Download Steady state solutions of an ecosystem mod

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Limnol. Oceanogr., 39(3), 1994, 597-608
0 1994, by the American Society of Limnology and Oceanography, Inc.
Grazing limitation and nutrient limitation in marine ecosystems:
Steady state solutions of an ecosystem model with
multiple food chains
Robert A. Armstrong
Program in Atmospheric
Jersey 08544-07 10
and Oceanic Sciences, Sayre Hall, P.O. Box CN7 10, Princeton
University,
Princeton,
New
Abstract
Here I develop a model with multiple phytoplankton-zooplankton
food chains where each food chain
is based on a different size class of algae. Model parameters were chosen to allow food chains based on
smaller algal size classes to dominate under oligotrophic conditions, with larger size classes being added
sequentially with increased nutrient loading. The model makes use of allometric relations among size
classes to minimize the number of free parameters, facilitating numerical exploration of its steady state
behavior. The model is simple, yet it is complex enough to allow simultaneous predator limitation
of
algal size classes (or species) and nutrient limitation
of total phytoplankton
biomass. This perspective,
not obtainable from models with single phytoplankton
and zooplankton size classes, is applied to the
particular case of elevated nitrate levels in the equatorial Pacific. The model shows clearly that it is possible
for each phytoplankton
size class to be limited by its herbivores, while at the same time micronutrient
(notably iron) dcliciency may limit the number of size classes that can exist in the community and hence
the total phytoplankton
biomass that can be supported.
Empirical evidence and theoretical argument suggest that a common ecosystem structure is one wherein the density of each plant
species is limited by its herbivores while total
plant biomass is limited by the availability
of
space or nutrients (Hairston et al. 1960; Power
1992; Strong 1992). Marine ecosystem models
containing a single “phytoplankton”
variable
and a single “zooplankton”
variable cannot
easily convey this distinction, however, since
the phytoplankton
variable in such models simultaneously represents a single phytoplankton species and the entire phytoplankton
trophic level, confounding
the limitation
of
species’ population sizes with the limitation of
total algal abundance. The inability of models
with single phytoplankton
(P) and single zooplankton (Z) variables (1 P 1Z models) to convey this distinction limits their utility in exploring important empirical questions.
Consider, for example, the discussion by
Frost and Franzen (1992) of simultaneous nutrient limitation
and grazing control of phyAcknowledgments
I thank Steve Carson, Jorge Sarmiento, and Robbie
Toggweiler for helpful suggestions on the manuscript.
This research was funded in part by grant NA26RGO 10201 from the National Oceanic and Atmospheric Administration. Additional support was provided by NASA grant
NAGW-3 137 and NSF grant OCE 90- 12333 to J. L. Sarmiento.
597
toplankton standing stock and growth rates in
the equatorial Pacific. Frost and Franzen rely
on a simple model of pelagic food chain to
observe that nutrient limitation
acting alone
would result in algal standing stocks that are
too high and algal growth rates that are too
low. Then they rely on verbal arguments to
further conclude that limitation
by micronutrients may be necessary to account for growth
rates that seem low by a factor of two (Cullen
et al. 1992) as well as to account for the absence
of large algal species, which are thought to be
more sensitive to low micronutrient levels than
species smaller are (Hudson and Morel 1990;
Morel et al. 1991). This reliance on verbal arguments to account for the absence of large
phytoplankton
size classes underscores the incompleteness of their 1P 1Z model: it would
be more satisfying, and more instructive, if the
model were capable of portraying not only the
simultaneous effects of herbivory and nutrient
limitation
on resident phytoplankton
populations but could also account naturally for the
absence of larger size classes.
Here, I develop a more complete model that
allows the interaction of nutrient control and
grazing control to be examined more directly.
This model is a simplified version of the model
of Moloney and Field (199 1). The model consists of multiple food chains, each based on a
different size class of algae. The parameters of
598
Armstrong
each size class are related to those of other size
classes by allometric relations, imparting a
unified structure to the model and limiting the
number of free parameters. Numerical steady
state solutions of this model show that as nutrient loading is increased, larger algae and
zooplankton
size classes are added sequentially until virtually all limiting nutrient is exhausted or until no larger algal size class can
grow fast enough to invade the existing community. In this context, the problem of high
ambient nitrate levels in the equatorial Pacific
is reduced to the question: What keeps larger
algal size classes from invading? (Miller et al.
199 1). On a broader scale, the model also shows
that while predators may limit the population
sizes of their respective prey, the total biomass
of the system and its productivity
may ultimately be set by the number of food chains
that can be supported at a given nutrient loading (Hairston et al. 1960; Power 1992; Strong
1992).
A model with multiple food chains
In a review of the effects of size on algal
physiology and phytoplankton
ecology, Chisholm (1992, p. 222) noted that the total amount
of chlorophyll in each of several successively
larger size fractions seems to have an upper
limit and that “beyond certain thresholds,
chlorophyll
can only be added by adding a
larger size class of cells.” Raimbault
et al.
(1988), for example, noted that there appears
to be a maximum of roughly 0.8 mg Chl a m-3
in the l-3-pm phytoplankton
size class and
0.7 mg Chl a m-3 in the 3-lo-pm size class.
