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A multi-nutrient model for the
description of stoichiometric modulation
of predation in micro- and
mesozooplankton
ADITEE MITRA
INSTITUTE OF ENVIRONMENTAL SUSTAINABILITY, UNIVERSITY OF WALES SWANSEA, SINGLETON PARK, SWANSEA SA2 8PP, UK
*CORRESPONDING AUTHOR: [email protected]
Received December 12, 2005; accepted in principle January 31, 2006; accepted for publication February 17, 2006; published online February 22, 2006
Communicating editor: R.P. Harris
Changes in predator behaviour when confronted with prey of disadvantageous composition have been
termed stoichiometric modulation of predation (SMP; Mitra and Flynn, 2005; J. Plankton Res. 27,
393–399). Through SMP, a predator may compensate for (positive SMP) or compound (negative
SMP) dietary deficiencies. While these responses are documented in experiments, albeit typically with
poor parameterization, previous zooplankton models contain no explicit description of these events. A
new multi-nutrient biomass-based generic zooplankton model is described, capable of handling SMP at
the levels of ingestion and assimilation, for the exploration of zooplankton growth dynamics in situations
where prey quality and quantity changes over time. SMP is enabled by configuring ingestion rate and
assimilation efficiency descriptors as functions of food quality (indexed here to prey N:C). Sensitivity
analysis of the new model shows the structure to be robust against variation in parameter (constant)
values. The form of the model enables its use in population dynamic studies of different zooplankton
groups; here, the model has been configured to represent micro- and mesozooplankton. It is shown that in
the absence of inclusion of SMP, fits of the model to experimental data can be poor with potential for
significant misrepresentation of trophic dynamics.
INTRODUCTION
Our developing knowledge of plankton system dynamics
provides the opportunity to replace previous black box
empirical descriptions of ecosystem components with
more mechanistic sub-models. To match the development of multi-nutrient (light–N–P–Si–Fe) and functional
group phytoplankton models (Flynn, 2001, 2003;
Anderson, 2005), multi-nutrient zooplankton models
are needed. Such zooplankton models should not only
be capable of handling the simple stoichiometric disparity between predator and prey but also be able to
describe changes in predator behaviour when confronted
with the prey of disadvantageous composition.
Experimental studies show that zooplankton are capable
of differentiating between dead and live prey (Landry
et al., 1991) and prey of different taxonomic groups
(Stoecker, 1988). They also show that zooplankton
respond differently towards the same prey species present
at different nutritional status (Flynn and Davidson, 1993;
Plath and Boersma, 2001; Jones and Flynn, 2005). Thus,
minor changes in food nutrient status, as reflected by
stoichiometric changes, can be associated with significant
changes in food palatability and potentially with changes
in prey selectivity (Flynn et al., 1996). Typically, multinutrient stoichiometric zooplankton models do not explicitly describe these important zooplankton functional
responses (e.g. Anderson, 1992; Anderson and Hessen,
1995; Sterner, 1997; Touratier et al., 1999; Sterner and
Elser, 2002; Anderson et al., 2005].
Responses of the predator to alteration in prey nutritional status (i.e. quality) have been termed stoichiometric modulation of predation (SMP; Mitra and Flynn
2005). Figure 1 demonstrates the different ways in which
a consumer could potentially respond to such changes.
doi:10.1093/plankt/fbi144, available online at www.plankt.oxfordjournals.org
Ó The Author 2006. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
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+ve SMPIng
1
Relative IgX
Relative IgC
8
4
+ve SMPIng
2
0 SMP
–ve SMPIng
0 SMP
–ve SMPIng
0
0.0
0.5
0
1.0
0.0
Prey X:C :: predator X:C (dl)
C
D
+ve SMPAE
1.0
+ve SMPIng
1
Growth rate
Relative AEX
1
0.5
Prey X:C :: predator X:C (dl)
0 SMP
+ve SMPAE
0 SMP
–ve SMPIng
–ve SMPAE
–ve SMPAE
0
0
0.0
0.5
1.0
0.0
Prey X:C :: predator X:C (dl)
0.5
1.0
Prey X:C :: predator X:C (dl)
Fig. 1. Stoichiometric modulation of predation (SMP). Panels (A) and (B) show the operation of +ve and –ve SMP at ingestion (SMPIng) in
comparison with the default expectation of neutral SMP (0 SMP). Panel (C) shows the impact of +ve and –ve SMP at assimilation efficiency of
nutrient X (SMPAE). Positive SMP acts to compensate for decline in prey X:C; –ve SMP exaggerates the impact. Panel (D) shows the effect of
SMP on growth. Note the impact of SMPIng and SMPAE is demonstrated alone and not in combination (see text for further explanation).
Here, the ratio of prey X:C relative to predator X:C has
been used as the driver for SMP (X could describe N, P,
fatty acid etc.); the effects of SMP on the relative rate of
ingestion, assimilation efficiency and growth rate are
shown. With a decrease in prey quality, the consumer
could simply maintain its rate of ingestion of carbon as
constant (IgC, Fig. 1A). Constant IgC would invariably
result in a decline in the rate of ingestion of the limiting
factor, X, with a decrease in the relative prey quality (IgX,
Fig. 1B), leading in turn to a decline in the growth rate of
the consumer (Fig. 1D); this is the default expectation
and represents neutral, i.e. 0 SMP. However, the consumer could try to maintain a constant uptake of X (i.e.
constant IgX, Fig. 1B) and thus retain a constant growth
rate (Fig. 1D). This could be achieved by increasing the
food intake (IgC ", Fig. 1A) with declining prey quality;
this is termed positive SMPIng (+ve SMPIng in Fig. 1A, B,
D; Darchambeau and Thys, 2005). Plath and Boersma
(Plath and Boersma, 2001) observed that Daphnia supplied with algal prey (at 1 mgC L1) exhibited higher
appendage beat rates and, therefore, presumably higher
filtration rates (ingestion rates), when those prey were
of poor quality (i.e. low P:C). On the contrary, stoichiometric changes in the prey nutrient status could result in
the food becoming deleterious or unpalatable. In such
instances, the predator may decrease its rate of ingestion
(IgC #) leading to an even greater decline in IgX and
hence in the growth rate; this is termed negative SMP
(–ve SMPIng in Fig. 1A, B, D; e.g. Flynn and Davidson,
1993).
