through
the mixed layer, does not change with the mixed-layer depth. However, zooplankton concentration does increase or decrease as animals are spread out or concentrated by the deepening or shoaling of the mixed layer.
FDM calculated the mixed-layer depth, M, as a function of time by fitting
straight-line segments to seasonal mixed-layer depths measured over a 3 year
period off Bermuda. We tried two approximations to the mixed-layer depth as a
function of time: their use of a series of straight-line approximations between data
points, and a smooth spline fit to the data points (Figure 1A).
Sub-mixed layer water is the source for nitrate when the mixed layer deepens.
FDM used a value for its concentration No = 2 JJLM. We also examined the case
where JV0 = 5 \LM.
Other interactions
The FDM model includes interactions with dissolved organic nitrogen (DON),
detritus, bacteria and zooplankton. Because our interest was in the effect of the
phytoplankton model formulation, we have not modified the FDM formulations
for these other compartments.
FDM examined the effect of two different settling rates for detritus. We have
standardized on one: 10 m day-1. In conjunction with this settling rate, FDM used a
phytoplankton mortality rate ji, = 0.045 day 1 .
FDM included DON loss as a factor in the phytoplankton population dynamic.
They set it at a small constant fraction of primary production -y, = 0.05.
Numerical solution
The model was solved numerically by coding in FORTRAN, using an AdamsBashford differential equation subroutine (DDEABM from the Slatec subroutine
library). Each variation was run for three simulated years, with only the results
from the last year presented here. The first two simulated years were used to initialize the model. Thefinalyear's results for simulations showed no appreciable difference with either 2 or 4 years of initialization.
Results
The base model
Phytoplankton in the basic FDM model has two concentration peaks during an
annual cycle (Figure 2A). The spring bloom, driven by nutrients entrained in submixed layer water, accounts for thefirstpeak (t ~ 100 days) with a phytoplankton
concentration of 0.42 u,M. This is followed by a rapid decline in phytoplankton as
summer shoaling of the mixed layer cuts nitrate input and increases zooplankton
grazing pressure. The second phytoplankton peak (0.48 n-M) is associated with the
start of autumn mixed-layer deepening (/ ~ 280 days). Phytoplankton growth is
low during the winter because of low winter light input,
70
Modeling phytoplankton growth rates
0
50
100
150
200
250
300
.150
Time (day of year)
Fig. 1. Representation of mixed-layer depth and its effects. (A) Mixed-layer depth through time using
straight-line segments and a cubic spline curve to interpolate between observations. (B) Phytoplankton, zooplankton and nitrate concentrations (P, Z, JV.) for different interpolation schemes between
measured mixed-layer depths. Straight line segments are used in FDM. Cases are differentiated by line
thickness, as indicated on the figure.
The peak nitrate concentration, 0.42 u,M, is also associated with the winter
mixed-layer deepening, although earlier (at —100 days) than that of the phytoplankton. Nitrate and phytoplankton concentrations show bumps at 230 and 260
days that are associated with how mixed-layer depth is calculated (see below). The
peak ammonia concentration is much smaller (0.05 u,M), occurring during the
winter mixed-layer deepening.
The net rate of phytoplankton change is the sum of all the processes that add and
subtract from the phytoplankton pool (Figure 2B). Growth rate (uP) shows strong
pulses that are associated with changes in the rate of mixed-layer deepening h+ that
are partially mirrored in the rate of loss to mixed-layer deepening and mixing. By
far the largest phytoplankton loss is to simply mortality u,,P. Dead phytoplankton
contribute to the detritus which could, in turn, settle from the euphotic zone or be
consumed by zooplankton. Loss to zooplankton is always small, having a primary
peak when the zooplankton is concentrated by mixed-layer shoaling and a secondary peak at the time of maximum phytoplankton concentration at the end
71
J.D.Haney and G.AJackson
0.5
50
100
150 200 250 300
350
Time (day of year)
Fig. 2. Simulation results for the standard multiplicative-Monod (FDM) formulation. Time is in days
from 1 January. (A) Concentrations of phytoplankton, zooplankton, nitrate and ammonia (P, Z, N,
Nt) through an annual cycle. (B) Terms composing — through time. Shown are the net rate of change
d(
(—), algal growth corrected for DON leakage [(1 - yt)<jP], algal death (ji,/1), losses to mixed-layer
At
changes and mixing (MLD), and zooplankton grazing loss (C,). Note that some of the terms are
subtracted rather than added to calculate —. Algal death is by far the largest loss term, nearly equal
df
to the growth term. Losses to zooplankton grazing are small at all times.