These size classes are of approximately
equal
width on a logarithmic scale, suggesting roughly equal maximum chlorophyll per logarithmic
increment in size.
This pattern can be produced quite naturally
by models with multiple food chains, where
the steady state phytoplankton
concentration
in each size class is set by the herbivorous
predator on that size class and where parameters are chosen to produce equal phytoplankton biomass in each logarithmic size interval.
Further, to allow food chains based on smaller
phytoplankton to dominate under oligotrophic
conditions, with chains based on larger phytoplankton appearing only at increased nutrient loadings, chains based on small phytoplankton and zooplankton
must be able to
invade at lower nutrient concentrations than
can chains based on larger size classes of algae.
These simple requirements define the structure
of the model developed below.
Independent food chains -The
simplest
model containing multiple food chains is one
in which the food chains are parallel and independent (Fig. la). In the most general case,
each food chain could include its own triplet
(phytoplankton,
zooplankton,
detritus), with
appropriate growth, mortality, remineralization, and export rates. However, for clarity of
presentation, I suppress the role of detritus and
assume instant remineralization
of dead organic matter. This model (Fig. la) is defined
by the equations
dPi
= PiPi - XiPi - ZiHi,
dt
(la)
dZ.
2 = YiZiHi
dt
(lb)
- GiZi,
and
S=T-CPi-CZi
(2)
for food chains i = 0, 1, 2, . . . , n - 1. Here
phytoplankton
biomass Pi, zooplankton biomass Zi, free nutrient (substrate) concentration
S, and total nutrient concentration
T are all
measured in terms of their nutrient (here nitrogen) equivalents (see notation list). Furthermore, in these equations pi is the growth
rate of phytoplankton
size class i as a function
of substrate concentration S; Xi is a background
loss rate of this phytoplankton
size class due
to predation by generalized herbivores or other
causes; Hi is the harvest rate (per unit zooplankton) of phytoplankton
size class i by the
zooplankton size class that feeds on it; yi is the
assimilation efficiency; and 6i is the death rate
(including excretory losses) of this zooplankton size class. Nutrient is assumed to be instantaneously
remineralized
upon phytoplankton loss or zooplankton death so that total
nutrient is conserved.
For purposes of numerical investigation, the
growth and harvest functions are specified to
have the following forms:
CL,= Pnx3x.iS/(Ks,i + AS);
W
Multiple
The Monod function (Eq. 3a), with maximum
growth rate pmax,iand half-saturation
constant
Ks,i, is a standard choice for describing algal
growth (e.g. DeAngelis 1992). However, the
piecewise-linear form (Eq. 3b) chosen for the
harvest function is not standard; it was chosen.
over the Monod form both because it simplifies the algebra (making the model considerably easier to solve numerically) and because
it yields a stable predator-prey relationship in
the single-chain case (see Armstrong
1976;
Wolkowicz 1989) without the necessity of adding a feeding threshold with its attendant additional parameter (cf. Frost and Franzen
1992). Note particularly that the parameter Kp,i
in Eq. 3b is the phytoplankton
concentration
at which zooplankton feeding is fully saturated, not half-saturated.
For numerical investigation,
we substitute
Eq. 3 into Eq. 1, yielding
(W
Note that only the ascending limb of Eq. 3b
has been used, since unique steady state values
of the Pi can exist only for YiHmax,i 1 6i and Pi/
Kp,i < 1.
The parameters of each food chain are next
specified by means of allometric (power law)
relations among food chains based on different
size classes of algae (Moloney and Field 1989,
199 1). In particular, rate constant Ri for the
ith food chain is related to the corresponding
rate constant R,, for a reference food chain by
Ri = Ro(LiILo)‘R
(5)
where, without loss of generality, we have taken
the reference chain to be the smallest food
chain, i = 0. In Eq. 5, PR is the allometric
constant for rate constant R, and Li and Lo are
the characteristic lengths of phytoplankton
at
the bases of food chain i and reference food
chain 0, respectively.
It is important to understand how parameter
values influence steady state phytoplankton and
zooplankton concentrations. Setting dPi/dt =
599
food chains
zo
Zl
25..
PO
PI
P2 . . .
zo -
Zl -
22 . . .
PO
Pi
P2= mm
(a)
w
Fig. 1. Food-web structures. [a.] A food web with multiplc indepcndcnt food chains. Each zooplankton size class
(Z”, 5, -G * * .) can only feed on a single size class of
phytoplankton
(P,,, P,, Pz, . . .). [b.] A food web where
larger zooplankton size classes can also prey on smaller
zooplankton size classes.
0 and Zi = 0 in Eq. 4a, we calculate the minimum substrate concentrations Smin,i at which
phytoplankton size class i can exist in the community in the absence of its herbivorous prcdator as
s.,=
KSi
m1n’2 (/dmax,i/Xi) - 1 *
(6)
The allometric constants associated with the
three parameters pmax,Ks,i, and Xi are then chosen SO that Smini increases with algal size, allowing smaller size classes to invade at lower
nutrient concentrations than are required for
the invasion of larger size classes.