SMP could also be displayed at the level of assimilation where with a decrease in prey quality the predator
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could either increase the assimilation efficiency of X
(AEX ") exhibiting +ve SMPAE or decrease AEX (#)
giving –ve SMPAE (Fig. 1C; e.g. Jones and Flynn,
2005). However, unlike +ve SMPIng, +ve SMPAE cannot
exceed beyond a maximum of 100% (Fig. 1A versus C).
It should be noted that SMPIng and SMPAE are not
mutually exclusive. The consumer may show various
combinations of the two; some combinations of which
may compensate for each other (e.g. –ve SMPIng with
+ve SMPAE).
Existing multi-nutrient zooplankton models consider
primarily the impact of stoichiometry on the assimilation of ingested prey into predator biomass (Anderson,
1992; Anderson and Hessen, 1995; Sterner, 1997;
Touratier et al., 1999; Sterner and Elser, 2002;
Anderson et al., 2005). Typically, in these models, the
effects of SMP on ingestion rate and assimilation efficiency are not considered explicitly, if at all. The aim of
this article is to present a generic biomass-based multinutrient zooplankton model, capable of handling SMP
at the levels of ingestion as well as assimilation, to
facilitate the study of zooplankton growth dynamics
and for incorporation within ecosystem food web
models.
The zooplankton model presented here is capable of
handling not only simple stoichiometric disparity
between predator and prey but also the behaviour of a
predator when confronted with prey of varying status.
This has been achieved by linking the ingestion rate and
assimilation efficiency to prey availability and nutritional
status. The form of the model enables its use in the
conduct of population dynamic studies of different zooplankton groups, representing micro- and also mesozooplankton populations.
efficiency of retention against respiratory losses, with
the balance being growth. Thus, the rate of change of
the zooplankton biomass can be described through
equation (1), where ingestion minus voiding equals
assimilation.
METHOD
Zooplankton could respond to changes in the quality
of food (R) by increasing or decreasing ingestion, or
make no change. R is a quotient defining food quality
described through equation (3). In this study, quality is
assumed to be based on stoichiometric differences
between the prey nutritional status and predator
N:C. However, this could be replaced by some other
index of quality, such as toxin content. Thus, a value
of R = 1 indicates high-quality food, while R = 0
represents food with no nutritional value, whether
that be due to a dietary deficiency or unpalatability.
SNC
R ¼ MIN
;1
ð3Þ
ZNC
Model description
The zooplankton model contains a single-state variable describing predator carbon (C) biomass (ZC). The
biomass of other zooplankton components (e.g. nitrogen, N) is assumed by fixed stoichiometric ratio
(Caron and Goldman, 1988; Anderson and Hessen,
1995; Jones et al., 2002; but cf. Ferrão Filho et al.,
2005). Throughout this article, reference is made to N
and C with the assumption that all other nutrients are
in excess; however, phosphorus (P) or any other nutrient (e.g. fatty acid) could be substituted for N or used
as a third nutrient if required. The physiological processes described are ingestion, the efficiency of assimilation (the excess material being voided) and
dðZCÞ
¼ ðingestion voidingÞ respiration
dt
ð1Þ
Arguments for the construction of the model appear in
the text below; Tables I and II list all the model parameters, identifying their type and units. Figure 2 shows a
schematic of the model identifying the role of the equations. The schematic shows the points at which food
quality (FQ) could impact on ingestion and/or assimilation. These FQ-links enable +ve and/or –ve SMP; if FQlink is switched off, the model displays 0 SMP at both
ingestion and assimilation levels and hence behaves like a
traditional stoichiometric model. Boolean logic terms in
equations take the value 1 if true and 0 if false.
Ingestion
Im is the maximum ingestion rate required to support the
maximum growth rate (m) of the zooplankton. The
value of m for a particular species will be fixed at a
given temperature and could be made a function of
temperature (together with respiratory costs) if required.
In order to attain m, Im for the predator needs to
account for losses incurred due to respiration (basal,
BR, and metabolic, MR) and voiding (i.e. relating to
assimilation efficiency, AEFQ; see also equation (9)); Im
is thus described by equation (2).
Im ¼
m ð1 þ MRÞ þ BR
AE FQ
ð2Þ
In response to poor-quality food, i.e. low R, a zooplankton may be expected to increase ingestion (low
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Table I: List of model-state variables and auxiliaries, their description and units
Parameter
Description
Units
DIC
Dissolved inorganic carbon
gC L–1
DIN
Dissolved inorganic nitrogen
gN L–1
VOC
Voided organic carbon
gC L–1
VON
Voided organic nitrogen
gN L–1
SC
Prey carbon biomass
gC L–1
ZC
Predator carbon biomass
gC L–1
AE
Assimilation/digestion efficiency
dl
AEFQ
Food-quality link on AE
dl
BRb
Basal respiration from predator body
gC (gC)–1 d–1
BRi
Basal respiration using excess C
gC (gC)–1 d–1
Cas
C-specific assimilation rate
gC (gC)–1 d–1
CR
C-specific structural respiration rate
gC (gC)–1 d–1
DINr
N-specific DIN regeneration rate
gN (gC)–1 d–1
dl
State variables
Auxiliary variables
GGEC
Gross growth efficiency of Z for C
GGEN
Gross growth efficiency of Z for N
dl
IgC
C-specific ingestion rate
gC (gC)–1 d–1
Im
Maximum ingestion rate
gC (gC)–1 d–1
MINUP
Control to select release of N for predator–prey
dl
stoichiometric differences
VOCt
C-specific defecation rate of predator
gC (gC)–1 d–1
VONt
Prey N voided in the particulate form
gN (gC)–1 d–1
VONtr
Prey N voided
gN (gC)–1 d–1
R
Relative food quality
dl
SNC
Prey N : C ratio
gN (gC)–1
VIg
Ingestion control parameter
dl
XSC
C available for digestion but not for
gC (gC)–1 d–1
ZC
Zooplankton growth rate
protoplasmic uses
gC (gC)–1 d–1
dl, dimensionless.
dietary quality; +ve SMPIng, Fig. 1) or decrease ingestion (toxin-associated unpalatability; –ve SMPIng, Fig.
