vol. 169, no. 6
the american naturalist
june 2007
E-Article
Paradoxes of Enrichment: Effects of Increased Light versus
Nutrient Supply on Pelagic Producer-Grazer Systems
Sebastian Diehl*
Department Biologie II, Ludwig-Maximilians-Universität
München, Grosshaderner Strasse 2, D-82152 Planegg-Martinsried,
Germany
Submitted December 13, 2006; Accepted January 12, 2007;
Electronically published April 19, 2007
abstract: Energy-based plant-herbivore models produce the “paradox of enrichment,” a destabilizing influence of enrichment on
population dynamics. Because many plants change their carbon : nutrient stoichiometry in response to the light : nutrient supply ratio,
enrichment with light can cause a mismatch between the elemental
compositions of plants and their herbivores. Herbivore growth rates
may then decrease with increased light supply, which is termed the
“paradox of energy enrichment.” I present a stoichiometric phytoplankton-grazer model that accounts for the dynamical vertical light
gradient and explore how algal and grazer densities, mineral nutrient
concentration, algal nutrient stoichiometry, and system stability respond to enrichment with light (through changes in irradiance, background turbidity, and water column depth) versus enrichment with
nutrients. Parameterized for Daphnia, the model produces several
“unusual” phenomena: multiple equilibria (with grazers extinct in
spite of high algal biomass at one equilibrium), inconsistent light
enrichment effects on stability (light enrichment first destabilizes and
then stabilizes), and the paradox of energy enrichment. These phenomena are restricted to the low end of realistic nutrient supplies
except in very shallow systems, where high sedimentation rates effectively deplete the water column of nutrients. At higher nutrient
supplies, light enrichment produces the classical paradox of enrichment, leading first to an increase in grazers at a stable equilibrium
and then to algae-grazer oscillations.
Keywords: carbon to phosphorus ratio, Daphnia, ecological stoichiometry, mixing depth, multiple stable states, paradox of energy
enrichment.
The ecological literature abounds with studies of enrichment. The term refers to any increase in supply of a
production-limiting resource, usually to a guild of basal
* E-mail: [email protected].
Am. Nat. 2007. Vol. 169, pp. E173–E191. 䉷 2007 by The University of
Chicago. 0003-0147/2007/16906-42276$15.00. All rights reserved.
species such as primary producers or detritivores. Empirical and theoretical evidence suggests that community
properties, such as the distribution of biomass among
trophic levels, the outcome of competitive interactions,
the stability of consumer-resource interactions, and species
diversity, are strongly affected by enrichment (Oksanen et
al. 1981; Tilman 1987; DeAngelis 1992; Abrams 1993; Rosenzweig and Abramsky 1993; Abrams and Roth 1994;
Leibold et al. 1997; Bohannan and Lenski 1999; Diehl and
Feißel 2000). Central to an understanding of enrichment
effects on ecosystems is an understanding of the dynamics
of its basic building blocks, that is, consumer-resource
interactions. Many population models predict that enrichment will destabilize consumer-resource dynamics and
produce large-amplitude oscillations where populations go
through bottlenecks of very low density (reviewed by Murdoch et al. [2003]), a phenomenon that has been confirmed in laboratory studies of simple predator-prey systems (Luckinbill 1974; Bohannan and Lenski 1997;
McCauley et al. 1999; Fussmann et al. 2000; Holyoak
2000). Rosenzweig (1971) pointed out that such bottlenecks put populations dangerously close to extinction and
coined the term “paradox of enrichment” to describe this
harmful influence of enrichment on population persistence.
Most consumer-resource models producing the paradox
of enrichment share three assumptions: (1) the dynamics
of the resource species are described by a single state variable (e.g., carbon or nutrient biomass), (2) enrichment
is quantified in a single currency (e.g., the carrying capacity
of the resource species or the supply with a single limiting
substrate), and (3) the efficiency with which consumed
resources are converted into consumer biomass is constant
and independent of population densities (Rosenzweig
1971; Armstrong 1976; Abrams and Roth 1994; Bohannan
and Lenski 1997; Fussmann et al. 2000). Collectively, these
assumptions imply that the biochemical composition of
the resource species is fixed and that the growth rate of
the resource species is limited by a single substrate.
Assumptions 1–3 are frequently violated in systems
where the resource is a primary producer, because many
E174 The American Naturalist
plants change both their growth rate and their biochemical
composition (e.g., their carbon : nutrient stoichiometry)
in response to changes in supply of multiple abiotic resources, such as light, carbon dioxide, and various nutrients (Rhee and Gotham 1981; Bridgham et al. 1995; Burkhardt and Riebesell 1997; Sterner et al. 1997; Curtis and
Wang 1998; Diehl et al. 2002). Consequently, primary production can often be stimulated with more than one abiotic resource, but enrichment with the “wrong” abiotic
resource may cause a stoichiometric mismatch between
the biochemical composition of producers and the nutritional requirements of their herbivores (reviewed by Andersen et al. [2004]). A well-documented example involves
freshwater zooplankton of the genus Daphnia feeding on
planktonic algae. Here an increase in light supply (enrichment with light) may decrease grazer production if the
benefits of increased food quantity (increased algal carbon
fixation) are outweighed by decreased food quality (decreased content of essential elements per unit algal carbon)
and, thus, decreased conversion efficiency of algal into
Daphnia biomass (Urabe and Sterner 1996; Sterner et al.
1998; Urabe et al. 2002b).
Flexible carbon : nutrient stoichiometry of plants has
long been recognized (Droop 1974; Vitousek 1982), but
not until recently has it been included in producer-grazer
models (Andersen 1997; Loladze et al. 2000, 2004; Muller
et al. 2001; Hall 2004). Common to these models are the
(empirically supported) assumptions that herbivores show
very little flexibility in the biochemical composition of
their biomass and that, consequently, the conversion efficiency of consumed plants into grazer biomass is a function of the concentrations of limiting elements in plant
tissue. Andersen (1997) was the first to recognize that such
stoichiometric constraints add an element of nutrient
competition to plant-herbivore interactions: an increase in
herbivore biomass may decrease plant production, because
limiting elements sequestered by herbivores are unavailable to plants; conversely, an increase in plant carbon biomass (e.g., in response to enrichment with light) may dilute the concentration of a limiting nutrient in plant tissue
so much that the nutrient intake rate by the herbivore and,
consequently, the herbivore’s growth rate go down as plant
carbon biomass increases. The latter phenomenon parallels
the above-mentioned responses of Daphnia to increased
algal light supply and has been termed the “paradox of
energy enrichment” (Loladze et al. 2000).
Interestingly, all of the numerical examples in the cited
stoichiometric producer-grazer models have been parameterized to mimic a phytoplankton-Daphnia system. Rigorous empirical tests of model predictions are, however,
lacking. It is thus unclear under which environmental conditions real producer-grazer systems might exhibit the paradox of energy enrichment. Probably the most relevant
conditions are the ones determining the abiotic resource
environment. Resource supply and resource dynamics,
however, are not well represented in existing models. For
example, all models make the unrealistic assumptions that
nutrient mineralization from detritus is instantaneous and
that mineralized nutrients are immediately taken up by
algae regardless of how nutrient replete these algae are.
Moreover, light has been either completely ignored (Andersen 1997; Muller et al. 2001), emulated by a (lightlimited) algal carrying capacity (Loladze et al. 2000, 2004),
or treated as a static input without feedback from algal
self-shading (Hall 2004). Aquatic systems, however, are
characterized by a dynamical vertical light gradient, which
is set by external input, water column depth, and attenuation by algae and nonalgal components (the latter being
termed “background attenuation” [Huisman and Weissing
1994]). As a consequence, there are three ways of enriching
an aquatic system with light—increased irradiation, decreased water column depth, and decreased background
attenuation—each of which affects the dynamics of algae,
nutrients, and light in a different way (see Diehl 2002 for
a detailed discussion).