Next, we set dZildt = 0 in zooplankton Eq.
4b to determine steady state phytoplankton
concentrations Pi”, which represent both the
minimum concentrations of phytoplankton
in
size class i that are necessary for zooplankton
size class i to invade and the steady state phytoplankton concentrations once invasion has
occurred. These relationships are defined by
pi” -
‘iKPPi
Yi
(7)
Hmax,i
and are independent of zooplankton densities
Zi”. Finally, the parameters of phytoplankton
600
Armstrong
Eq. 4a determine steady state zooplankton
concentrations
(when zooplankton
are present) as
zi* = dKPi
H max,i
S
- Xi m (8)
pmaxdKs,i + s
>
Consider now the simplest case (the case
explored by Frost and Franzen 1992), where
only a single food chain (i = 0) is present. These
equations generated a simple pattern that has
three parts (e.g. see DeAngelis 1992). First,
when T is less than the critical value Smin,O(Eq.
6) for phytoplankton
invasion, neither phytoplankton nor zooplankton will be present,
and S = T. In the range Smin,O< T < Smin,O+
PO* (Eq. 6 and 7), the steady state value of S
is pinned at Smin,O,and phytoplankton concentration PO steadily rises as T - Smin,O.In this
range, the phytoplankton
concentration is less
than that required for zooplankton to invade,
and Z,* = 0. Above T = Smin,O+ PO*, however,
zooplankton are present, their grazing fixes the
phytoplankton density at PO*, and S* and Z,*
rise together (Eq. 8) with increasing total nutrient concentration
T. Finally, as the algal
growth rate approaches pmax,O,Z. approaches
its maximum value (Eq. 8), and all additional
increase in T goes to free nutrient S.
Next, consider a case with multiple food
chains (Fig. 2). Assume that the Smin,i (Eq. 6)
increases with i. Then as nutrient input is increased, phytoplankton density in the smallest
size class will increase until PO* is reached,
when the first predator (Zo) will invade and
present further increase. At this point, Zo* and
nutrient concentration S both start to rise until
of the
S min, 1 is reached, where phytoplankton
next larger size class P, start to increase in
density until they in turn arc limited by their
predator. Phytoplankton and zooplankton size
classes are then added sequentially in pairs in
order of size, and the contribution of each size
class to biomass is constrained in accordance
with Chisholm’s (1992) model.
A generalization:
The food-web model of
Moloney and Field-The
model developed
above can be generalized by allowing each zooplankton to graze on multiple zooplankton size
classes as well as on multiple phytoplankton
size classes. Moloney and Field (199 1) proposed such a general model but restricted their
attention to the situation in which each zoo-
plankton size class grazes only a single size
class of phytoplankton
(as in the model of the
previous section) and also grazes only the next
smaller zooplankton size class (Fig. 1b). In this
generalized model, Eq. 1 for phytoplankton
and zooplankton must be replaced by
dP,
pi
L = PiPi - XiPi - Zi
Hi,
dt
Pi + 4Zi-1
(94
and
dZ,- - TiZiHi - SiZi - Zi+I
dt
4zi
H,
Pi+1 + +Zi ‘+l’
Pb)
In this equation, the parameter 4 characterizes
the relative efficiencies of grazing on zooplankton and phytoplankton;
in particular, for 6 =
0, the model reduces to the independent food
chains model of the previous section.
For numerical investigation,
the harvest
function Hi is defined in analogy with Eq. 3b as
Hi =
Hmax,i(Pi
+
@Zi-
1 YKp,i,
i H maxJ7
Pi
+
+Zi-
1 5
Pi + $Zi-,
Substituting
Kp,i
> Kp,i.
(10)
Eq. 3a and 10 into 9 yields
dP.
S
L = Pi p-L,a,iv Ks,i f S
dt
(1 la>
and
dZ.
(=
dt
H
zi yiL ,““y i (Pi + @Zi-1)
f,l
- 6i - Zi+,$y
H
.
(llb)
P,r+l
Parameter values-The
pattern of invasion
of phytoplankton and zooplankton size classes
is affected both qualitatively and quantitatively by the choice of model parameters. To illustrate the range of possible behaviors, I will
use representative parameter values from Moloney and Field (1989, 199 I), Chisholm (1992),
Ducklow and Fasham (1992), and Frost and
Franzen (1992). Parameter definitions and their
units are summarized in the list of notation.
Multiple
601
food chains
10
IO
8.
8
6
6
(w
'4
4
2=22
22z2
...-,& f .*’
22 ...-..
22122=
*......-#Z
.
.
.
.
y;,.........
-mm
$,..
,?fff~~::;:::::::::::::::::::::
22.
.;PPPPPPPPPPPPPP
Zr::.ppPPpPPbBb~iiiie...............
2
2
..........................................
..............................................
tn
2
8
0
0
0
2
4
6
8
-IO
.
zg+~PPQ
pP$P . . . . . . . . .
... .. ... ... ..
,I’
... ... .. ... ... ..
0
2
..
..
..
..
. . . ..a......................
..*.........................
..... ... ......... ... ... .... .