1). The response of ingestion to changes in R is
described in the model by reference to parameter
VIg [equation (4)], with pVIg defining the slope of the
curve and a the direction. Figure 3A shows the different forms of VIg depending on the values of these
constants where increasing values of pVIg increase the
curvature of the responses. Values of a > 0 give
increased ingestion compensating for the decline in
R, demonstrating +ve SMPIng (Fig. 1A). In contrast,
values of a < 0 give –ve SMPIng with a decrease in the
zooplankton ingestion rate in response to poor-quality
food (Fig. 1A). The food-quality link to ingestion can
be disabled (i.e. switched off) by setting a = 0.
VIg ¼ 1 þ ð1 RpVIg Þ a
ð4Þ
Ingestion rate (IgC) is then described by equation (5), as
a hyperbolic function (Holling Type II) of prey concentration (SC, with a half saturation constant of KIng), with
the operational maximum rate of ingestion set by the
product of Im and VIg.
IgC ¼ Im VIg SC
SC þ KIng
ð5Þ
For handling multiple prey items, the ratio-based
selectivity function of Fasham et al. (Fasham et al., 1990)
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Table II: Definition, units and default values for constants
Constants
Definition
Unit
Value
Guide references
a
Defines direction of VIg
dl
0
AEmax
Maximum value that AE can attain
dl
0.75
AEmin
Minimum value of AE
dl
0.25
BR
Basal respiration rate
gC (gC)–1 d–1
0.1
KAE
Half saturation constant for assimilation
dl
1
KIng
Half saturation constant for ingestion
mgC L–1
MR
Respiration cost associated with
gC (gC)–1
0.2
Flynn (2004)
gN (gC)–1
0.3a
Derived from the N:C ratio in the
0 to 1
Fasham et al. (1990)
Fasham (1993)
Fenchel (2004)
50
Evans and Garçon (1997) assuming N:C = 0.2
metabolic functions
NCmax
Maximum mass ratio of N:C for
voided PON
upper quartile values of protein amino acids
pI
Grazing preference index
dl
pVig
Defines slope of VIg
dl
1
SCIni
Initial prey concentration at start of
gC L–1
b
SCm
Algal growth rate
simulation
SNCmax
gC (gC)–1 d–1
–1
Maximum prey N:C ratio
gN (gC)
–1
d
–1
b
0.25a
Ingraham et al. (1983)
1
Evans and Garçon (1997)
m
Target predator growth rate
gC (gC)
ZCIni
Initial predator concentration
gC L1
b
ZNC
Predator N:C ratio
gN (gC)–1
0.2
at start of simulation
Goldman et al. (1989); Davidson
et al. (1995a)
dl, dimensionless.
a
Values not subjected to tuning.
b
See Table V for values used in simulations presented in Fig. 5.
can be used. IgCi (Gi in Fasham et al., 1990) is the
ingestion rate for prey item i, ImVIg is the maximum
grazing rate (g in Fasham et al., 1990) and pi
is the preference index for prey type i present at
concentration SCi (Pi in Fasham et al., 1990), such that
n
P
pi ¼ 1; equation (5) is then replaced by equation (6)
i¼1
where IgC ¼
n
P
IgCi .
i¼1
FQ-link
IgCi ¼
Predator
3–4
5/6*
Prey
Ingestion
FQ-link
5iii
ð6Þ
15 &17
Voiding
9
18ii
2
7/8* & 10
Ingested material may be assimilated into biomass,
respired/regenerated through metabolic processes or
voided, as described below (see also Fig. 2).
Regeneration
18i
Assimilation
ðIm VIg Þ pi SCi2
KIng ðp1 SC1 þ p2 SC2 þ p3 SC3 þ Þ þ p1 SC12 þ p2 SC22 þ p3 SC32 þ 11–12
Respiration
13–14
Assimilation
19
Growth
Fig. 2. Schematic of the model. Prey are ingested, followed by assimilation with losses through respiration, regeneration and voiding. The link to
food quality (FQ-link) is incorporated at the levels of ingestion and assimilation enabling SMP. Numbers identify equations describing the functions. i, ii or iii denotes first, second or third term in the equation indicated.
Feedbacks (dashed lines) from prey quantity [term 3, equation (5)] and
assimilation have the potential to effect ingestion. * denotes alternative
equations.
As zooplankton biomass is considered to have a fixed
stoichiometry, the incorporation of nutrients from
ingested material proceeds according to its own
elemental ratio (here N:C, described by ZNC).
Equation (7) describes this stoichiometric interaction
between the predator and ingested prey by reference
to a minimal threshold control (MINUP) that accounts
for differences in values of SNC and ZNC.
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a = 1; pVIg = 1
a = 1; pVIg = 5
a = –1; pVIg = 1
VIg
1
0.5
1.0
Food quality (R)
AEFQ (quotient)
B1.0
KAE = 0.01
0.5
KAE = 1
0.1
0.2
Fig. 3. Different functional forms for the auxiliaries (A) VIg and (B)
AEFQ. See text associated with equations (4) and (9), respectively.
SNC
;1
ZNC
ð7Þ
It is at this point that the different nutrients within a
multi-currency scenario (e.g. N and P) would be apportioned. In this instance, MINUP would be controlled by
the nutrient in lowest proportion (i.e. most limiting) in
the prey, relative to the predator. For example, equation
(7) could be replaced by equation (8) where SPC is the
prey P:C ratio and ZPC the predator P:C ratio.
MINUP ¼ MIN
SNC SPC
;
;1
ZNC ZPC
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ð9Þ
ð10Þ
Respiration
SNC [gN (gC) ]
–1
MINUP ¼ MIN
PAGES
The actual assimilation efficiency (AE ) is thus given by
AEFQ . MINUP.