In this article, I explore theoretically the resource supply
space in which phytoplankton-grazer systems show the
paradox of energy enrichment and other dynamical behaviors. I present a stoichiometric phytoplankton-grazer
model (parameterized for Daphnia) that explicitly accounts for the vertical light gradient and nutrient recycling.
I use a simplified version of the model to gain an understanding of how algal and grazer biomass, mineral nutrient
concentration, algal nutrient stoichiometry, and system
stability are affected by enrichment with light (through
changes in irradiance, background turbidity, and water
column depth) versus enrichment with nutrients. Of these
factors, water column depth has the most complex influence, because decreasing water column depth enhances
specific algal production (enrichment with light) but also
increases the removal rate of nutrients from the water
column through sedimentation (depletion of nutrients). I
therefore explore the influence of water column depth on
algae-grazer dynamics in greater detail by including algal
sinking in the model and considering the consequences of
nutrient recycling through a pool of sedimented nutrients.
Model Structure
The model describes a grazer population with fixed carbon : nutrient stoichiometry feeding on a phytoplankton
population with flexible carbon : nutrient stoichiometry in
a well-mixed water column. It assumes that phytoplankton
and grazer production can be limited by energy (as light
and food carbon, respectively) and a single mineral nutrient and describes the dynamics of phytoplankton, graz-
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
ers, light, and various nutrient compartments. The system
is assumed to be closed for nutrients. Thus, nutrient supply
to the mixed water column originates exclusively from
excretion and remineralization of biomass. The total
amount of nutrients in the system (Rtot) is then a measure
of nutrient enrichment. Inorganic carbon is assumed to
never limit phytoplankton production. Carbon supply and
carbon recycling are therefore not considered. Four differential and two algebraic equations determine the dynamics of six state variables, that is, the concentrations of
algal (A) and grazer (G) carbon biomasses and of dissolved
mineral nutrients (R) in a mixed water column of depth
z, the pool of sedimented (detrital) nutrients (Rs), the
nutrient content (quota) per algal carbon biomass (Q),
and the light intensity (I(s)) at depth s of the water column:
冕
z
dA
A
p
dt
z
v
p(I(s), Q) ds ⫺ l m A ⫺ A ⫺ JAG,
z
(1a)
0
dG
p [g ⫺ (m ⫹ d)]G,
dt
(1b)
dR s
v
p Q Az ⫹ qdGz ⫺ rR s ,
dt
z
(1c)
dR
r
p R s ⫹ qmG ⫹ (QJA ⫺ qg)G ⫺ r(Q, R)A,
dt
z
Qp
R tot ⫺ qG ⫺ R ⫺ R s /z
,
A
I(s) p I ine⫺(kAs⫹K bg s).
(1d)
(1e)
(1f)
The definitions and units of all variables and parameters
are listed in table 1. Grazers are assumed to have a fixed
nutrient content q per unit carbon biomass and to use
“synthesizing units” (Kooijman 2000) to produce new biomass from assimilated food, which gives the specific biomass production rate of grazers as
gp
cJA
,
(cJA/g max) ⫹ (cq/Q) ⫹ [Q/(cq ⫹ Q)]
(2)
where JA is the specific ingestion rate of algal carbon per
grazer biomass, c the (fixed) fraction of ingested carbon
that is assimilated, and gmax the maximum specific biomass
synthesis rate. Equation (2) implicitly assumes that nutrient assimilation is 100%. The quantity JA is a saturating
functional response,
JA p
fmax A
,
Ks ⫹ A
(3)
E175
with maximum specific ingestion rate fmax and half-saturation constant Ks. The synthesizing-unit (SU) concept
developed by Kooijman (1998, 2000) assumes that the rate
of biosynthesis is limited by the arrival rates of multiple
substrates (here carbon and nutrient atoms from assimilated food) to the binding sites of biosynthesis units and
thus describes growth as a smooth function of multiple
limiting factors (see Muller et al. 2001 for a fuller description in the context of a Daphnia model). Following Muller
et al. (2001), I assume that the theoretical maximum rate
of biosynthesis is much larger than the rate of food ingestion (g max k JA), so the first term in the denominator
of equation (2) is set to 0. It is then easily seen that the
SU converges to Liebig’s minimum rule if the nutrient
content in algae is either much higher (Q k q) or much
lower (Q K q) than that in grazers. In the first case, equation (2) collapses to g p cJA and grazer growth is limited
only by ingested carbon. In the second case, equation (2)
simplifies to g p (Q/q)JA and grazer growth is limited only
by ingested nutrients. When algal and grazer nutrient contents are more similar, grazer growth is colimited by the
ingestion of both carbon and nutrients. Equation (1b),
furthermore, assumes that energetic costs of food processing and biosynthesis are a constant fraction of ingested
carbon that is included in the carbon assimilation efficiency c. Finally, grazer biomass is lost through death at
rate d and metabolism at rate m. To keep nutrient content
per grazer biomass at the fixed level q, metabolic losses
are accompanied by the excretion of nutrients at rate qm
and excess nutrients—that is, the difference between ingested nutrients and nutrients incorporated into new biomass (QJA ⫺ qg)—are excreted (eq. [1d]). The latter conforms with the observation that nutrient excretion by
grazers decreases with decreasing algal nutrient content
(Elser and Urabe 1999).
Algal growth is assumed to be colimited by internal
nutrient stores and light. Following empirical evidence
(Senft 1978) and to comply with earlier work (Diehl 2002;
Diehl et al. 2005; Berger et al. 2006), I describe algal growth
as a multiplicative function of these two resources rather
than with an SU or a minimum rule (see also Huisman
and Weissing 1995; Gurney and Nisbet 1998). Specific algal
growth rate (p) is thus an increasing, saturating function
of light intensity and algal nutrient quota described by
p(I, Q) p pmax
(
)
I
Q min
1⫺
,
I⫹H
Q
(4a)
which, averaged over a mixed water column of depth z,
yields
E176 The American Naturalist
Table 1: Definitions and units of parameters and state variables and basic set of parameter values
Symbol
A
c
d
fmax
g
gmax
G
H
I
Iin
Iout
JA
k
Kbg
Ks
lm
m
M
p
pmax
q
Q
Qmin
Qmax
r
rmax
r
R
Rs
Rtot
s
v
z
冕
Definition
Value and/or units
Algal carbon density
Fraction of ingested carbon assimilated by grazer
Specific grazer death rate
Maximum ingestion rate of algae by grazer
Specific grazer growth rate
Maximum specific biosynthesis rate
Grazer carbon density
Half-saturation constant for light-dependent algal production
Light intensity (photon flux)
Light intensity at surface
Light intensity at bottom of mixed layer (depth z)
Ingestion rate per grazer
Specific light attenuation coefficient of algal biomass
Background light attenuation coefficient
Half-saturation constant for ingestion of algae by grazers
Specific algal maintenance respiration rate
Specific grazer maintenance respiration rate
Half-saturation constant for nutrient uptake
Specific production rate of algae
Maximum specific production rate of algae
Grazer nutrient quota
Algal nutrient quota
Algal minimum nutrient quota
Algal maximum nutrient quota
Specific algal nutrient uptake rate
Maximum specific algal nutrient uptake rate
Specific mineralization/recycling rate of sedimented nutrients
Dissolved mineral nutrient concentration
Pool of sedimented (detrital) nutrients
Total nutrients in the system (expressed per volume, assuming
resuspension of sedimented nutrients)
Depth below water surface
Algal sinking velocity
Depth of mixed layer
z
1
z
1 pmax
H ⫹ I in
Q
p(I(s), Q) ds p
ln
1 ⫺ min .