..... ... ......... ... ... .... .
4
6
8
IO
a,
10
TTT
TTTT
TT
8
TTTT
No
TT
TTTT
6
TT
TT’
TT
TT
TT
2222222222222222222222222222222
*.,..............m...........
z#fff~. *,............*......,............
*
.8
0
6
8
TTTT
I4
I2
I
10
4
TT
TT
F. .-*
$ 3 pPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
. .. . .. . . . . .. . .. . .. . . . .. . . . . . . . .. . .. . . . . .
#PPP
. . . . . . . ..mm....m..............................
2
TT
10
I
I
TT
TT
TT
~222222222222222222222222222222222222222
,,?Jf
#
pPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
.. . . . . . . . .. . . . .. . .. . . .. . .. . . . . . .. . .. .. . .. .. .
7
0
Total Nitrogen [mmol
2
4
6
8
IO
m”)
Fig. 2. Plots showing nutrient concentrations
in each phytoplankton
and zooplankton size class for the case of
multiple independent food chains (Fig. la). The row of P’s in each panel denotes the nutrient in total phytoplankton
(on the ordinate) vs. total nutrient concentration (on the abscissa).! The dotted lines beneath the P rows then denote the
cumulative concentrations of nutrient in successive phytoplankto
’ size classes. For example, the distance between zero
7
(on the ordinate) and the first dotted line (at 0.6 mmol N m-3) represents
the nutrient in the smallest algal size class;
the distance between this line and the second dotted line (at 1.2 mmol N m-3) represents the nutrient (also 0.6 mmol
N m-3) in the second size class; and so forth. Next, the distance between the P row on each plot and the row of Z’s
on the same plot represents the total nutrient in zooplankton;
the distances between successive dotted lines again
represent the amount of nutrient in individual zooplankton size classes. Finally, the distance between a Z row and a
diagonal row of T’s (total nutrient) represents ambient nutrient concentration.
[a.] The first standard case (p,,,,, = 1.4
standard case (p,,,,, = 1.4 d-l; BP,,. = -0.4). [c.] The first (“ligand
d- ‘; Plh,, = -0.75). [b.] The second (“diatom”)
limited”) micronutrient
case (pm,, = 0.7 d I; ppmoX
= - 1.0). [d.] The second (“diffusion
limited”) micronutrient
case
b4n,x = 0.7 d-‘; PPmor= -2.0).
We begin by dividing the phytoplankton size
spectrum into discrete size classes. Following
Moloney and Field (1991), we define phytoplankton size classes that are of equal width
on a logarithmic scale. I found it most convenient to define size classes so that the nominal size of algae in adjacent size classes differs
by a factor of 4. If we then choose the upper
limit of the smallest size class to be 2 pm, upper
limits on size classes will be 2 pm, 8 pm, 32
,um, and so forth, and the corresponding nom-
inal algal sizes in these classes will be 1 pm, 4
,um, 16 I.cm, etc.
Note next that the data of Raimbault et al.
(1988) show that the picophytoplankton
(< lpm diam) have a maximum of 0.5 mg Chl a
m-3 while the size fraction < 10 pm in diameter has a maximum of 2.0 mg Chl a m-3,
implying a concentration of 1.5 mg Chl a m-3
in this lo-fold range of diameters. Size classes
whose boundaries differ by a factor of 4 should
therefore contain roughly 1.5 x log,,4 z 0.9
602
Armstrong
Notation
PO z,
S, T
:;y-’
A,
Smin.r
H max,,
KP,,
YI
iI7
Phytoplankton
and zooplankton concentrations, mmol N m-3
Ambient and total nutrient concentrations,
mmolNm
3
Maximum phytoplankton
growth rates, d I
Half-saturation
nutrient concentrations for
phytoplankton
growth, mmol N m-’
Intrinsic phytoplankton
loss rates, d-l
Minimum ambient nutrient concentrations for
invasion of phytoplankton
size classes, mmol
N m-3
Maximum zooplankton harvest rates, mmol P
(mmol a-’ d-l
Zooplankton full-saturation
constants for harvest and growth, mmol P m-’
Zooplankton growth efficiencies, dimensionless
Intrinsic zooplankton loss rates, d -I
Allometric constant for rate process or saturation constant R, dimensionless
mg Chl a m-3. Assuming that there is a maximum of roughly 1.6 mg Chl a to 1 mmol N
(Frost and Franzen 1992), each size class should
contain 0.6 mmol N m-3 at steady state.
Zooplankton parameter values for the reference chain were set to make Pi* = 0.6 mmol
N me3 for all i in the case of independent food
chains (4 = 0). Values for the maximum harvest rate of the smallest food chain HmaX,O=
1.4 d-l) and for growth efficiency (yO = 0.4)
were taken from Frost and Franzen (1992).