of summer. The relatively slow algal specific growth rate can be inferred from the
fact that phytoplankton specific growth rate, which has a maximum of \Ln = 2.9
day 1 , actually has a value almost equal to the phytoplankton specific mortality rate
u., = 0.045 day-1.
Mixed-layer model
Using straight-line segments to approximate the mixed layer (Figure 1A) introduces transients that using a smoothed curve does not. There are shght changes in
the timing and size of phytoplankton concentrations as a result (Figure IB).
72
Modeling phytoplankton growth rates
Ammonia inhibition of nitrate uptake
Ammonia inhibition of nitrate uptake is increased by larger i|; values, resulting in
higher nitrate concentrations, and delayed but higher peak phytoplankton concentrations in the spring bloom (Figure 3). The maximum spring nitrate concentration
is 0.58 u,M for »»| = 5.59 u,M~' and 0.38 n-M for i|i = 0.02 n-M'1, compared to 0.44 fiM
for the FDM \\i = 1.5 JJLM"1 case. The larger \\i results in a 30% increase in maximum
spring nitrate concentration relative to the FDM case. The associated peak spring
phytoplankton concentrations are 0.41 and 0.46 JAM compared to 0.42 ^M for
FDM. There is virtually no difference in either nitrate or phytoplankton concentrations during summer (t = 180-270 days), when there is essentially no nitrate or
ammonia in the water.
The Michaelis-Menten inhibition formulation of ammonia inhibition gives
results very similar to those for the FDM model (not shown). The slightly higher
nitrate and ammonia concentrations during the winter help to create a slightly
higher phytoplankton maximum during the spring bloom.
Half-saturation constants for ammonia uptake
A smaller half-saturation constant for ammonia uptake [K, = 0.05 u,M; equation
(4)] has little effect on simulation results (Figure 4). Ammonia concentrations are
lower and uptake greater than with the standard K, = 0.5 u.M. The resulting phytoplankton and nutrient concentrations are similar to those for the case of lower
ammonia inhibition of nitrate uptake (Figure 3), a greater spring phytoplankton
concentration occurring earlier. The second half of the year shows essentially no
differences between the two formulations.
Nutrient-growth model
Nutrient and phytoplankton concentrations are very similar during summer and
early fall (/ = 180-300 days), but very different during winter and spring for the
Droop and Monod (FDM) growth formulations (Figure 5A). Maximum winter
nitrate concentrations, 0.43 JJLM for FDM, are almost 10 times greater than the 0.05
H,M maximum for the Droop case. The cell quota for the Droop model ranged from
2.5 x 10-7 to 8 x 10-7 u,mol-N cell1 (not shown).
The integrated net primary production (NPP) is higher in the Droop model during almost all of the year (Figure 5B). The higher phytoplankton concentrations
for the Droop model in the winter contribute to an earlier spring bloom peak, with
the maximum primary production being greater than for FDM. Both formulations
show strong stepping in NPP and new net primary production (NNPP) during the
latter half of the year that is associated with the sudden changes in the rate of
mixed-layer depth deepening (/i+).