z kA ⫹ K bg H ⫹ I out
Q
(
)(
)
0
(4b)
Here, pmax is the maximum specific production rate, Qmin
the algal nutrient quota at which growth ceases, k the
specific light attenuation coefficient per unit algal biomass,
Kbg the background light attenuation coefficient (describing attenuation by nonalgal components), H the halfsaturation constant of light-dependent production, and Iin
the (constant) light intensity at the water surface. Vertical
light attenuation is described by Lambert-Beer’s law. The
light intensity at the bottom of the mixed water column,
Iout (Huisman and Weissing 1994), is then defined by equation (1f) for s p z. Light attenuation by grazers is assumed
mmol C m⫺3
.5
.03 day⫺1
1.0 day⫺1
day⫺1
day⫺1
mmol C m⫺3
120 mmol photons m⫺2 s⫺1
mmol photons m⫺2 s⫺1
300 mmol photons m⫺2 s⫺1
mmol photons m⫺2 s⫺1
day⫺1
.0036 m2 mmol C⫺1
.25 m⫺1
13 mmol C m⫺3
.1 day⫺1
.09 day⫺1
.05 mmol P m⫺3
day⫺1
day⫺1
.0125 mol P mol C⫺1
mol P mol C⫺1
.00154 mol P mol C⫺1
.0154 mol P mol C⫺1
mol P mol C⫺1 day⫺1
.08 mol P mol C⫺1 day⫺1
.05 day⫺1
mmol P m⫺3
mmol P m⫺2
mmol P m⫺3
m
.1 m day⫺1
m
to be negligible. Algae are assumed to respire carbon for
maintenance at a rate lm and to settle out of the mixed
water column at a rate proportional to their sinking velocity v and inversely proportional to mixing depth z (Ptacnik et al. 2003).
Algae that have settled out of the water column turn
into detritus and enter the sedimented nutrient pool. I
assume that dead grazers sink so fast that nutrients in dead
grazers immediately become part of the sedimented nutrient pool (eq. [1c]). Sedimented nutrients are mineralized and released back into the water column at rate r.
Nutrients excreted by grazers go directly into the pool of
dissolved mineral nutrients (eq. [1d]). Finally, following
Andersen (1997), algal nutrient uptake is assumed to be
a decreasing function of internal algal nutrient stores and
an increasing function of the external nutrient concentration:
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
r(Q, R) p rmax
Q max ⫺ Q
R
.
Q max ⫺ Q min M ⫹ R
(
)
(5)
Here, rmax is the maximum specific algal nutrient uptake
rate, M the half-saturation constant of nutrient uptake,
and Qmax the algal nutrient quota at which uptake ceases.
Algal nutrient quota Q increases through nutrient uptake
and carbon respiration and decreases through growth as
z
dQ/dt p r(Q, R) ⫹ l mQ ⫺ (Q/z) ∫0 p(I(s), Q). Because the
system is closed for nutrients, Q is, however, more conveniently calculated using mass balance (total nutrients
must add up to Rtot; eq. [1e]). The term in parentheses in
equation (5) implies that nutrient uptake approaches 0
when the quota approaches Qmax and thus ensures that the
quota is bounded by Q ! Q max (Qmax is reached asymptotically when algal production and biomass are strongly
limited by light). This assumption on nutrient uptake is
not critical to the results. The system behavior is very
similar when nutrient uptake is independent of Q (i.e.,
when [Q max ⫺ Q]/[Q max ⫺ Q min] is deleted from eq. [5]; see
app. A).
Results
Below, I explore how light and nutrient supply influence
algae and grazer dynamics. The system is not analytically
tractable, and so I rely on numerical simulations, using
realistic parameter values. Because the growth rates of algae
and grazers are set by the availability of both light energy
(directly or fixed as organic carbon) and nutrients, the
system’s responses to enrichment with light strongly depend on the level of nutrient enrichment. I therefore explore responses to changes in light supply over a range of
total nutrient contents (Rtot) while keeping most other
parameters constant. Parameter values (listed in table 1)
roughly match environmental conditions (Rtot, Iin, and Kbg)
in a nutrient-poor, temperate lake in late summer. The
potentially growth-limiting nutrient is assumed to be
phosphorus. Algal parameter values reflect average traits
(see chap. 3.5 of Andersen 1997), most of them being
identical to the ones used in Diehl et al. (2005). To characterize the grazer, I took parameters values for Daphnia
pulex from McCauley et al. (1996) and Nisbet et al. (1997).
Simulations with a very different set of grazer parameter
values for Daphnia galeata (from the same sources) yielded
qualitatively similar results and are therefore not shown.
Isocline Analysis of a Simplified System
Much insight can be gained from the exploration of a
structurally simpler limiting case of the full model. This
simplified model shows all types of algae-grazer dynamics
E177
exhibited by the full system, but its behavior is much easier
to visualize and understand. I therefore introduce and analyze the simplified model before I explore the full model.
When the recycling rate r of sedimented nutrients and
the maximum algal nutrient uptake rate rmax are set to
infinity, the pool of detrital nutrients approaches 0
(R s p 0) and the concentration of dissolved mineral nutrients is measurably above 0 only when algae are nutrient
replete (i.e., Q approaches Qmax). In the limit, this corresponds to a system where detrital nutrients are instantly
mineralized and all mineral nutrients are immediately
taken up by algae as long as Q ! Q max. In addition, I initially assume that algae are mobile or neutrally buoyant
and thus do not experience sinking losses. Under these
assumptions, the system simplifies to a two-dimensional
set of differential equations describing the dynamics of
algae (eqq. [1a], [4] with v p 0) and grazers (eqq. [1b],
[2], [3]), the remaining state variables being determined
by the algebraic equation (1f) (for I(s)) and
(
Q p min Q max,
R tot ⫺ qG
,
A
R p R tot ⫺ QA ⫺ qG.
)
(6a)
(6b)
The dynamics of this system can be understood from
an analysis of zero-net-growth isoclines plotted against the
carbon densities of algae and grazers (fig. 1). For a given
nutrient level Rtot, water column depth z, and background
turbidity Kbg, any point on this plot not only describes the
state of the system in terms of the carbon densities of algae
(A) and grazers (G) but also provides full information on
the algal nutrient quota (Q), the concentration of mineral
nutrients (R), and the light gradient (Iout). System dynamics can be projected from the current state of the system
relative to the isoclines, which show the combinations of
algal and grazer densities for which algae (algal isocline)
and grazers (grazer isocline) do not change over time.