KAE = 10
0.0
j
Cas ¼ IgC AEFQ MINUP
KAE = 0.1
0.0
6
KAE is the half saturation constant; the higher the
value of KAE, the more powerful is the –ve SMPAE (Figs
1C and 3B). This FQ-link to assimilation can be switched
off by setting KAE 0.01 and AEmin = AEmax, thus
invoking neutral stoichiometric modulation of assimilation (Fig. 3B; 0 SMP in Fig. 1C). An alternative equation
would be required if it were necessary to consider +ve
stoichiometric modulation of AE.
The C-assimilation rate (Cas) is then described through
equation (10):
0
0.0
NUMBER
AEFQ ¼ AEmin þ ðAEmax AEmin Þ
SNC=SNCmax
ð1 þ KAE Þ
ðSNC=SNCmax Þ þ KAE
a = –1; pVIg = 5
2
j
prey nutritional status (i.e. SNC) relative to the maximum
status of the prey (SNCmax).
a = 0; pVIg = 1
A
28
ð8Þ
Respiration is differentiated into two types, a basal respiration rate (BR) and a metabolic respiration cost (MR)
(Flynn, 2004). BR accounts for all non-anabolic respiration, including maintenance of homeostasis (i.e. osmotic
and ionic gradients) and recycling and repair mechanisms
(e.g. enzyme turnover and DNA repair). The cost of
hunting as functions of prey availability and water turbulence (Caparroy and Carlotti, 1996) could also be included
within BR. MR is the respiration cost associated with
metabolic functions [including synthesis of new biomass
and specific dynamic action (Kiørboe et al., 1985)].
It has been argued (Roman, 1983; Anderson, 1992)
that any excess C in the prey is utilized to meet all or, at
least, part of the basal respiratory costs. BR has thus been
formulated to use as a priority the digestible excess C
(e.g. sugars) ingested by the predator, defined here as
XSC in equation (11). The component of BR so supported (BRi) is described by equation (12), with the balance (BRb) given by equation (13).
The efficiency with which ingested food is assimilated
may vary as a function of the quality of the prey species;
only –ve SMPAE is considered here (e.g. Jones and Flynn,
2005). To account for such an effect on the consumption
of prey of different qualities, AEFQ is allowed to vary
between a minimum (AEmin) and a maximum (AEmax)
value. The functional form of AEFQ between these two
values could take various forms; it is described here
[equation (9)] as a normalized hyperbolic function of
602
XSC ¼ IgC AEFQ ð1 MINUP Þ
BRi ¼ ðBR XSCÞ BR þ ðBR > XSCÞ XSC
BRb ¼ BR BRi
ð11Þ
ð12Þ
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The total catabolic respiration cost (including BR not
supported by XSC), CR, is given by equation (14), as the
sum of respiration associated with catabolism (Cas MR)
and BRb.
CR ¼ Cas MR þ BRb
ð14Þ
This demand for C, from respiration of C previously
assimilated into the body biomass, must inevitably be
associated with a stoichiometric regeneration of N [equation (18)].
Voiding (defecation)
Excess nutrients over those assimilated into predator
biomass and used for respiration must be voided. Thus,
C within the ingested material (IgC), which is neither
assimilated (Cas) nor respired (BRi), is voided as organic
material [VOCt; equation (15)].
and particulate organics as functions of assimilation and
ingestion rates.
Regeneration and excretion
The release of dissolved inorganic N (DIN, e.g. ammonium) occurs through two processes. The first is that in
conjunction with the release of dissolved inorganic C
(DIC, i.e. CO2) as a consequence of respiratory processes
[first term in equation (18); see also equation (14)]. In
addition, there is that part of N associated originally with
voided organic material that, due to limitations set by the
chemistry of organic structures, cannot actually be
voided as VON [equation (17)]. The total release of DIN
by the predator (DINr) is thus given by equation (18).
DINr ¼ CR ZNC þ ðVONtr VONt Þ
ð18Þ
Growth rate and growth efficiencies
VOCt ¼ IgC Cas BRi
ð15Þ
Associated with this voided C is an amount of N to be
voided [VONtr; equation (16)]; BRi utilizes XSC, and
hence, there is no associated term with BRi for the voiding of N in equation (16).
VONtr ¼ IgC SNC Cas ZNC
The net growth rate (ZC) and gross growth efficiencies
for C (GGEC) and N (GGEN) of the predator are outputs
in this model and are given by equations (19)–(21),
respectively.
ZC ¼ Cas CR
ð16Þ
IgC VOCt CR BRi
IgC
ð20Þ
IgC SNC VONt DINr
IgC SNC
ð21Þ
GGEC ¼
However, chemical constraints on the structure of
organic molecules dictate an upper limit on the N:C of
voided material; given that this N:C cannot increase
beyond a critical value (described here by NCmax), the
surplus N must be lost in the inorganic form (DIN). The
rate of voiding of excess N as organic nitrogen (VON) is
calculated through equation (17), with the balance (VONtr
– VONt) being released as inorganic N [see equation (18)].
VONtr
VONt ¼
> NCmax VOCt NCmax
VOC
t
VONtr
þ
NCmax VONtr
VOCt
ð19Þ
GGEN ¼
Steady-state sensitivity analysis
ð17Þ
The voided organic material (VOM; VOC, VON etc.)
will include both particulate and dissolved components.
The distribution of dissolved organics versus particulate
organics in VOM will depend on numerous factors
including the balance of digestion and assimilation of
ingested material and the gut/digestive vacuole transit
rate. This interaction is not described explicitly here.
Mechanistically, such processes could be described
using approaches similar to Jumars (Jumars, 2000a,b)
or empirically by dividing VOMs into dissolved organics
Steady state of the model was achieved by providing a
constant saturating concentration of prey (SC) to the
zooplankton and running simulations until the rate
processes within the zooplankton model achieved constant values (constant GGE, growth rate etc.). Values
were assigned to the constant parameters as summarized in Table III. Steady-state sensitivity analyses,
giving S values according to equation (22) (Haefner,
1996), were performed on all the constant parameters
by typically doubling and halving the base (nominal)
values.