Light-nutrient growth model
There are major differences, some subtle, some striking, between the results from
using the multiplicative light-nutrient interactions (FDM) and those using the
73
J.D.Haney and G.A Jackson
1
1
0.5 » /
*/
\
0.4
/
/
p
\
\
—
Z - Nn
Nr
V
\
1)1=15
¥=5 59
^.02 ^T
A
J \ ''
c
o 0.3
/
.—^_X
^b
0.2
4>
o
c
o
U
-1
j
0.1
v,
>.•/*- + /.•
"-•••••v---
0
H
0.05
1
H
dP/dt
(1-TlXJP
1——i—i—i—
¥=u
- -
0.04
B
MLD
GiP
0.03
/ >
/ \ !
u 0.02
te of chan
—
yfc5 59
i(»002
i
^
.•-•-.
•v>...
0.01
0
-
^
O.01
0
50
100
150
200
250
300
350
Time (day of year)
Fig. 3. Effect of different ammonia inhibition formulations. (A, B) as in Figure 2. Shown are the results
for the standard FDM (if = 1.5 M-M"')- and for higher and lower values of I\I of 5.59 and 0.02 fj.M"'.
Differences are greatest at the beginning of the year when the mixed layer is deepening. By / = 210 days,
there is little difference in the results for the different values of I|I.
threshold formulation. For the threshold formulation, the spring phytoplankton
bloom occurs earlier during the time of maximum mixed-layer thickness and lasts
longer, but has a slightly smaller maximum concentration (Figure 6A). Phytoplankton concentrations fall rapidly when the mixed layer shoals. Nitrate and
ammonia concentrations are small all year, ~0.01 u-M, compared to the winter
ammonia and nitrate concentrations of ~0.05 and 0.4 jtM in the multiplicative
formulation. Zooplankton concentrations during the winter are ~0.1 jtM,
approximately 10 times those for the multiplicative model. The maximum zooplankton concentration is ~0.18 u.M, about three times that of the multiplicative
model.
Values of relative light and nutrient phytoplankton specific growth rates / and Q
help explain the difference (Figure 6D). In FDM, phytoplankton growth is light
limited during the winter, with Q ~ 0.5 and / ~ 0.05. Cell division is nutrient limited in the shallow mixed layer of summer, with Q ~ 0.5 and / ~ 0.05. The product
QJ determines the actual specific growth rate and ranges from —0.02 to 0.025. In
contrast, the threshold formulation has Q almost always less than J, although the
74
Modeling phytoplankton growth rates
p
—
—
0.5
Kr=*05
N,
0.4
\
I
\
\
1
;,
^
0.2
I
/
\
rm
X
/
/
\
o
c
/
/
V
\
o
O
;
V '
I
\
0.3
1)
\
/
"</ ^ f
o
A
KH=0J
N.--
X
\X r
0.1
.V,
0
1—
/-^
\
1—
1
0.05
H
B
dPAh'
0
Ml?* *"
0.04
-a
-
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G|P
0.03 '•
3.
\^\
1
<u
Vy
ed
.
.
.
-
•
•
.
1
.-
-
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JC
U
-
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" >•
0.01
1
.
•
mi-
•
•
-
.
J
•
X
-
-0.01
50
•
- f\A'
&''''
Rat
.
100
150 200
250
300 350
Time (day of year)
Fig. 4 Effect of half-saturation constant for ammonia uptake (K,). (A, B) as in Figure 2. The standard
FDM uses K, - 0.5 JAM; the variant uses K, = 0.05 M-M. The lower K, results in a higher winter growth
rate [(1 - y,)uP] and grazing rate (G,P). The overall rate of change (—) shows little difference
d(
except for a slightly earlier spring peak.
two have close to the same values during the winter. The values of J are similar for
the two formulations. The difference is in the value of Q, which ranges only from
0.01 to 0.04 for the threshold case and is always less than Q for the multiplicative
case. As a result of the lower nitrate concentrations, the threshold case is nutrient
limited essentially all year. Despite the lower nutrient concentrations, the phytoplankton division rate is usually greater for the threshold case.