However, the half-saturation constant for zooplankton feeding (0.2 mmol N m-3) used by
Frost and Franzen (1992) was much lower than
values obtained from other sources; indeed, it
was too small to allow Pi” = 0.6 mmol N m-3
with the functional response used here (Eq. 3b
and 9b), since the piecewise-linear
form requires Pi < Kp,i at steady state. I therefore
adopted the higher half-saturation
value, 1.0
mmol N mB3 (used by Ducklow and Fasham
1992 for both their size classes of zooplankton), doubling it to Kp,i = 2.0 mmol N m-3 for
use as the full-saturation
value in the piecewise-linear functional response. These values
imply a combined death-excretion rate of 6, =
0.168 d-l to achieve the desired value of Pi*.
Allometric coefficients for growth and respiration were taken from Moloney and Field
(1989, 199 l), multiplied by 3 to convert from
mass to diameter (under the assumption that
mass is proportional
to the cube of the diameter); the values chosen were PNmax= -0.75
and & = - 0.75. Finally, I assumed that assimilation efficiencies yi and feeding saturation
constants Kp,i are constant with size, implying
in the case of independent food chains (4 = 0)
that the Pi” will be equal for all i.
Phytoplankton parameters were chosen next.
Maximum phytoplankton
growth rate and its
allometric coefficient were allowed to vary
across cases. Following
Frost and Franzen
(19%
1 set pmax,o= 1.4 d-l in the standard
case; and following Moloney and Field (199 1)
and Chisholm (1992), I set pW,,, = -0.75 in
this case also. However, Chisholm (1992, p.
222) also notes that when additional algal size
classes are added, “Diatoms play a particularly
important
role in this ‘additional’
chlorophyll.” Since maximum algal growth rate declines less fast within taxa than across taxa, I
also considered a second “diatom”
standard
case where pmaX,O= 1.4 d-l, but also where
-0.4, the value given by Chisholm
PGmax
(1992; for diatoms; this case represents a situation in which larger algal size classes contain
mostly diatoms.
Two cases representing micronutrient
(e.g.
iron) limitation were also considered. In both
of these cases, pmax was reduced to 0.7 d-’
(where the reduction represents a nutrient stress
on even the smallest size classes: see Cullen et
al. 1992; Frost and Franzen 1992) while
Phoaxwas taken alternately to be - 1 (representing a case where micronutrient
uptake is
limited by ligand density on the cell surface:
Morel et al. 199 1) or -2 (for uptake limited
,by diffusion to the ccl1 surface: Morel et al.
199 1). In all cases, the half-saturation constant
for nutrient uptake Ks,i was taken to be 0.1
mmol N m-3, an appropriate value for small
(“5 pm) oceanic species from oligotrophic
waters (table 2 of Epplcy et al. 1969). Finally,
1 assumed a constant background loss rate Xi
of 0.0 16 d-l and applied it equally to all size
classes. Although Moloney and Field (199 1)
suggested a higher sinking rate, and hence a
higher loss rate, for larger size classes (see also
Michaels and Silver 1988), such an assumption would be inappropriate
in the present context, where nutrients are assumed to be instantly regenerated.
The constancy of Xi and K,i across phytoplankton size classes implies that minimum
nutrient concentrations Smin,l required for invasion of algal size class i in the absence of
Multiple
predation depend only on maximum phytoplankton growth rates ,umaxi (Eq. 6). In particular, ifk,ax,i declines with size, then Smin,imust
increase with size, so that smaller size classes
can invade at lower nutrient levels as required.
Although one could include size dependence
of Xi (Moloney and Field 199 1) and K,i (Eppley
et al. 1969; Aksnes and Egge 1991; Maloney
and Field 199 1) to produce more complicated
patterns, the requirement that Smin,i values increase with size dictates that these patterns will
not be qualitatively
new. It may be desirable
to allow these parameters to vary for quantitative comparison with data, but constancy will
suffice for present purposes.
603
food chains
Table 1. The four cases explored in the presentstudy
are each characterized by a maximum algal growth rate
for the standard chain P,,,,,,~ (d-l) and by the allometric
coefficient pp,., for this growth rate (dimensionless). For
each case, the minimum free nutrient concentration Sm1n.i
(mmol N m-3) needed for invasion of size class i at an
intrinsic loss rate of A, = 0.016 d-l is listed next to the
nominal size of individuals in that size class. Size classes
whose entries are dashes have values of Sn1in.rthat exceed
10 mmol N m-3; these classes cannot invade in the current
study, since free nutrient S is always less than total nutrient
7’ I 10 mmol N m- J.
Case a
&nax,o
PLhox
1.4
-0.75
Nominal size (pm)
I
0.0012
4
0.0033
16
0.0101
64
0.0349
256
0.2723
1,024
4,096
-
Case b
1.4
-0.4
0.0012
Case c
0.7
-1.0
0.0023
Case d
0.7
-2.0
0.0023
0.0577
0.0020
0.0101
Results
0.0036
0.0577
For each of the two standard cases and two
0.0064
micronutrient-limited
cases and for values 6
0.0117
0.0224
= 0 and 4 = 1, the model equations were solved
0.0467
numerically over a range of total nutrient concentrations T I 10 mmol N m-3-the
value
used by Frost and Franzen (1992) for their
subsurface nutrient concentration.