603
S¼
ðRa Rn Þ=Rn
ðPa Pn Þ=Pn
ð22Þ
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Table III: Steady-state sensitivity analysis conducted on the constants
Parameters
Responses
Constants
Values
SNC: 0.1
GGEC
a
Base value
P1
0.26
0.42
0.42
0.42
0.46
0.36
0.42
0.13
0.42
0.42
0.42
0.46
0.36
0.42
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.20
0.40
0.78
0.42
0.42
0.42
0.46
0.36
0.42
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.42
4
0.1
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.05
0.20
0.40
0.25
0.47
0.47
0.45
0.51
0.41
0.45
0.00
0.00
0.08
0.23
0.23
0.14
0.23
0.23
0.14
0.2
0.20
0.40
0.28
0.33
0.33
0.36
0.36
0.29
0.36
0.00
0.00
0.08
0.20
0.20
0.14
0.20
0.20
0.14
Base value
0.75
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.42
P1
0.4
0.13
0.26
0.26
0.24
0.24
0.42
0.25
0.20
0.42
0.75
0.75
0.00
0.92
0.92
0.00
0.97
0.97
0.00
0.23
0.46
0.26
0.50
0.50
0.42
0.54
0.43
0.42
0.75
0.75
0.00
0.92
0.92
0.00
0.97
0.97
0.00
0.9
Base value
0.25
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.42
P1
0.1
0.17
0.34
0.26
0.40
0.40
0.42
0.45
0.36
0.42
0.25
0.25
0.00
0.08
0.08
0.00
0.03
0.03
0.00
0.23
0.46
0.26
0.44
0.44
0.42
0.46
0.37
0.42
0.25
0.25
0.00
0.08
0.08
0.00
0.03
0.03
0.00
0.42
S1
P2
0.4
S2
Base value
1
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
P1
0.01
0.30
0.59
0.26
0.48
0.48
0.42
0.48
0.39
0.42
0.49
0.49
0.00
0.15
0.15
0.00
0.06
0.06
0.00
S1
P2
10
S2
0.17
0.34
0.26
0.38
0.38
0.42
0.43
0.35
0.42
0.02
0.02
0.00
0.01
0.01
0.00
0.00
0.00
0.00
Base value
50
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.42
P1
25
0.20
0.40
0.35
0.45
0.45
0.59
0.48
0.39
0.59
0.00
0.00
0.67
0.12
0.12
0.83
0.12
0.12
0.83
0.20
0.40
0.17
0.37
0.37
0.25
0.40
0.32
0.25
0.00
0.00
0.33
0.12
0.12
0.41
0.12
0.12
0.41
S1
P2
100
S2
Base value
0.2
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.42
P1
0.4
0.15
0.30
0.23
0.30
0.30
0.35
0.33
0.26
0.35
0.25
0.25
0.13
0.28
0.28
0.17
0.28
0.28
0.17
0.10
0.20
0.17
0.18
0.18
0.24
0.20
0.16
0.24
0.25
0.25
0.17
0.28
0.28
0.21
0.28
0.28
0.21
0.42
S1
P2
0.6
S2
Base value
1
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
P1
0.1
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.42
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
0.42
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.42
S1
P2
6
S2
m
ZC
0.40
S2
pVIg
GGEN
P1
P2
MR
GGEC
0.40
S1
King
ZC
0.20
S2
KAE
GGEN
Base value
P2
AEmin
GGEC
0.20
S1
AEmax
ZC
0
S2
BR
GGEN
SNC: 0.25
1
S1
P2
SNC: 0.2
Base value
1
0.20
0.40
0.26
0.42
0.42
0.42
0.46
0.36
P1
0.4
0.20
0.40
0.12
0.30
0.30
0.13
0.32
0.26
0.13
0.00
0.00
0.92
0.49
0.49
1.14
0.49
0.49
1.14
0.20
0.40
0.50
0.47
0.47
0.90
0.51
0.41
0.90
0.00
0.00
0.92
0.11
0.11
1.14
0.11
0.11
1.14
S1
P2
S2
2
See Table II for definitions. In equation (22), Pn took the base value, while Pa took either the P1 or P2 values given in the table. S values derived using
equation (22) illustrated in bold face. See text for equation (22) for further explanations.
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Ra and Rn are the model responses for the altered and
nominal parameters, while Pa and Pn are the altered and
nominal parameters, respectively. Values of S = 0 indicate no change in model output (Ra) with changes in the
values of parameters (Pa), S = 1 indicates a pro rata change
(i.e. doubling in parameter value doubled the value of the
output variable), whereas S = –0.5 indicates a halving of
output for a doubling of the parameter value. Analysis
was conducted for prey of low (SNC = 0.1), medium (SNC
= 0.2) and high (SNC = 0.25) nutritional status compared
with the predator (for these tests, ZNC = 0.2).
Parameterization and tuning
The evolutionary search method supported by Powersim
Solver v2 (Isdalstø, Norway) was used to fit the model to
experimentally derived data. All the zooplankton-constant
parameters (Table II) were tuned simultaneously; these
constants were allowed to vary within the range given in
Table III. During the tunings of the model to micro- and
mesozooplankton data, the benefit of employing the foodquality link (FQ-link, Fig. 2) was tested by invoking VIg
and AE functions [equations (4) and (9), respectively] to
account for SMP at ingestion and assimilation (SMPIng
and SMPAE in Fig. 1, respectively). Disabling the FQ-links
gives outputs from the model with neutral modulation
(0 SMP in Fig. 1), analogous to the operation of previous
zooplankton models (i.e. with constant ingestion rate and
assimilation efficiency; e.g. Anderson, 1992; Anderson
and Hessen, 1995; Touratier et al., 1999; Sterner and
Elser, 2002; Anderson et al., 2005).
The model was fitted to experimental data for gross
growth efficiency of egg production (GGE) and growth
rate of the mesozooplankton Acartia tonsa (Kiørboe,
1989). Using the constant values obtained in these fits,
the model was then fitted to the egg production dataset of
the copepod Paracalanus parvus (Checkley, 1980) for validation. As both these studies supplied the copepods with
saturating amounts of different quality prey, the model
was configured to run to steady state.