The effect of continual nutrient limitation in the threshold case can be seen in
the high growth rates aP during the winter mixed-layer deepening (Figure 6B).
Furthermore, sharp changes in the mixed-layer depth representation cause an
even greater saw-toothed pattern in aP. Phytoplankton dying, JJL,/), is still a major
term removing phytoplankton, although zooplankton grazing is nearly comparable during the winter and greater during the spring zooplankton pulse.
Net primary production is significantly greater during the winter for the threshold case (Figure 6C). Because NPP is the product of phytoplankton growth per
unit volume uP and the mixed-layer depth (A/), low rates of concentration during
75
J.D.Haney and G.A Jackson
'
50
100
150
200
250
300
350
50
100
150
200
250
300
o
350
Tune (day of year)
Fig. 5. Effect of using Droop and Monod (FDM) nutrient-growth models. (A) Effects on nutrient pools
(P, Z, JVW N,). (B) Effects on terms of phytoplankton rate of change and net change for different lightnutrient interactions through time for threshold formulation. (C) Effects on rates of vertically integrated primary productivity measures, including net primary production (NPP), net new primary production (NNPP) and net regenerated primary production (NRPP). (D) Effect on relative nutrient (Q),
light (/) and joint [JQ for FDM, min(<r, (?) for threshold] through time. The standard Monod case is not
shown in (C), but is in Figure 2B.
the winter are balanced by the large values of M. Most of the difference between
the threshold and the multiplicative cases is in the greater importance of production resulting from regenerated production (NRPP) in the threshold case.
Subsurface nitrate concentration
Changing the subsurface nitrate concentration, No, has a great effect on the FDM
formulation. Using No = 5 jiM rather than 2 u.M resulted in peak nitrate concentrations during the spring bloom of 2.2 |i,M, ~5 times that of the standard case
(Figure 7A). Peak ammonia and zooplankton concentrations are also much
greater, ~20 and 30 times higher. Maximum phytoplankton concentrations are
about the same, although there are pronounced oscillations after the peaks. All
compartments show oscillations after the spring bloom peak. These are intensified
for higher values of No (not shown). Phytoplankton population regulation changes
from predominantly natural mortality to predominantly zooplankton grazing with
the increase in Ng, although natural mortality and zooplankton grazing are comparable during the pre-spring bloom winter deepening of the mixed layer.
The oscillations in phytoplankton concentrations, as well as those of other compartment, are the most dramatic effect of higher N^ Excepting these large oscillations, it is surprising how small the changes in phytoplankton concentrations in
76
Modeling phytoplankton growth rates
0.05
dPAh
-
Thresh
U-YIXJP •
05
0.04
MLD
G|P
0.03
OJ
0.02
§0.2
00
0.01 "o
C
o
3.
-0.01
~
6
0J
04
a
-5 3
§
03
>
fi 2
02 g
•a
0
1,
0
01
50
100
150
200
250
300
350
50
100
150
200
250
300
350
0
Time (day of year)
Fig. 6. Effect of formulation of nutrient-light interactions through time (A-C) as in Figure 5. Shown
are the results for multiplicative (FDM) and threshold formulations of the Monod growth model. (D)
Equivalent relative growth rates for the two formulations.
time are. There is a slight increase in pre-spring bloom concentrations with larger
No, but remarkably little else. A decrease in No to 1 ^.M does dramatically decrease
P to about half the values when No = 2JJLM.
The results for the threshold formulation with the higher value of No are similar
for the multiplicative formulation (Figure 7B). Winter nutrient concentrations are
slightly lower, 1.9 versus 2.2 |xM, and peak zooplankton concentrations during the
spring are slightly higher, 2.25 versus 1.6 (xM. The Droop formulation shows similar results to those of the threshold formulation at this higher value of No, as it did
for the case of No = 2 u,M. However, the Droop formulation shows little oscillation
in phytoplankton concentration.