Because the nutrient quota in grazer biomass is fixed
at q, algal nutrient quota is uniquely defined by equation
(6a). At a given Rtot, solving equation (6a) for G at fixed
levels of Q produces straight isolines of equal algal nutrient
quota Q that fan out from the upper left corner (A p 0,
G p 12 in fig. 1), with isolines farther away from the
origin indicating lower algal nutrient quota. The uppermost isoline is the system’s mass balance constraint, given
by G p (R tot ⫺ Q min A)/q. A given amount of nutrients Rtot
can yield a combination of algal and grazer carbon densities only on or below this line. The lowest isoline (given
by G p [R tot ⫺ Q max A]/q) is the upper bound of a region
where mineral nutrients are present (R 1 0). Above this
line, all nutrients are locked up in biomass and the mineral
nutrient concentration is 0. Below this line, algal nutrient
E178 The American Naturalist
Figure 1: Isocline plot in algae-grazer state space of the simplified algaegrazer system. Red, grazer zero-net-growth isocline (grazer biomass increases below and decreases to the left of and above the isocline). Green,
algal isocline (algal biomass increases below and decreases above the
isocline). Also shown are lines of equal algal nutrient quota Q. Upper
thick black line, Q p Qmin (molar C : P p 650 ); lower thick black line,
Q p Qmax (molar C : P p 65); thin black line, Q p 0.0033 mol P mol
C⫺1 (molar C : P p 300). Gray shading indicates combinations of algal
and grazer biomass that are impossible because of the mass balance
constraint imposed by the limiting nutrient. Light blue shading indicates
combinations of algal and grazer carbon biomass for which dissolved
mineral nutrients R are present. Blue diagonal lines are isolines of equal
R, with R increasing toward the origin. Parameter values are Rtot p
0.15 mM P; v p 0 m day⫺1; z p 3 m. The remaining parameters are as
in table 1. Upper horizontal axis scales in units of photosynthetically
active radiation at the bottom of the mixed water column Iout.
quota is at Qmax and algae cannot sequester more nutrients.
Consequently, the concentration of mineral nutrients increases toward the origin (fig. 1), where fewer and fewer
nutrients are bound up in biomass (eq. [6b]).
Because algal and grazer growth depend on algal nutrient quota, the zero-net-growth isoclines differ from the
familiar isoclines of purely energy-limited resource-consumer systems. Most conspicuously, the grazer isocline is
a straight vertical line only where Q p Q max. In this region,
Q is constant and specific grazer growth rate depends solely
on the ingestion rate of algal carbon (eq. [2]). In contrast,
in the region where Q ! Q max, the grazer isocline has an
initially steep, positive slope but bends over to the right;
the slope becomes negative at higher algal carbon densities.
This phenomenon has been described in previous work
and is a consequence of increasing nutrient limitation of
the grazer (Andersen 1997; Loladze et al. 2000; Muller et
al. 2001; Diehl 2003). In this region, the algal nutrient
quota (food quality) and, consequently, the conversion
efficiency and specific growth rate of grazers all decrease
in the vertical direction as a higher proportion of total
nutrients is locked up in grazers (eq. [6a]). Grazers must
then increase their ingestion rate to balance their losses,
which requires a higher algal carbon density and thus explains the positive slope of the grazer isocline. At sufficiently high algal densities, algal nutrient quota may become so low that the grazer isocline bends downward and
may even intersect the A-axis (fig. 1). Above and to the
right of a downward-sloping grazer isocline, algal quality
in terms of Q is so low that grazer net population growth
is negative irrespective of algal carbon density.
The algal isocline has either an entirely negative slope
or a unimodal shape (figs. 1, 2). The slope of the isocline
changes (sometimes abruptly) at the transition from fixed
to flexible algal nutrient quota (isoline Q p Q max) because
of a shift in algal self-limitation. To the right of the isoline
Q p Q max, algal density limits specific algal production
through both self-shading and decreased nutrient quota,
whereas to the left of the isoline, the quota is constant
and only self-shading occurs (eq. [4]). Unlike the grazer
isocline, the algal isocline can never cross the mass balance
constraint line, because algal production goes to 0 at
Q p Q min (eq. [4]).
Equilibrium Patterns of the Simplified System
Intersections of the isoclines mark equilibrium states with
both algae and grazers present. In the following, I describe
how enrichments with nutrients and light interact to influence equilibrium biomass patterns and the stability of
the algae-Daphnia system. I first explore how enrichment
with light affects the positions of the algal and grazer isoclines relative to each other at three levels of nutrient
enrichment (Rtot) and plot the resulting biomass patterns
as functions of light supply. I then summarize the results
in bifurcation graphs of asymptotic system behavior plotted against nutrient and light enrichment parameters.
Because grazer growth and death rates are independent
of light, enrichment with light does not affect the grazer
isocline. In contrast, because grazer growth depends on
algal nutrient quota (which depends on Rtot; eq. [6]), nutrient enrichment moves the grazer isocline up. To understand exactly how nutrient enrichment affects the
grazer isocline, we need to consider how nutrient enrichment affects the isolines of equal Q. Nutrient enrichment
moves these isolines up in parallel along the G-axis, with
two effects. First, the isoline Q p Q max intersects the Aaxis at increasingly higher algal densities. Consequently,
beyond a threshold level of Rtot, an increasingly larger part
of the grazer isocline comes to lie in the region Q p
Q max, where the isocline is a vertical straight line (see top
panels in fig. 2). Second, the part of the grazer isocline
that lies in the region of flexible algal nutrient quota
(Q min ! Q ! Q max) also moves up in parallel with the iso-
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
E179
Figure 2: Effects of enrichment with nutrients (increasing total nutrients Rtot) and light (decreasing mixed water column depth z) on the simplified
algae-grazer system. Columns of panels have increasing levels of nutrient enrichment from left to right (left: Rtot p 0.08 mM P; middle: Rtot p
0.15 mM P; right: Rtot p 0.19 mM P). In each column of panels, the top panel shows an isocline plot similar to fig. 1, with the grazer isocline (red
line), three algal isoclines for increasing levels of light enrichment (green lines; shown are isoclines for z p 20 , 10, and 1 m, from bottom to top),
and three lines of equal algal nutrient quota Q (black lines, values of Q as in fig. 1). The bottom three panels in each column show the asymptotic
densities of algal biomass (A), grazer biomass (G), and algal nutrient quota (Q). Solid lines represent stable equilibria. Dashed lines show upper
and lower bounds of stable limit cycles. Light supply decreases from right to left with decreasing z. Note the alternative stable states at z ! 2.16 m
in the middle panels, where the disconnected lines represent the no grazer–high algae–low Q state. Algal sinking velocity v p 0 m day⫺1. The
remaining parameters are as in table 1.
lines of equal Q (note that the maxima of the grazer isoclines in fig. 2 always occur at A p 8.97 and G p
G max ⫺ 3.55, where G max p R tot /q is the intercept of the
isoline Q p Q min with the G-axis). This must necessarily
be so, because for a given algal density A, grazer growth
rate can be 0 only at a uniquely defined Q, which must
be the same for all levels of nutrient enrichment Rtot.
Enrichment with light (increased irradiation, decreased
mixing depth, decreased background turbidity) and enrichment with nutrients (increased Rtot) affect the algal
isocline in similar ways: the isocline moves up and to the
right because the positive effects of more light or nutrients
on specific algal production must be offset by more grazing
and/or increased algal self-limitation to yield zero population growth (fig. 2). Note that nutrient enrichment affects the algal isocline only in the region where nutrient
quota is flexible (Q min ! Q ! Q max). Below, I explore in
detail how light enrichment affects the algae-grazer system
and illustrate it with a change in mixing depth (fig. 2).
Changes in irradiation or background turbidity have qualitatively similar effects (see fig. 3).