(A C-lanclasses can invade (though only the first seven
guage program for solving these equations is are included in Table 1). In this case, ambient
available on request.)
nutrient concentration at the highest nutrient
Minimum
substrate concentrations
Smin,i loadings is low enough (0.03 mmol N m-3) at
necessary for each phytoplankton
size class to T = 10 mmol N m-3 that the seventh algal
invade at a loss rate of 0.0 16 d- l in the absence size class (Smin,T= 0.0467 mmol Nm-3) cannot
of herbivory were computed with Eq. 6 for invade, and only six of the nine possible food
each of the four cases; these values are listed
chains are present.
in Table 1 beside the nominal size of each size
The two micronutrient-limited
cases (Fig.
class. Differences in the numbers of phyto2c,d) contrast sharply with the standard cases.
plankton size classes that can invade in each The number of chains that can invade is limcase (Table 1) lead to dramatic differences in ited to three and two chains, respectively. Betotal phytoplankton
density and ambient nu- cause the number of phytoplankton
size
trient concentration among the four cases.
classes than can invade is limited, total bioIndependent food chains- Consider first the mass cannot increase enough to reduce nutricase with independent food chains (4 = 0) (Fig.
ent levels to near-zero levels, and they remain
la). In the first standard case (P,,.,~~,~
= 1.4 d-l;
high in both simulations (6.3 and 7.7 mmol N
m-3, respectively) when T = 10 mmol N rnm3.
Pknnx= - 0.75; Fig. 2a), five chains can potentially invade (Table l), and all five that can
Connected food web -The parameter values
invade do invade. At T = 10 mmol N mW3, used in the previous section were chosen in
each extant phytoplankton
size class contains
part to produce the Chisholm (1992) pattern
the same amount of nutrient (0.6 mmol N m-3)
in the independent chains model (Fig. la).
per model design, accounting together for 3.0 However, a food web comprised of multiple
mmol N m-3, and the zooplankton taken toindependent food chains is empirically
ungether account for 6.4 mmol N me3, leaving
realistic, so it is of considerable interest to see
0.6 mmol N me3 as free nutrient. In the second
how much or how little the model and its pa(“diatom”)
standard case (pmaX,o= 1.4 d-l;
rameters must be changed to produce this patP+Mx = -0.4; Fig. 2b), algal growth rate de- tern in a connected food web (Fig. lb). I thereclines more slowly with size than in the first
fore also investigated a food-web model with
standard case; here the first nine algal size $ = 1, where phytoplankton
and smaller zoo-
604
Armstrong
plankton are equally available to their zooplankton predators.
When the food chains are connected by the
grazing of larger zooplankton on smaller zooplankton, this grazing becomes an additional
source of mortality for the smaller zooplankton which in turn increases the steady state
phytoplankton
concentrations of the smallest
algal size classes. To offset this tendency and
retain the values Pi* E 0.6 mmol N rnm3, the
values 6i of the intrinsic death-respiration terms
must be decreased. Since zooplankton will now
have two food sources of roughly equal abundance, the first change I made was to halve the
intrinsic death-respiration
rate to 6, = 0.084
d-l. This change allows zooplankton to exist
at lower total nutrient loadings (T - 0.3 mmol
N m-3 rather than 0.6 mmol N mV3), which
in turn allows coexistence of phytoplankton
and zooplankton at lower total nutrient loadings.
The resulting system exhibits undesirable
behavior. Since little nutrient is lost at each
zooplankton size class, the number of zooplankton size classes can become very large;
use of the lowered mortality (6, = 0.084 d-l)
in combination with parameters from the first
standard case, for example, produced a system
with two phytoplankton
size classes and 11
zooplankton size classes at T = 10 mmol N
me3. To eliminate this behavior we must make
the zooplankton death-respiration
rate fall off
less rapidly with size than does the harvest
rate.
To accomplish this change, we separate the
zooplankton death-respiration
term into two
parts, one part representing respiration (retaining its allometric coefficient of -0.75) and
a second part representing background mortality (with a separate allometric coefficient).
The smallest zooplankton size classes are so
small that they are effectively immobile, so
that if size-independent
predation by gelatinous predators or larger zooplankton is the
cause of the background mortality on the phytoplankton, it would seem reasonable to make
zooplankton subject to this same background
mortality. We therefore assign the same background mortality to the smallest zooplankton
classes as to the phytoplankton (0.0 16 d-l) and
assign the residual part of the death-respiration
term (0.068 d- l) to respiration. We could leave
the allometric coefficient for the death term at
zero, as for the phytoplankton,
or we could let
this coefficient decline with size, reflecting an
increased ability of larger zooplankton to escape this generalized predation. An allometric
coefficient for the zooplankton death term of
-0.4 was used to produce the results shown
in Fig. 3.
The resulting model produces patterns that
are superior in many ways to those from the
independent chains model. First note that in
the standard cases, the model no longer automatically
produces equal phytoplankton
densities in each phytoplankton size class (Fig.
3a), although with careful selection of parameter values, rough equality can be obtained
(Fig. 3b). Next note that for 4 > 0 the number
of zooplankton size classes may exceed the
number of phytoplankton
size classes, since
the ability of zooplankton to use smaller zooplankton when Cp> 0 allows new trophic levels
to be added without adding supporting phytoplankton size classes; this behavior is clearly
impossible at 4 = 0 and adds realistic new
possibilities to the model. This behavior is most
evident in the micronutrient
cases (Fig. 3c,d),
where both cases converge to a food chain with
a single phytoplankton
size class and five zooplankton size classes.