Discussion
We wanted to know whether the formulation of phytoplankton growth rate
relationships affects simulation results. The answer is clearly yes. For this particular system, the biggest difference is during the deep mixing of early winter. There
are two different types of model response. The first has phytoplankton nutrient
uptake and growth rates that are too low to effectively deplete nutrient concentrations during the winter. The lower nutrient usage formulations include those
with higher ammonia suppression of nitrate uptake and the multiplicative growth
and light interaction formulation. The second response has higher uptake and
growth rates that result in a depletion of nutrients, larger phytoplankton concentrations and a smaller decrease in winter zooplankton populations. The extra winter plant growth and higher phytoplankton concentrations resulting from more
77
J.D.Haney and G.AJackson
2
0
50
100
150
200
250
300
350
Time (day of year)
Fig. 7. Effect of higher sub-mixed layer nitrate concentration, No = 5 p.M. The FDM standard value of
No = 2 JJ.M. (A) Effect on concentrations of phytoplankton, zooplankton, nitrate and ammonia (P, Z,
N* ^r) through time for the standard FDM model. (B) Effect on phytoplankton and zooplankton
concentrations for higher sub-mixed layer nitrate concentration for the FDM, threshold and Droop
formulations.
efficient nutrient usage keep the zooplankton population from having as large a
winter decrease and allow it to consume more of the plant production during the
spring. The threshold growth model and the Droop formulation are examples of
models with this effect.
That the ammonia suppression of nitrate uptake results in low winter uptake is
not surprising, but it might be surprising that the multiplicative formulation has a
similar effect. To see why, consider a case where light and nutrient concentration
would each limit the algal specific growth rate to 0.25 of maximum. With the multiplicative approach, the effective algal specific growth rate would be 0.0625, a quarter of the rate of the threshold approach. As noted earlier, the relative specific
growth rate for the non-limiting factor in the multiplicative case is always less than
one and frequently less than 0.5 (Figure 6D). As a result, the algal specific growth
rate is always significantly less than the rate of the lower, controlling process, i.e.
the multiplicative formulation yields specific growth rates for phytoplankton significantly lower than those determined by the more limiting factor.
78
Modeling pbytoptankton growth rates
Despite the formulation of growth rate as a multiplicative function of light and
internal cell quota, the Droop formulation has an effect similar to that of the
threshold formulation. Adding the threshold to the Droop formulation results in
little additional change (not shown). The decoupling of nutrient uptake and
growth [equations (9-14)] allows the cell quota to become large enough during the
winter deepening of the mixed layer so that the relative specific growth rate for
nutrients becomes almost one, with a similar effect to that of the threshold formulation. Probably more important is the fact that the maximum specific rate of nutrient uptake is greater than that for growth when the cell quota is small. This
decoupling of nutrient and growth rates is in the spirit of the ideas advanced by
Goldman and McCarthy (1978), Morel (1987) and McCarthy (1981).
Of all their parameters, FDM found that their model results are most sensitive to
the choice of subsurface mixed-layer nitrate concentration. We find that this
extreme sensitivity results in noticeable transients in the phytoplankton/nutrient/
concentrations when the depth of the thermocline is approximated using straightline segments to interpolate between observations. Similar transients can be seen
in the FDM results. When we use a smooth spline fit to approximate the mixedlayer depth, the oscillations disappear.
Platt et al. (1989), Goldman (1993) and Goldman et al. (1992) have argued that
storms can be important periodic nutrient sources, particularly for large diatoms in
the Sargasso Sea. Such pulsing of nutrient input would also have a large effect on
this model and suggests that it might be important to include in future models.