At very low nutrient levels (R tot ≤ 0.08 mM P), the system always settles to a locally stable equilibrium. Moving
from low to high light supply (from high to low z), the
E180 The American Naturalist
Figure 3: Effects of enrichment with nutrients (increasing total nutrients Rtot) and light (increasing irradiation Iin in A, decreasing background
turbidity Kbg in B, and decreasing mixed water column depth z in C) on asymptotic states of the simplified algae-grazer system. Solid lines represent
boundaries between system states. A p algae-only system (grazers extinct); AG p stable algae-grazer equilibrium; AG osc. p algae and grazers
oscillate on a limit cycle. Numbers indicate regions of alternative system states. 1 p grazers extinct or stable algae-grazer equilibrium (this region
is hardly visible in A); 2 p grazers extinct or algae and grazers oscillate on a limit cycle. Dashed red lines bound regions of the paradox of energy
enrichment from below; that is, along a gradient of enrichment with light (moving from bottom to top through a plot), grazer biomass increases
to a maximum on the dashed line but decreases with further light enrichment. Dashed blue lines mark levels of light enrichment beyond which
further light enrichment decreases the amplitude of persistent grazer oscillations. Unless varied in the panel, parameter values are v p 0 m day⫺1;
Iin p 300 mmol photons m⫺2 s⫺1; Kbg p 0.25 m⫺1; z p 5 m. The remaining parameters are as in table 1.
following sequence is observed (left-hand panels in fig. 2).
At very low light supply (z 1 45 m), algal losses exceed
production, and algae cannot establish a population. At
slightly higher light supply (45 m 1 z 1 42 m), algal biomass increases with increasing light supply, but algal production cannot support a grazer population. Once a
grazer-algae equilibrium is feasible (z ! 42 m), further
light enrichment moves the equilibrium from left to right
along the grazer isocline toward lower algal quality Q.
Initially, both grazers and algae increase. However, once
the equilibrium passes the hump in the grazer isocline
(z p 12.5), the system exhibits the paradox of energy enrichment, that is, Q becomes so low that grazers decrease
while algae increase. At sufficiently high light supply
(z ! 9.8 m), the algal isocline moves entirely above the
grazer isocline, and grazers go extinct.
At higher nutrient enrichment (0.08 mM P ! R tot !
0.19 mM P; middle panels in fig. 2), parts of the grazer
isocline are in the region Q p Q max and therefore vertical.
For low to intermediate light supply (42 m 1 z 1 32 m),
the algal isocline intersects this vertical part of the grazer
isocline, and enrichment with light moves the equilibrium
up vertically. Equilibrium grazer biomass therefore increases, while algal biomass and cell quota remain constant. Further light enrichment (z ! 32 m) destabilizes the
equilibrium and algae, grazers, and algal nutrient quota
oscillate on a limit cycle. This happens when the slope of
the algal isocline in the intersection with the vertical grazer
isocline turns positive. Thus, the effects of light enrichment
have so far been as in the Rosenzweig-MacArthur (1963)
model, where enrichment destabilizes and produces increasingly vigorous population cycles. In contrast to the
Rosenzweig-MacArthur model, further light enrichment
(z ! 5.3 m) stabilizes the system again, because algal nutrient limitation turns the slope of the algal isocline negative in the region Q ! Q max. Thus, with increasing light
supply, the system moves from a stable equilibrium,
through a limit cycle, and back to a stable equilibrium. If
light supply increases further (z ! 2.1 m), the algal and
grazer isoclines intersect twice, producing one unstable
and two stable, alternative equilibria: at the left-hand stable
equilibrium, algal quality is relatively high, and algal biomass is kept low by a dense grazer population; at the righthand stable equilibrium, algal biomass is at its maximum,
algal quality is very low, and grazers are extinct.
Finally, at even higher nutrient enrichment (R tot ≥
0.19 mM P; right-hand panels in fig. 2), algal and grazer
isoclines intersect in the now fairly large region of algal
nutrient saturation (Q p Q max) even when algal growth
is light saturated. Consequently, all equilibria are on the
vertical part of the grazer isocline, and light enrichment
has effects similar to those of enrichment in the Rosenzweig-MacArthur (1963) model over the entire range of
light supplies: initially (42 m 1 z 1 32 m), grazer biomass
increases with light enrichment and controls algal biomass
and nutrient quota, but further light enrichment (z ! 32
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
m) destabilizes the system and leads to persistent oscillations. In contrast to the Rosenzweig-MacArthur model,
enrichment with light can, however, have a stabilizing influence far from equilibrium and can dampen oscillations
around unstable equilibria. This is because grazer population peaks cannot exceed the hump in the grazer isocline,
which puts a ceiling (11.6 mM C in the right-hand panel
of fig. 2) on grazer biomass (and thus grazing pressure)
during cycles. While light enrichment does not affect this
ceiling, it stimulates algal growth. Consequently, cyclic algal populations rebound earlier (at higher own and grazer
density) with light enrichment, and both algal and grazer
biomass troughs are increasingly bounded from below
(middle and right-hand panels of fig. 2). The mechanisms
behind this phenomenon are explained in detail in appendix B.
Figure 3 summarizes the interactive effects of nutrient
and light supply on the simplified grazer-algae system.
Enrichment with nutrients is plotted on the X-axes as
increasing total nutrients. Enrichment with light is plotted
on the Y-axes as increasing irradiation, decreasing background turbidity, and decreasing mixing depth, respectively. In each panel, algae are unable to persist at too low
a light supply (labeled “empty system”) but can (for the
assumptions of infinitely fast nutrient regeneration and
uptake rates) maintain a viable population even at extremely low nutrient supply. The persistence of a grazer
population (labeled “AG”) requires a somewhat higher
light supply but also a minimum nutrient supply. To the
left of the diagonal boundary separating the AG region
from the “A” region (where only algae can persist), grazers
are primarily limited by low food quality (low Q), whereas
below the horizontal boundary, grazers are primarily limited by low food quantity (low A). In the rounded corner
of the boundary, colimitation by food quality and quantity
occurs. Depending on light and nutrient supply, a persistent algae-grazer system may either settle to a stable equilibrium (labeled “AG”) or oscillate on a limit cycle (labeled
“AG osc”).
For a given nutrient supply Rtot, enrichment with light
is equivalent to moving vertically through the panels of
figure 3. At low nutrient supply (e.g., R tot p 0.07 mM P),
light enrichment produces the paradox of energy enrichment: algae increase and grazers decrease (eventually to
extinction) with light enrichment above the dashed red
line. There is also an intermediate range of nutrient supplies, where light enrichment produces a sequence from
stability to limit cycles and back to stability (often ending
with grazer extinction). Note that an alternative stable
equilibrium with grazers extinct exists in parts of this range
of nutrient supplies. Finally, at sufficiently high nutrient
supply (R tot 1 0.18–0.20 mM P), light enrichment produces
the classical paradox of enrichment: initially, grazers in-
E181
crease and control algal biomass, but further light enrichment destabilizes the algae-grazer system. Again, an alternative stable equilibrium with grazers extinct exists in
parts of this range of nutrient supplies. Note that a total
phosphorus concentration of 0.20 mM P is fairly low
(Wetzel 1983). Thus, figure 3 suggests that the paradox of
energy enrichment should be restricted to nutrient-poor
water bodies, whereas enrichment with light should produce the classical paradox of enrichment over a much
broader range of nutrient supplies. Note, however, that a
stabilizing influence of light enrichment on the amplitude
of population cycles (above the dashed blue line) occurs
over a larger range of nutrient enrichment (0.08 mM
P ! R tot ! 0.4 mM P).
Equilibrium Patterns of the Full System
Returning to the full model (relaxing the unrealistic assumptions of no algal sinking losses and infinitely fast
nutrient recycling) does not introduce new system dynamics. Rather, it changes the parameter space in which specific
dynamics are observed. This is because algal sinking and
delayed nutrient regeneration contribute to the buildup of
a nutrient pool in the sediments, which reduces the nutrient content in the water column and thus effectively
constitutes a depletion of nutrients. Because sinking losses
are inversely proportional to mixing depth (eq. [1a]), system dynamics show a strong dependence on mixing depth
(figs. 4, 5; app. C). With decreasing mixing depth, an
increasing proportion of total nutrients (up to 90% in fig.