The model was also configured for a dynamic system
and fitted to the experimental results of Flynn and
Davidson (Flynn and Davidson, 1993) describing microzooplankton—phytoplankton population dynamics. For
this simulation, the growth of the phytoplankton prey on
ammonium was described with a simple normalized
quota model (Flynn, 2003), yielding prey N:C (SNC) as
well as C-biomass (SC). Ammonium regenerated by the
predator was available for use by the prey. The experimental data (Flynn and Davidson, 1993) show that the
microzooplankton Oxyrrhis marina reverts to cannibalism
when the prey phytoplankton Isochrysis galbana is of poor
nutritional status (i.e. low SNC ). To enable this, the
prey-selectivity function of Fasham et al. (Fasham et al.,
1990) was incorporated [equation (6)] such that the predator was provided with the option of feeding on two
prey items, the phytoplankton I. galbana [SC1 in equation
(6)] and/or itself [i.e. cannibalism; SC2 in equation (6)].
The model was tuned to the dataset from Flask S and
validated against the dataset from Flask T (Flynn and
Davidson, 1993); the only difference between the two
experimental flasks was that the initial prey concentration in Flask T was double that in Flask S; all other
conditions were the same at the start of the experiments.
Accordingly, for the validation of the model to Flask T,
the values of the constant parameters obtained from the
tuning to Flask S were retained, with only the initial prey
and predator biomass being different.
RESULTS
Steady-state sensitivity analysis
Sensitivity analyses showed the model constants to be
robust to alterations of the values of these parameters
over a wide range (most giving –0.5 S 0.5; Table
III). The constants, which displayed greatest sensitivities,
were m and AEmax. However, these S values were wholly
in accordance with expectation; thus, doubling m (from 1
to 2) with prey of high nutritional status (SNC = 0.25)
doubled the growth rate (ZC = 0.42–0.9), and
when AEmax was halved, the GGEC and GGEN were also
halved.
Tuning and validation
For tuning of the model to data for copepod egg production [fitted to Kiørboe (1989) and validated against that
of Checkley (1980)], no obvious difference was found
between the model fits with the FQ-links switched on
or off (Fig. 4). With the FQ-links functioning, however,
the model displayed +ve SMP at ingestion level and –ve
SMP at assimilation level (note the values of a, KAE and
pVIg in Table IV).
The fit for the Isochrysis–Oxyrrhis dynamic system for
the dataset from Flask S (Flynn and Davidson, 1993)
required the inclusion of the FQ-links (Fig. 5; Table V).
However, this link was only required at the ingestion
level, displaying –ve SMPIng by decreasing the ingestion
rate for poor-quality prey (see also Fig. 1). The link was
not operational at the assimilation level, as indicated by
the low KAE values in Table V (see also Fig. 3B).
With FQ-links switched off, only 34% of the variation
in the phytoplankton biomass data (Fig. 5A) is explained
by the model output (R2 = 0.34), whereas with the
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GGEC
0.25
0.00
GGEN
0.50
0.25
0.00
ZCµ (d–1)
0.50
0.25
0.00
0.10
0.15
0.20
–1
SNC [gN (gC) ]
Fig. 4. Model outputs (lines) tuned to data (open circles; Kiørboe,
1989) for growth efficiencies (GGE) and egg production rate ( growth
rate; ZC) of Acartia tonsa. The growth efficiency dataset of Paracalanus
parvus of Checkley (closed circles; Checkley, 1980) is also shown. Thick
line, link to food quality (FQ-link) [equations (4) and (9)] operational;
thin line, FQ-link off. For the growth rate plots, the use of FQ-link had
no discernible effect on model outputs; line types overlap. See also
Table IV.
Table IV: Tuned values used for Fig. 4
Constants
Units
Tuned values in Fig. 4
FQ-link off
FQ-link on
0.505
a
dl
0a
AEmax
dl
0.496
0.499
AEmin
dl
{=AEmax}
0.400
BR
gC (gC)1 d1
0.196
0.039
KAE
dl
0.01a
0.107
MR
gC (gC)1
0.200
0.200
pVIg
dl
1a
3.501
m
gC (gC)1 d1
0.378
0.433
ZNC
gN (gC)1
0.200b
0.200b
dl, dimensionless. See Table II for definitions.
a
Fixed constant value during tuning to switch the FQ-links off.
b
Constant values taken from experimental dataset (see Fig. 4 legend).
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FQ-link operational this value was 91% (R2 = 0.91).
Furthermore, the sum of absolute deviations decreased
four-fold when the FQ-links were used (8.52 versus a
value of 32.76 without FQ-link); as the number of data
points for the comparisons is the same, the summed
deviations show the relative fit of the models to the
data. For the microzooplankton (Oxyrrhis) biomass data,
with the FQ-links operational, 90% of the variation in
data was explained by the model (R2 = 0.9). Validation of
the model with the FQ-links to Flask T showed that the
model tuned to Flask S was capable of simulating the
experimental outputs (Fig. 5B; Table V) for Isochrysis (R2 =
0.92) and Oxyrrhis (R2 = 0.94).
Figure 6 shows the steady-state relationships between
prey N:C GGEs and growth rate for the Flask S predator
configurations, with and without FQ-links. The impact
of –ve SMPIng (Fig. 1) in response to declining prey N:C
is clear in the model runs employing FQ-links; Isochrysis
SNC values of <0.07 cannot support growth of the predator. The GGEN plot for the model without FQ-links
shows the flat response expected with neutral stoichiometric modulation (0 SMP in Fig. 1; cf. GGEN in Fig. 3).
The absence of GGE values at low SNC (Fig. 6) is due to
low ingestion rates resulting in the predator (Oxyrrhis)
utilizing all ingested material for respiration.
0.50
0.05
28
DISCUSSION
The multi-nutrient, biomass-based generic zooplankton
model presented here is a development from a preliminary study (Mitra et al., 2003) in which it was shown that
including an effect of prey quality on ingestion [akin to
VIg here, equation (4)] and/or assimilation [akin to AE
here, equation (9)] could have a profound impact on the
predator–prey relationship. The abilities of the model to
describe micro- and mesozooplankton activities have
been demonstrated here (Figs 4 and 5).
It is important that the performance of any new model
should be compared with that of existing ones. Earlier
zooplankton stoichiometric models (e.g. Anderson, 1992;
Anderson and Hessen, 1995; Touratier et al., 1999) have
been fitted to data for egg production efficiency (GGE) of
A. tonsa (Kiørboe, 1989) and P. parvus (Checkley, 1980);
the new model was subjected to the same test (Fig. 4).