A comparison of the various annual productivity values highlights similarities
and differences for the different formulations (Table II). The annual new production is essentially unchanged in all of the simulations with No = 2 \iM because
essentially all of the nitrate that arrives in the surface mixed layer is taken up by the
phytoplankton. There is a small, 5%, increase for the threshold Monod formulation. The annual regenerated productions of the threshold and Droop formulations are 2.6 and 2.3 times that of the standard case. The resulting /-ratios are
0.58, 0.60 and 0.77. Thus, the enhanced zooplankton population associated with
the higher winter growth of the threshold and Droop formulations gives rise to
greater nitrogen recycling and greater annual total production.
FDM were interested in understanding the seasonal cycle in the plankton off
Bermuda. They used historical data both to 'tune' their model and to test it. They
observed the effects of varying algal mortality rate u.,, subsurface nitrate concentration N& and detrital sinking rate V on the annual total net and new primary
production. They ultimately chose tofixNo = 2 u.M and to choose the value JJL, that
best fit the productivity data for a given value of V.
It could be argued that a better comparison of the original FDM model with any
of the variants that we have discussed would involve the same parameter selection
procedure used by FDM. This would involve the selection of a specific death rate
that gave similar values for the annual total net and new production rates. Contour
plots of the values of annual total production and/-ratios show that this technique
can be used for the threshold and Droop formulations only over a more limited
range of/-ratios (Figure 8). Such a procedure yields a value for p., = 0.054 day-',
18% higher than the previous value. However, the range of/-ratios that can befitis
79
J.D.Haney and G.A Jackson
Table II. Integrated annual phytoplankton production. New production is production resulting from
nitrate uptake; regenerated production is that resulting from ammonia uptake; total production is the
sum of new and regenerated production, / i s the ratio of new to total production
Case
Production
(mol-N nr 2 year 1 )
/
Total
% change
/
% change
Total
New
Regenerated
Multiplicative Monod
Standard
Smoothed mix layer
<\i = 5.59 M.M1
I|I = 0.02 M-M-'
K, = 0.05
No = 5 (JIM
0.52
0.52
0.49
0.53
0-57
1.61
0.40
0.40
0.39
0.40
0.40
0.80
0.12
0.12
0.10
0.13
0.17
0.81
0.77
0.77
0.80
0.76
0.71
0.49
_
_
0
-5.8
1.9
9.6
209.6
0
3.9
-1.3
-7.8
-36.4
Multiplicative Droop
Standard
/V 0 =5(iM
0.70
1.89
0.41
0.85
0.28
1.04
0.60
0.45
34.6
263.5
-22.1
^U.6
Threshold Monod
Standard
u., = 0.054 day-'
iV 0 =5nM
0.72
0.57
1.77
0.42
0.42
0.85
0.31
0.15
0.92
0.58
0.73
0.48
38.5
9.6
240.4
-24.7
-5.2
-37.7
smaller for the Droop and threshold formulations. For these, the maximum value
of/is 0.78, compared to 0.88 for the FDM case, over the region of interest. Note
that the fact that essentially all of the nitrate is consumed fixed the annual NNPP.
Choosing parameters to match NPP then also determines NRPP and the/-ratio.
FDM noted that the results suggested that algal death was the dominant loss for
the phytoplankton growth, although they also noted that the inclusion of sizebased algal-grazer interactions would alter the model results. It is disconcerting
that the predominant loss of phytoplankton is to solitary death, whose rate
depends on a parameter used to tune the model. The importance of this term is
partially the result of the low specific growth rates calculated using the multiplicative growth formulation and the low zooplankton populations. While Walsh (1983)
has argued that such death should be important in the sea, there is little experimental evidence that this is so. However, p., should be a measurable quantity. Using a
threshold growth formulation with the same specific death rate (0.045 day 1 )
increases the average specific algal growth rate and makes the zooplankton grazing
a comparable or, occasionally, a greater loss term for the algae. Fasham (1994) has
derived a new expression for algal death which decreases its role as the major loss
of phytoplankton.