5) is locked up in the sediment, except for extremely shallow mixing depths (!0.2 m), where algae are too strongly
limited by sinking losses to sustain a population. Decreasing mixing depth (enrichment with light) is thus inseparably linked to decreasing concentrations of total nutrients
in the water column (nutrient depletion). The latter can
be strongly stabilizing at low mixing depths. For example,
while at R tot p 0.19 mM P, a system with nonsinking algae
and fast nutrient recycling oscillates at all mixing depths
of !32 m (right-hand panels of fig. 2), the amplitudes of
oscillations start to decrease in the full model at mixing
depths of !15.5 m and give way to stable dynamics and
the paradox of energy enrichment at mixing depths of !7.4
m (fig. 5). In the full model, the region where decreasing
mixing depth reduces the amplitude of population cycles
and the region of the paradox of energy enrichment are
therefore both moved toward deeper mixing depths and
higher nutrient supplies Rtot (cf. figs. 3C, 4).
The lower the mineralization rate r of sedimented nutrients, the stronger the nutrient depletion caused by algal
sinking. For example, when r p 0.02 day⫺1, up to 95% of
total nutrients are locked up in the sediment at shallow
depths (app. C). Slower mineralization then shifts the
E182 The American Naturalist
Figure 4: Effects of enrichment with nutrients (increasing total nutrients Rtot) and light (decreasing mixed water column depth z) on asymptotic
states of the full algae-grazer model with algae sinking out of the water column at v p 0.1 m day⫺1. Solid lines represent boundaries between system
states. A p algae-only system (grazers extinct); AG p stable algae-grazer equilibrium; AG osc. p algae and grazers oscillate on a limit cycle. In A,
the number 1 indicates a region of alternative stable states with grazers either extinct or at a stable algae-grazer equilibrium. Dashed red lines bound
regions of the paradox of energy enrichment from below; that is, along a gradient of enrichment with light (moving from bottom to top through
a plot), grazer biomass increases to a maximum on the dashed line but decreases with further light enrichment. Note that the dashed red line
coincides with the boundary of the AG osc. region for Rtot 1 0.25. Dashed blue lines mark levels of light enrichment beyond which further light
enrichment decreases the amplitudes of persistent grazer oscillations. Parameters are r p 0.05 day⫺1 in A and r p 0.02 day⫺1 in B. The remaining
parameters are as in table 1.
regions exhibiting decreasing cycle amplitudes and the paradox of energy enrichment, respectively, toward deeper
mixing depths and higher nutrient supplies (fig. 4B). Conversely, if algae do not sink, dead grazers are the only
source of sedimented nutrients, and the full model behaves
very similarly to the simplified model (see app. D).
Discussion
The production of biomass at all levels of a food web
ultimately depends on the supply of energy and nutrients.
The purpose of this article was to explore the influences
of light and nutrient supply on pelagic producer-grazer
dynamics using a stoichiometric model where flows and
transformations of carbon are constrained by concurrent
flows, transformations, and recycling of nutrients. In contrast to other stoichiometric producer-grazer models, my
model explicitly considers the dynamics of the vertical light
gradient, allowing a mechanistic description of light enrichment through changes in external supply, background
attenuation, and water column depth. The model thus
enables a characterization of the environmental parameter
space in readily measurable units. In addition, reduction
of the full model to a two-dimensional limiting case greatly
facilitates an understanding of the mechanisms underlying
specific system behaviors.
Nutrient Supply, Nutrient Recycling, and the
Paradox of Energy Enrichment
Several novel phenomena discovered in simpler stoichiometric plant-herbivore models also arise when light is
treated mechanistically. Notably, the system may exhibit
the paradox of energy enrichment, where increasing light
supply causes herbivore equilibrium density to decline,
which contrasts starkly with the opposite pattern produced
by nutrient enrichment (Rosenzweig 1971; Loladze et al.
2000). With nonsinking algae, the parameter space producing the paradox of energy enrichment is restricted to
systems at the very low end of realistic nutrient supplies
(R tot ! 0.2 mM P), while unstable dynamics prevail at
higher nutrient supplies (app. D; see also fig. 3). As pointed
out by several authors (Nisbet et al. 1991; Kuijper et al.
2004), delayed nutrient recycling stabilizes plant-herbivore
dynamics by effectively depleting the water column of nutrients. With sinking algae, changes in mixing depth therefore represent a special case of enrichment with light, because the concurrent sinking losses withdraw nutrients
from the water column. The proportion of nutrients stored
in sediments depends positively on algal sinking velocity
and negatively on mixing depth and mineralization rate.
In very shallow systems (z ! 2 m), algal sinking losses can
therefore reduce the total nutrient concentration in the
water column by 190% (fig. 5; app. C). Consequently, the
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
E183
paradox of energy enrichment may occur at somewhat
higher nutrient supply levels Rtot in these systems (fig. 4).
Still, at only moderately higher nutrient supplies of
R tot 1 0.5 mM P, the paradox of energy enrichment requires
very shallow depths, !1 m. In theory, if phytoplankton
biomass is dominated by fast-sinking taxa (v 1 0.1 m
day⫺1), the paradox of energy enrichment could also occur
in deeper waters. This is, however, an unlikely scenario,
because fast-sinking algae are usually outcompeted in shallow layers by buoyant or motile taxa (Reynolds et al. 1983,
1984; Huisman et al. 2004).
Stabilizing and Destabilizing Influences of
Light Enrichment
While empirical and conceptual work on the role of light
versus nutrient supply has focused on the paradox of energy enrichment (Urabe and Sterner 1996; Sterner et al.
1997), my model suggests that, over most of the range of
realistic nutrient supplies, enrichment with light should
produce a behavior similar to the traditional paradox of
enrichment, where light enrichment leads first to an increase in grazers at a stable equilibrium and then to algaegrazer oscillations. In the simplified system, light enrichment tends to be destabilizing at nutrient levels R tot 1
0.18 mM P (fig. 3). Adding algal sinking and realistic rates
of nutrient recycling does not greatly change this pattern.
Excluding extremely shallow systems (z ! 1 m), a destabilizing role of light enrichment also seems to be the rule
in the full model unless recycling rate is extremely low
(fig. 4).
The most complex consequences of light enrichment
arise in an intermediate range of nutrient supply, where
enrichment with light initially destabilizes and eventually
stabilizes the system (e.g., fig. 5). The graphical analysis
of the simplified model shows that a destabilizing effect
of light enrichment requires the algal isocline to intersect
with a vertical part of the grazer isocline, where algae are
nutrient replete (Q p Q max). The destabilizing effect of
light enrichment is therefore fully analogous to the destabilizing effect of nutrient enrichment. In contrast, the stabilizing effect of light enrichment operates through algal
nutrient deficiency (Q ! Q max), which reduces specific algal and grazer growth rates and therefore tends to dampen
oscillations.