The new model provided good fits not only to the GGE
datasets but also to the growth rate data of Kiørboe
(Kiørboe, 1989). While these fits were achieved through
the implementation of SMP at both ingestion and assimilation levels (Table IV), the presence of the FQ-links
(invoking SMP) does not obviously improve the model
outputs compared with those when the FQ-links are
disabled (0 SMP in Figs 1 and 4). The fact that it appears
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A
B
SC (mg C L–1)
10
5
5
0
0
9
9
ZC (mg C L–1)
ZC (mgC L–1)
SC (mgC L–1)
10
6
3
0
6
3
0
0
5
10
15
20
0
Days
5
10
15
20
Days
Fig. 5. Model outputs tuned to the predator–prey interaction data for Oxyrrhis marina (ZC) feeding on Isochrysis galbana (SC). Data (symbols) for
tuning are from Flask S with validation against data from Flask T of Flynn and Davidson (Flynn and Davidson, 1993). (A) Model outputs tuned to
Flask S. (B) Model with the link to food quality (FQ-link) validated to Flask T. Thick lines, model outputs with FQ-links [equations (4) and (9)]
operational; thin lines without FQ-links. C-biomass as mgC L–1. See also Table V.
unnecessary to invoke SMP in the simulation of the
copepod egg production datasets of Checkley
(Checkley, 1980) and Kiørboe (Kiørboe, 1989) may
explain why the importance of SMP was unrecognized
in earlier zooplankton models that were tuned/validated
against these same data.
Whilst SMP has not been explicitly described previously in models, it has been shown to be important in
the real world. Responses of zooplankton to the presence
and/or ingestion of poor-quality food include changes in
filtration rate, ingestion rate and/or assimilation efficiency (Flynn and Davidson, 1993; DeMott et al., 1998;
Plath and Boersma, 2001; Jones et al., 2002;
Darchambeau and Thys, 2005; Jones and Flynn, 2005).
Hence, and contrary to the prediction given by traditional stoichiometric based zooplankton models, the GGE
for the limiting nutrient (here GGEN) need not be maintained constant or at a high value but may decline
significantly. DeMott et al. (DeMott et al., 1998) have
shown that the GGE of phosphorus in Daphnia is also
not constant but declines, being low at both very high
and low P:C. To consider such processes further, the new
model was tuned to GGE and growth rate data of populations of A. tonsa at the extremes of nutrient-replete and
nutrient-deplete prey (Fig. 7). The two data points in Fig.
7 represent the net results of many experiments (Jones
et al., 2002; Jones and Flynn, 2005) conducted with prey
of contrasting nutrient status, all reporting the same
observation namely that the ingestion of poor-quality
prey results in –ve SMP. Unless the FQ-links were
used, the model failed to describe the events seen experimentally. Thus, the inclusion of the FQ-links (Fig. 7)
invoking –ve SMP at both ingestion and assimilation
levels showed the expected decline not only in GGEC
but also in GGEN. In contrast, without the FQ-links, the
model overestimated the GGEC (Fig. 7A) and growth rate
values (Fig. 7C) when consuming nutritionally deplete
prey and underestimated these values when the copepod
(Acartia) was supplied with nutritionally replete prey.
While more data are required to fully parameterize
the model, there is no doubt that the consumption of
poor-quality food impacts on ingestion rates and assimilation efficiencies not only in copepods (Jones et al.,
2002; Jones and Flynn, 2005) but also in other animals
(see review by Yearsley et al., 2001). It may also be
noteworthy that while SMP was not required for models
fitted to the egg production data (Fig. 4), SMP was
essential for the comparison with population growth
data (Fig. 7). This may be due to the gross differences
between the processes of egg production and whole
population growth dynamics. This difference may also
warn against extrapolating data between such different
physiological processes; population growth models
should ideally be parameterized with data from population growth experiments.
The need to account for SMP also becomes apparent
when the model is tuned to the phytoplankton–microzooplankton population dynamics (Flynn and Davidson,
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Validated to Flask T
Fig. 5A:
Fig. 5B: FQ-link on
GGEC
Tuned to Flask S
Fig. 5A:
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0.8
Table V: Tuned values used for Fig. 5
Constants Units
j
0.4
FQ-link off FQ-link on
a
dl
0a
1.000
b
SCm
gC (gC)1 d1
0.359
0.402
b
SCIni
mgC L1
1.197
1.2
1.514
AEmax
dl
0.900
0.900
b
AEmin
dl
{=AEmax}
0.372
b
BR
gC (gC)1 d1
0.020
0.057
b
KAE
dl
0.01a
0.010
b
KIng
mgC L1
1.000
0.371
b
MR
gC (gC)1
0.200
0.230
b
0.0
P1
dl
0.450
1.000
b
0.6
pVIg
dl
1a
3.453
b
m
gC (gC)1 d1
0.600
0.647
b
ZCIni
mgC L1
0.092
0.076
0.120
ZNC
gN (gC)1
0.150
0.150
b
0.0
ZCµ (d–1)
GGEN
0.8
dl, dimensionless. See Table II for definitions.
The model was tuned or validated as indicated to the data for Flasks S
and T from Flynn and Davidson (Flynn and Davidson, 1993). For Flask
T, only the start prey and predator biomass values were tuned.
a
Fixed constant value during tuning to switch FQ-link off.
b
Other values were as for ‘Fig. 5A FQ-link on’.