It is intriguing that model zooplankton dynamics are so sensitive to changes
in algal growth formulations, that relatively small changes in phytoplankton
concentrations, —30%, can cause such large changes in winter zooplankton
concentrations. Although the zooplankton directly respond only to phytoplankton
concentration, they are able to effectively respond to large changes in phytoplankton growth rates and productivity that show only as small changes in phytoplankton concentration.
80
Modeling phytoplankton growth rates
/-ratio
Annual total production
1.4
1 6
1.8
2.0
2.2
1.2
1.4
1.6
1.8
2.0
2.2
Sub-mixed layer nitrate concentration (uM)
Fig. 8. Values of annual total production (A, C, E) and/-ratios (B, D, F) for standard Monod-multiplicative (A, B), threshold-Monod (C, D) and Droop-multiplicative (E, F) formulations as functions of
/Voand(i.,.
In a similar study of the zooplankton mortality term formulation, Steele and
Henderson (1992) have shown that the zooplankton/phytoplankton dynamics are
influenced by the form of the zooplankton mortality term. Changes in the description of zooplankton grazing rates can lead to pronounced predator-prey oscillations in a simple NPZ model. FDM also noted that their results were very sensitive
to the choice of grazing model. Of the terms and constants that we examined,
predator-prey oscillations were most sensitive to the subsurface nitrate concentration (e.g. Figure 7).
While an algal preference for the reduced nitrogen form of ammonia rather than
nitrate has been justified on energetic grounds, Dortch (1990) has argued that the
experimental evidence that this actually occurs is weak. On the contrary, Levasseur et al. (1993) have observed that nitrate can be taken up by diatoms preferentially to ammonia under the low-light conditions where energy should be limiting.
Usage of the ammonia inhibition factor can result in total nitrogen uptake rates
being lower in the presence of the additional nutrient source of ammonia than the
uptake of nitrate alone without ammonia present. That adding a nutrient source to
a nutrient-limited situation can decrease total uptake does not seem reasonable. A
81
J.D.Haney and G.A Jactsod
similar preference for ammonia uptake can be obtained by using a smaller value of
Kt. For these reasons, we suggest that ammonia inhibition be dropped until such
time as there is better experimental evidence to describe it.
The Droop formulation for nutrient-limited growth has several nice properties
that the Monod one does not. Its decoupling of nutrient uptake and growth rate
does agree with experimental observations. Furthermore, the adaptation of algal
nutrient content allows a modicum of organism response to changing environmental conditions. This additional flexibility comes at the cost of an additional variable
and several additional parameters needed to describe the system.
Despite these nice properties, it is not clear the Droop formulation is optimal.
Morrison et al. (1987) examined the effect of algal growth formulation on the ability of a water quality model to predict phytoplankton concentrations in a Canadian
lake and reservoir system. For each formulation, they used an optimization procedure to calculate coefficients that provided the best fit during a 3 year period at
one of three stations. They compared simulation results to observations for the
remaining data set,findingthat storage models such as that of Droop did a poor job
of predicting concentrations. Their models differed significantly from those of
FDM because they imposed grazing losses on the phytoplankton using field
measurements of zooplankton populations rather than let zooplankton populations respond to phytoplankton concentrations. However, they do indicate that
deciding the best phytoplankton growth formulation will require the comparison
of observations and predictions at multiple sites.
The threshold formulation for incorporating both nutrient and light limitation
of algal growth appears to be better than the multiplicative one because the latter
has the algae growing at rates that are substantially less than that expected for the
single most limiting factor.
In either case, there is a more subtle problem in combining photosynthesis and
nutrient uptake which should be manifest in the composition of the cells (Eppley,
1981; Cullen et al., 1993). If a cell is taking nutrients up at a rate equivalent to 0.25
day"1, but photosynthesizing at 0.5 day"1, its carbon content should continue to
increase relative to that of nitrogen, easily to unrealistic levels. In fact, the photosynthesis rate also needs to fall to be in balance with the nutrient uptake. This lack
of balance between photosynthetic and nutrient uptake rates is another sign that
the present formulations for algal growth have been patched together rather than
developed as a unified idea.