Third, the ability of zooplankton size classes
to use smaller zooplankton as food when 4 >
0 causes phytoplankton
size classes to “wink”
in and out at different nutrient densities. (Note
in Fig. 3b the appearance of phytoplankton
size class 5 at T = 2.9 mmol N m-3, its disappearance at T = 4.4 mmol N m-3, and its
reappearance at T = 6.4 mmol N m-3.) As T
is increased, algal productivity
increases, leading to a higher concentration
of small zooplankton. This increased zooplankton concentration in turn allows larger zooplankton to
survive at steady state on lower concentrations
of phytoplankton.
When small zooplankton
become abundant enough, larger zooplankton
are able to survive without their corresponding
phytoplankton
size classes, and the latter disappear from the steady state solution. Finally,
total phytoplankton
biomass rises more
smoothly (less stepwise) when 4 = 1.
The connected food-web model produces the
Chisholm (1992) pattern with low (0.03 mmol
N md3) residual nitrate at T = 10 mmol N
rnH3 in a reasonable standard case (Fig. 3b)
while producing high (5.3 mmol N m-3) re-
Multiple
605
food chains
10
8
-8
(b)
T
E
0
E
6
6
4
4
2
,$f
$T’
gz . . ,. *z= .*
.
zzz=
..* ..-- . ..’ . . .
z~?+/.....~ ..a.9 ,...*- ...*. ..--. . .
ZZ
. . . . . .. -.
ZI ..-*.---. ...’
zz,...‘m’
*........z.
z* ,...‘:.*#,..’/p~~~PPPPPPPPPP
,,s;~~~~~~-~~~~~~~~~~~~~~~~~~::~::::::::::;
2
_E _
. . ..*.*.........................
.. .. . . . . . . . . .. . . . .. . . . .. .. . . . . . .
,&y,::.......
&......,....“..’
0
0
2
4
6
8
10
10
8.
-TTTTT
TT
(d)
8.
TT
TT
6.
0
2
4
6
8
10
-TT
0
2
4
TT
6
T”
TT
TT
TT
8
10
Total Nitrogen (mmol mm3)
Fig. 3. As Fig. 2, but for a case (4 = 1) where larger zooplankton
classes (Fig. lb).
sidual nitrate at the same nutrient loading in
both micronutrient
cases (Fig. 3c,d); this is especially reassuring since the changes that were
needed to allow this to happen (splitting the
zooplankton respiration-mortality
term into
components and identifying
the background
mortality term on the smallest zooplankton
size class with the loss rate of phytoplankton)
increases model realism rather than decreasing
it.
Discussion
Ecosystems are highly structured. Two critical dimensions of pelagic ecosystem structure
are size (Moloney and Field 199 1; Chisholm
1992) and nutritional
requirements (e.g. silicon for diatoms). Yet often it is supposed that
the essence of ecosystem function can be captured by extremely simple models in which the
size classes can also eat smaller zooplankton
size
entire grazing system is represented by a single
predator-prey
pair (e.g. Evans and Parslow
1985; Fasham et al. 1990; Frost and Franzen
1992), by a single predator on multiple prey
(Armstrong 1979; Evans 1988), by a linear food
chain (Thingstad and Sakshaug 1990), or by
two size classes of predators and prey interacting in a way not easily generalized to multiple size classes (Taylor and Joint 1990; Ducklow and Fasham 1992). When more complete
ecosystem structures involving multiple food
chains are proposed, they are often so complicated that their behaviors can only be explored in specialized symmetric cases (Armstrong 1982, 1983) or by simulation (Moloney
and Field 1991).
Here, I proposed a model that includes
asymmetric interactions among multiple food
chains, yet is structured in a way that mini-
606
Armstrong
mizes the number of free parameters, facilitating numerical exploration of its steady state
behaviors. The model does not include explicit
consideration of detritus or other parts of the
decomposition
system. This omission limits
its use as a quantitative model, since typically
a large part of the nutrient in the system will
be either sequestered in the decomposition
system or exported as detritus and other components, where export itself may have a strong
size dependence (Michaels and Silver 1988).
The present model would, however, be particularly useful in driving a decomposition model
with multiple detrital size classes derived from
the multiple size classes of phytoplankton
and
zooplankton.