1993). This is consistent with the findings of Davidson
et al. (Davidson et al., 1995b), whose simple predator–
prey model could not explain the experimental results for
the interaction between O. marina and I. galbana. With the
inclusion of FQ-links, the new model correctly predicts
(using –ve SMPIng) the development of a residual Nstarved (low N:C) algal (Isochrysis) bloom as seen in the
experimental Flask S (Fig. 5A) with the invocation of the
processes shown in Fig. 6. In Flask T, the algae do not
become N-stressed due to nutrient recycling, and the
traditional predator–prey cycle is seen (Fig. 5B). A biological explanation for the development of –ve SMPIng
leading to prey rejection in Flask S is thought to be the
accumulation of secondary metabolites (toxins) in nutrient-stressed algal cells; Oxyrrhis invariably rejects
N-deprived Isochrysis (Flynn et al., 1996). Although the
predictions of the initial phases of the predator–prey
interaction are correct using the model without FQlinks (i.e. 0 SMP, Fig. 5A), the latter part of the simulation is not consistent with observations. Hence, with 0
SMP (a situation akin to the operation of traditional
stoichiometric models), the model incorrectly predicts
the demise of the algal bloom. Within an ecosystem
model, this failing could have major ramifications, such
as the inability to predict harmful algal bloom events.
0.4
0.3
0.0
0.05
0.10
0.15
0.20
SNC [gN (gC) ]
–1
Fig. 6. Steady-state model outputs for growth efficiencies (GGE) and
growth rates (ZC) for Oxyrrhis marina as tuned to the data shown in
Fig. 5A. Thick line, link to food quality (FQ-link) [equations (4) and (9)]
operational; thin line, FQ-links off.
The model described here, however, has been used successfully to simulate predator–prey interactions leading
to bloom formation (Mitra and Flynn, 2006).
It should be noted that the decline in zooplankton
growth when consuming prey of low N:C (Fig. 6) is not
a phenomenon associated with the so-called hard threshold element ratio (Sterner and Elser, 2002) but is specifically related to the prey (Isochrysis) becoming
unpalatable and hence not being ingested. That the
event is not seen when O. marina is fed with the alga
Dunaliella primolecta even when D. primolecta N:C is lower
than that of I. galbana (K. J. Flynn, Swansea, personal
communication) also indicates the importance of developing prey-selectivity functions linked to prey nutritional
quality as well as quantity. Thus, when feeding on a
mixed population of prey, the predator could display
+ve SMPIng towards one prey type and –ve SMPIng
towards another type.
While the best fits of the model to the microzooplankton experimental results were achieved through the
implementation of SMP at only the level of ingestion
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MODELLING ZOOPLANKTON RESPONSES TO PREY QUALITY
GGEC
0.15
–ve SMPIng, –ve SMPAE
0.10
0 SMP
0.05
0.00
0.15
GGEN
0 SMP
0.10
–ve SMPIng, –ve SMPAE
0.05
0.00
0.20
ZCµ (d–1)
0.15
0.10
0 SMP
–ve SMPIng, –ve SMPAE
0.05
0.00
0.05
0.10
0.15
SNC [gN (gC)–1]
Fig. 7. Model outputs (lines) compared with the data for growth
efficiencies (GGE) and growth rates (ZC) for populations of Acartia
tonsa (see text for further description). Thick line, link to food quality
(FQ-link) [equations (4) and (9)] operational; thin line, FQ-links off.
With FQ-links switched on, model uses –ve SMPIng (a = –1; pVIg = 0.84)
and –ve SMPAE (KAE = 8.91; AEmax = 0.26; AEmin = 0.01) to achieve
the fits.
(Fig. 5; Table V), SMP may be required at both ingestion and assimilation levels for the description of the
type of copepod behaviour shown by Jones and Flynn
(Jones and Flynn, 2005) in which not only poor-quality
prey are discriminated against but assimilation efficiency of what is ingested declines remarkably. It may
be more cost-effective for protists, which lack a gut that
can be voided rapidly and hence display a relatively
long digestive period (Öpik and Flynn, 1989), to be
most selective at the point of prey capture. In comparison, with the relatively short gut passage time of copepods, minimizing exposure to toxic or unpalatable
secondary metabolites is achievable by shortening the
gut passage time further; this affects assimilation efficiency (Paffenhöfer and
Van
Sant, 1985;
Darchambeau, 2005). In the model presented here,
while gut passage time is not simulated explicitly, the
consequential impact of food quality on AE is included
[via equation (9)]. There are also important interactions
between food quantity and quality on assimilation
efficiency operating via changes in gut passage time
(Jumars, 2000a,b; Tirelli and Mayzaud, 2005) which
may warrant further examination. Another factor
which could be more explicitly described but awaits
further parameterization is the relative contribution of
dissolved versus particulate material currently described
as ‘voided’ in equations (15) and (17); excess C may also
be voided through respiration as CO2 (Darchambeau
et al., 2003). The knowledge of such partitioning of
voided material is important because of the trophic
implications (primarily osmotrophy by microbes using
dissolved forms versus phagotrophy/ingestion of particulates by zooplankton).
Unfortunately, there are very few studies describing
the effect of variation in prey species nutritional status on
predator dynamics that yield data suitable for modelling.
Furthermore, typically ecologists have defined quality as
differences in species composition (e.g. Paffenhöfer,
1976; Mayzaud et al., 1998) and not as prey nutritional
status. For the purpose of the model described here, prey
quality has been described as the variation in the stoichiometric ratio within a prey species as given by the
quotient R [equation (3)]. However, R could be made a
function of the species composition available such that
R = 0 would indicate that only inedible or indigestible
species are available (e.g. filamentous or colonial forms;
Genkai-Kato, 2004), while R = 1 would indicate the
presence of optimal sized prey for ingestion. R could
also be related directly, rather than indirectly as in this
study, to the presence of unpalatable material (e.g. toxins
and mucus) in the prey that causes rejection by the
predator; for Figs 5 and 6, algal nutrient status was
used as an index for secondary metabolite content
which commonly accumulates in N- or P-stressed phytoplankton (Granéli et al., 1998).
In conclusion, this study describes a generic multinutrient zooplankton model whose operation demonstrates the importance of zooplankton behavioural
processes. These processes, such as the alteration of
ingestion rates and assimilation efficiencies with variation in food quality, have hitherto only been documented with poor parameterization with little or no explicit
description in previous models. There is a clear need for
more data linking prey quality as well as quantity to
zooplankton growth dynamics. This model represents a
base on which to develop research further on this topic.
ACKNOWLEDGEMENTS
This work was supported by the Natural Environment
Research Council (UK). The author thanks Kevin
J. Flynn and Paul Tett for commenting on previous
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versions of this manuscript and four anonymous
reviewers for their comments and time.
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