O'Neill et al. (1989) examined the ability of different expressions for nutrient
limitation by two factors to fit experimental observations. There are methodological problems with using the same formulation to predict rate of cell division for
algae grown in chemostats and with accumulated harvest yield in terrestrial crops.
Nevertheless, the comparisons are informative. O'Neill et al. found that a Blackman formulation using two Monod relationships did notfitthe data well when both
elements might be limiting, although it worked well when only one element was
limiting. Expressing growth as a product of two Monod relationships, in what they
called a 'Baule model', provided one of the betterfitsto the data, but the constants
derived from fitting all the data were different than those derived by using data
with only one limiting nutrient. The implication for oceanic models is that the con82
Modeling phytoplankton growth rates
stants which would make a multiplicative expression work are not the ones derived
from typical laboratory measurements. Rather, the constants need to be developed from experiments examining interactions between limiting factors.
This paper has avoided making comparisons to experimental observations to
focus on the differences that choices in model formulation can make. Furthermore, the results of Menzel and Ryther (1960) that FDM used to calibrate their
model have been superseded by newer, presumably more accurate, measurements
made as part of the Bermuda time series (Michaels et al., 1994; Siegel et al., 1995).
The newer measurements, made using trace metal-clean techniques, provide estimates of annual primary production twice as high as that of Menzel and Ryther
(1960) (Siegel etal., 1995). Were FDM to have used this data for their model, they
would have made different parameter choices and presumably have done a better
job fitting the new data, just as we would have also. It would not, however, have
changed the fact that choices in model formulation affect the results in ways that
need to be resolved.
When model predictions are compared against field measurements, it is important to remember that there are a range of measurements whose values must be
consistent with model predictions. J.Cullen (personal communication) has noted
that C:Chl data provide a powerful check, both directly, and also because the
assimilation index (g-C g-Chl"1 day 1 ) divided by the C:Chl ratio (g-C g-Chl-1)
should equal the algal specific growth rate (day 1 ). With the specific growth rate
calculated in the model and the assimilation number measured in the field, a value
of C:Chl that is consistent with them can be calculated and compared to the range
of values found in culture studies (e.g. Geider, 1993). Such values need to be
consistent.
The proper formulation of algal growth rate as a function of combined nutrientlight limitation is an experimental problem. There has been extensive work on the
role of nitrogen concentration on growth rates (e.g. Carpenter and Capone, 1983),
and extensive work on the interaction of light and photosynthetic rate (e.g. Platt et
al., 1977), but remarkably little progress in uniting the two. Doing so will require
describing the interaction of cellular carbon and nitrogen concentrations and light
absorbance, as well as understanding the specific growth rate for a given combination of irradiance and nutrient concentration (Cullen etal., 1993). Models such
as that of Laws and Chalup (1990) may help describe some of the nutrient-light
interactions, but are too complicated to be used directly in large ecosystem models.
They may be useful, though, to test the errors associated with computationally
simpler growth models, such as the threshold formulation. Although being able to
construct the relevant interactions will not be easy, it will be important for the
development of models describing planktonic ecosystems without the present
arbitrariness and uncertainty.
As computers become bigger and faster, as society focuses on global change, it is
natural and right that oceanographers try to use our understanding of oceanic ecosystems by building global biological models. The grandeur of such efforts should
not blind us to the continuing need for experimental and theoretical work to refine
the relationships upon which the large models are built.
83
J.D.Haney and G.AJackson
Acknowledgements
We were helped by conversations with P.Eldridge, A.Murray and P.Harrison,
J.Cullen and B.Zakardjian provided useful and helpful comments on the manuscript. This work was supported by Office of Naval Research Contract N00014
87-K0005 and by the DOE Computational Science Graduate Fellowship Program.
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Received on March 8, 1995; accepted on September 22, 1995
85