Models with multiple size classes provide a
natural way to capture the essential difference
in phytoplankton
size between regions dominated by small cells (domains 1 and 2 of Banse
1992) and regions having at least seasonal
blooms of much larger cells (domain 3 of Banse
1992). In domain 1 (subtropical gyres), nutrients are scarce, limiting the number of food
chains that can be supported. In domain 2 (the
North Pacific, equatorial Pacific, and Southern
Ocean), macronutrient
(e.g. nitrate) concentrations are high, but phytoplankton
abundances are low. In this region, the model suggests either that background mortality (Xi) on
larger size classes must be large (e.g. because
of generalized grazing by unselective predators, Banse 1992) or that low maximum growth
rates (p.,,.J of larger cells in this domain limit
the extent to which they can invade; low pm,,,i
may in turn be due to micronutrient-imposed
limits on fundamental growth rate P,~~,~,to an
increased rate of decrease of growth rate p@,,,
with size, or both. Finally, domain 3 is characterized by abundant nutrient delivery that is
virtually all used during the growing season;
here, multiple size classes can survive at steady
state or under bloom conditions. The fundamental characteristics of these regions cannot
be captured easily in a model without size
structure,
since in one-plankton-one-zooplankton (1PlZ) models, steady state phytoplankton density is completely determined by
the grazing characteristics of the (single) zooplankton size class and is independent of nutrient loading. In contrast, the present model
is well suited to capturing the essential features
of all three domains.
A
particularly important difference between
1P 1Z models and models based on multiple
food chains is that 1P 1Z models with constant
mortality predict that ambient nutrient concentrations must reach high levels at high nutrient loadings, while multiple-phytoplankton
multiple-zooplankton
models predict that ambient nutrient concentrations will rise appreciably only when the next available size class
of phytoplankton
cannot grow fast enough to
invade the existing community.
Ultimately,
therefore, high ambient nutrient concentrations during the growing season must be due
to low growth rates, high background death
rates, or high half-saturation
constants of the
next higher (missing) phytoplankton size class.
This perspective offers insight into the hypothesis (Martin et al. 199 1) that micronutrient (notably iron) limitation is responsible for
elevated nitrate levels in the equatorial Pacific.
The resident phytoplankton
in these regions
are small (Chavez 1989; Banse 1992; Frost and
Franzen 1992 for review), suggesting that the
larger size classes that would normally use the
nitrate are missing for some reason. It is possible, of course, that the (missing) “next larger”
phytoplankton size classes in these systems are
missing due to abnormally high background
death rates (Miller et al. 199 1) or because the
half-saturation
constants for nitrate uptake in
the equatorial Pacific are abnormally high or
increase more rapidly with size than do those
elsewhere in the world ocean. However, the
appearance of larger size classes of algae upon
addition of iron in “grow-out”
experiments
(Price et al. 199 1; Frost and Franzen 1992 for
review) in which cultures are allowed to incubate for several days argues strongly for micronutrient
limitation
of maximum
growth
rates of larger size classes.
I have modeled the action of micronutrient
(iron) limitation
as a generalized debilitating
effect on algal growth (Harrison and Morel
1986; Greene et al. 199 I), and I have modeled
its greater effect on larger algae as being caused
by transport limitation
of micronutrient
uptake (Morel et al. 1991). Additional
complications arise when the distinct roles of different
nitrogenous substrates are recognized. Many
algae appear to prefer ammonium as a growth
substrate (Dortch 1990), and iron is needed
more comfor the enzyme nitrate reductase.
plete description of the effect of size on growth
A
Multiple
rate might therefore need to include a general
debilitating effect of iron limitation,
competition for scarce supplies of ammonium (where
smaller algae are expected to be superior competitors: Hudson and Morel 1990), and a combination of effects on uptake and nitrate reduction that tends to make nitrate relatively
unusable, again disproportionately
so for larger cells. Because each of these mechanisms
might have its own characteristic allometric
constant, the total response of algal growth rate
to size might be considerably more complicated that the situation modeled above.
From the perspective of ecosystem modeling, the tight constraint imposed by 1P 1Z
models with constant zooplankton mortality
on phytoplankton
densities can lead to undesirable behaviors. For example, the 1PlZ
structure of the Fasham et al. (1990) ecosystem
model, used by Sarmiento et al. (1993) in their
study of nutrient cycling in the North Atlantic,
fixed steady state summer chlorophyll concentrations at high latitudes at constant values and
likely caused at least part of the mismatch betwecn model predictions and satellite chlorophyll measurements (Armstrong et al. in press).
This model also produced ammonium
concentrations one to two orders of magnitude
higher than reasonable when Armstrong et al.
(in press) attempted to assimilate satellite data
into it; multiple-predator/muItiplc-prey
models alleviated these problems by allowing chlorophyll concentrations to increase to required
levels (through the addition of larger phytoplankton size classes) with increased nutrient
loading.
From a broader perspective, models with
multiple phytoplankton
and zooplankton size
classes provide a natural framework for disentangling limitation of particular size classes
from limitation of total plant and animal biomass. Although alternative ways exist within
the 1P 1Z structure to allow total phytoplankton and zooplankton biomass to rise with increases in nutrient loading (e.g. Ginzburg and
Argakaya 1992; Steele and Henderson 1992),
none allows limitation
of species abundances
to be considered separately from limitation of
trophic levels. This ability to disentangle limitation of whole trophic levels from limitation
of their constituent parts (while simultaneously producing observed patterns in algal size
distributions and other community propcrtics)
food chains
607
provides a strong conceptual reason to prefer
models based on multiple phytoplankton
and
zooplankton size classes.
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Submitted: 19 April 1993
Accepted: 7 June 1993
Amended: 9 July 1993