The higher the nutrient supply, the higher the light supply
must be to yield a stabilizing degree of algal nutrient deficiency (as indicated by the positive slopes of the regions
where enrichment with light reduces cycle amplitudes or
produces the paradox of energy enrichment; blue and red
lines, respectively, in figs. 3, 4). In the simplified model and
in the full model in absence of algal sinking, there is an
upper bound to the nutrient enrichment at which light
Figure 5: Effects of enrichment with light (decreasing mixed water column depth z) on the asymptotic values of algal (A) and grazer (G)
biomass, algal nutrient quota (Q), and the ratio of sedimented nutrients
to total nutrients (1/z · Rs/Rtot) in the full algae-grazer model with algae
sinking out of the water column at v p 0.1 m day⫺1. Solid lines represent
stable equilibria. Dashed lines represent upper and lower bounds of stable
limit cycles. Light supply decreases from right to left with decreasing z.
Rtot p 0.19 mM P; the remaining parameter values are as in table 1.
supply can stabilize dynamics. At nutrient supplies R tot ≥
0.4 mM P, only the destabilizing influence of light enrichment is expressed, and the highest amplitude of grazer fluctuations occurs at infinitely shallow z (fig. 3C). In contrast,
with sufficiently fast-sinking algae and slow nutrient recycling, a stabilizing influence of decreasing mixing depth
occurs even at high levels of nutrient enrichment (fig. 4),
because the proportion of nutrients removed to the sediment fluctuates in synchrony with algal biomass. Consequently, total nutrients in the water column and algal nutrient quota are at minimum during algal peaks, limiting
algal and grazer growth rates and dampening oscillations at
shallow depths (app. C).
E184 The American Naturalist
Multiple Equilibria
Common to plant-herbivore models in which plant quality
decreases with plant density is the existence of multiple
equilibria, because the grazer population may equilibrate
at either low or high plant density, where grazer growth
is limited by plant quantity or quality, respectively (van
de Koppel et al. 1996). Similar to other stoichiometric
models (Andersen 1997; Loladze et al. 2000; Muller et al.
2001), my model produces multiple equilibria where either
a stable or a cyclic algae-grazer system coexists with a stable
algae-only equilibrium (fig. 3C). In spite of high algal biomass, grazers cannot invade the latter equilibrium, because
algae are too nutrient deficient. This region of alternative
attractors is limited to nutrient-poor systems (R tot ! 0.2
mM P in fig. 3). The reason is that algal biomass at the
algae-only equilibrium increases less than proportionally
with increasing nutrient supply (increased biomass enhances light limitation). Consequently, as nutrient supply
increases, the algal nutrient quota increases and will eventually become high enough to allow invasion of the algaeonly equilibrium by grazers.
The region of alternative states shrinks considerably in
the full model, as compared with the simplified model (figs.
3C, 4), largely because the parameter space allowing for the
algae-only equilibrium shrinks when algae sink. Sinking has
a positive influence on algal quality, because sinking losses
reduce algal density while a fraction of the sedimented nutrients is recycled back to the remaining algae. At sufficiently
high sinking losses, algal quality will therefore be high
enough to allow grazers to invade the algae-only equilibrium. A similar empirical phenomenon was described by
Sommer (1992), who reared Daphnia together with algae
in chemostats at different dilution rates: Daphnia could establish significant populations only at relatively high dilution rates, where dilution losses prevented the buildup of a
high density of nutrient-deficient algae.
Parameter Space of “Unusual” System Behavior
The analyses suggest that unusual dynamics described
from other stoichiometric algae-grazer models (multiple
equilibria, sign changes of light enrichment effects on stability, the paradox of energy enrichment) are also possible
in my model but should be restricted to the low end of
nutrient supplies Rtot. The parameter space producing specific system behaviors can be characterized by the “realized” total nutrient concentration in the water column,
that is, total nutrients minus nutrients in sediments
(R tot ⫺ R s /z p R ⫹ QA ⫹ qG). This quantity is readily
measured as total phosphorus (TP) and routinely used to
characterize lake nutrient status. The numerical examples
(fig. 5; app. C) predict that the paradox of energy enrich-
ment should be restricted to values of !0.15 mM realized
TP in the water column at equilibrium. This is at the very
low end of TP concentrations of real lakes. In contrast,
the stoichiometric plant-herbivore model of Loladze et al.
(2000), which emulates light supply as an algal carrying
capacity, produces the paradox of energy enrichment over
a more than 70-fold larger nutrient range (beyond 10 mM
P). This quantitative discrepancy is unlikely to be a matter
of parameter choices. Using Loladze et al.’s grazer parameters in my model does not significantly extend the parameter space of the paradox of energy enrichment toward
higher nutrient levels. Rather, the quantitative differences
in model predictions seem to be related to structural differences in how algal growth is modeled. This suggests that
a mechanistic representation of light supply and light use
strongly constrains the nutrient supply space in which unusual algae-grazer dynamics are to be expected.
Empirical Patterns
What is the empirical evidence? Traditionally, enrichment
has been equated with nutrient fertilization, and there
is a wealth of data relating plant and herbivore biomass
to nutrient supply (Sarnelle 1992; Mazumder 1994; Leibold et al. 1997). Nutrient enrichment experiments with
phytoplankton-Daphnia systems have frequently supported model predictions on equilibrium biomasses but
not on stability; that is, with nutrient enrichment, Daphnia
biomass increased and algal biomass often remained constant, but persistent oscillations have only rarely been observed (Murdoch et al. 1998; McCauley et al. 1999). My
analyses suggest that stoichiometric constraints cannot explain the lack of population cycles at high nutrient supply.
An increase in the proportion of mechanically inedible
algae with nutrient enrichment seems to be a more likely,
though not universal, explanation (Leibold 1989; Murdoch
et al. 1998; Vos et al. 2004).
Although the importance of nutrient stoichiometry to
plant-herbivore interactions has been widely perceived, data
relating plant and herbivore biomass to light enrichment
are scarce. Three experiments have been performed with
Daphnia-phytoplankton systems in which irradiance was
manipulated at two levels (30–60 vs. 300–400 mmol photons
m⫺2 s⫺1) in shallow water columns (!1–4 m). Two studies
at very low nutrient levels (!0.15 mM seston P at termination) reported negative effects of light enrichment on
Daphnia population growth (Sterner et al. 1998; Urabe et
al. 2002b). Because of their short duration (≤32 days), these
studies captured only transient dynamics, making them difficult to compare to asymptotic model behaviors. Long transients were indeed observed in the third experiment at a
nutrient level of 0.5 mM TP, where light enrichment delayed
the buildup of a high Daphnia density by 30 days, compared
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
to a low light treatment (Urabe et al. 2002a). Long transients
occurred also in an experiment at 0.33 mM TP, where the
time for Daphnia pulex to reach a peak correlated positively
with initial algal density and could last 190 days if initial
algal density was high (Nelson et al. 2001). At similar nutrient levels, the same phenomena occur in my model, where
the time for a grazer inoculum to reach high densities increases with light enrichment and may last 1200 days, if the
starting condition is a high density of low-quality algae. The
dependence of transient dynamics on light and nutrient
supply is, however, complex and is treated elsewhere (C.
Jäger, S. Diehl, C. Matauschek, C. Klausmeier, and H. Stibor,
unpublished manuscript).
Urabe et al. (2002b) observed negative effects of light
enrichment on Daphnia growth only at very low nutrient
levels (!0.1 mM particulate P). Fertilization with as little
as 0.1 mM P turned the effect of light enrichment into a
positive one. Jäger and coworkers manipulated mixing
depth over a range of 1–12 m at a nutrient level of 0.3
mM TP. They found that (transient) biomass peaks of
Daphnia hyalina were highest at the shallowest mixing
depths (C. Jäger, S. Diehl, C. Matauschek, C. Klausmeier,
and H. Stibor, unpublished manuscript). The data thus
seem to support the model expectation that the paradox
of energy enrichment requires very low nutrient supply.
Evidence of nutrient limitation of Daphnia has, however,
been reported from lakes exceeding 2.5 mM TP (DeMott
and Gulati 1999). In these lakes, planktivorous fish imposed high mortality on Daphnia. Grazers must counter
increased mortality by increased growth, which requires
increased food density and/or quality. In the simplified
model, increased mortality moves the left leg of the grazer
isocline toward higher densities of more nutrient-deficient
algae. Increased planktivory may thus create a “double
whammy” for grazers through a negative effect on algal
nutrient quota. A recent analysis of a stoichiometric food
chain model does indeed predict that carnivores may drive
grazers into increased nutrient limitation (Hall et al. 2007).
Mixed water column depth varies substantially among
lakes. Studies relating plankton biomass to mixing depth
are, however, scarce. Using multiple regression, Berger et
al. (2006) found that phytoplankton and zooplankton biomasses were related negatively to mixed surface layer depth
and positively to TP content in thermally stratified lakes.
Negative relationships of phytoplankton and zooplankton
to mean lake depth (which weakly correlates with mixed
surface layer depth) and positive ones to TP were also
found by Hanson and Peters (1984) and Jeppesen et al.
(1997, 2005). Together, these data suggest that phytoplankton and zooplankton biomasses increase with light en-
E185
richment across the full range of water column depths,
which seems to contradict the more discriminating picture
predicted by my model. Note, however, that multiple regressions across large ranges of mixing depth and nutrient
supply may be insensitive to deviant patterns occurring in
very limited parameter space, such as the paradox of energy enrichment at low nutrient supply. Also, producergrazer dynamics in stratified water columns may not be
adequately captured by my model, which assumes sediments to be in contact with the mixed water column. As
a first cut, one might approximate the effects of stratification by reducing the return rate of mineralized nutrients
to the mixed water column (i.e., by reducing r; fig. 4). In
versions of the model lacking grazers, the physical water
column structure (i.e., stratification vs. mixing to the bottom) does, however, have more complex influences on the
parameter space in which particular phenomena are observed (Diehl 2002; Diehl et al. 2005; Berger et al. 2006).
Predictions for stratified lakes should therefore be properly
derived from an algae-grazer model that explicitly accounts
for the physical structure of the water column.
If empirical and theoretical research are to mutually
guide each other, realism and testability should be one
goal of modeling. My model analyses suggest that a mechanistic treatment of light supply and light use is necessary
to properly delineate the parameter space in which particular algae-grazer dynamics should be observed. Because
light and nutrient supply are expressed in readily measurable units, the model predictions lend themselves to
experimental testing. A version of the model excluding
grazers has already been successfully tested in a field enclosure experiment (Diehl et al. 2005). It should be pointed
out, however, that the asymptotic model behaviors described in this article are sometimes approached first after
a period of transient dynamics exceeding the length of a
growing season in a temperate lake. In such cases, it may
be neither possible nor relevant to run experiments over
sufficiently long periods to approach long-term dynamics
(Hastings 2004). Asymptotic attractors may then be of
interest for their influence on transient dynamics rather
than as predicted endpoints of system behavior.
Acknowledgments
I thank R. Nisbet for numerous discussions and B. Kooijman, R. Nisbet, and an anonymous reviewer for comments
on a draft of the manuscript. The research was supported
by grants DI 745/3 and DI 745/5 from Deutsche Forschungsgemeinschaft.
E186 The American Naturalist
APPENDIX A
Influence of Different Nutrient Uptake Functions on System Behavior
Figure A1: Example of the influence of different nutrient uptake functions on asymptotic behavior of the full model, plotted against the nutrient
and light enrichment parameters Rtot and z. Plot A (which is identical to fig. 4A) assumes the nutrient uptake term as described in equation (5),
that is, r(Q, R) p rmax[(Q max ⫺ Q)/(Q max ⫺ Q min)][R/(M ⫹ R)]. Plot B assumes that nutrient uptake is independent of internal nutrient stores Q, that
is, r(Q, R) p rmax[R/(R ⫹ M)]. Comparison of A and B clearly shows that the exact choice of the nutrient uptake function has only a marginal
influence on the qualitative system behavior. Solid lines represent boundaries between system states. A p algae-only system (grazers extinct);
AG p stable algae-grazer equilibrium; AG osc. p algae and grazers oscillate on a limit cycle; the number 1 indicates a region of alternative stable
states with grazers either extinct or at a stable algae-grazer equilibrium. Dashed red lines bound regions of the paradox of energy enrichment from
below; that is, along a gradient of enrichment with light (moving from bottom to top through a plot), grazer biomass increases to a maximum on
the dashed line but decreases with further light enrichment. Note that the dashed red line coincides with the boundary of the AG osc. region for
R tot 1 0.25. Blue lines mark levels of light enrichment beyond which further light enrichment decreases the amplitudes of persistent grazer oscillations.
Parameters are as in table 1.
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
E187
APPENDIX B
Phase Plots of the Simplified Algae-Grazer System
Figure B1: Phase plots illustrating the stabilizing influence of light enrichment on the amplitude of limit cycles in the simplified two-dimensional
model. Each plot shows zero-net-growth isoclines and population trajectories for two different levels of light enrichment (decreasing mixing depth
z) plotted against algal (A) and grazer (G) biomasses. Light enrichment is color coded as follows: low light, dark green algal isoclines and purple
population trajectories; high light, light green algal isoclines and blue population trajectories. Grazer isoclines (red lines) are independent of light
enrichment. Unstable equilibria are marked by filled circles. Also shown are lines of equal algal nutrient quota Q. Upper thick black line, Q p
Q min; lower thick black line, Q p Q max ; thin black line, intermediate Q (see fig. 1 for details). Parameters not varied for plots are as in table 1, except
that v p 0 m day⫺1. Initial conditions are A p 10 mM C, G p 0.5 mM C. After one transient algal biomass peak, the system quickly homes in on
a limit cycle in all cases. As described below, light enrichment starts to reduce the limit cycle amplitudes when light supply is sufficiently high for
the population trajectories to cross the grazer isocline in the positively sloping part. In A (low nutrient supply, R tot p 0.15 mM P; low light, z p
14 m; high light, z p 7 m), light-enriched population trajectories (blue lines) cross the grazer isocline in the positively sloping part (where food
quality [Q ! Q max] limits grazer growth) at levels of z ! 13.5 m. Compared to the low-light situation (purple line), grazer biomass therefore starts
to decline at a higher algal biomass A, which limits grazing pressure on algae. Algae, in turn, have a higher growth rate in light-enriched systems,
in particular to the left of the Q p Q max line, where algae are nutrient replete and their growth depends only on light. As a consequence of marginally
increased grazing pressure but strongly increased growth rate, algal populations rebound earlier (at higher grazer and own densities) in light-enriched
environments. Because their food recovers faster, grazers also rebound earlier (at higher own density). Higher grazer densities during grazer population
troughs subsequently prevent algae from escaping grazer control, which reduces algal peaks in light-enriched environments. In B (higher nutrient
supply, R tot p 0.25 mM P; low light, z p 5 m; high light, z p 1 m), light-enriched population trajectories (blue lines) cross the grazer isocline in
the positively sloping part at levels of z ! 6.4 m. The mechanisms reducing cycle amplitudes are the same as in A, but the algal population troughs
are so close to the Y-axis that it is hard to see the differences between the blue and purple trajectories.
E188 The American Naturalist
APPENDIX C
Algal Sinking, Nutrient Mineralization Rate, and the Influence of Mixing Depth on Stability
Figure C1: Numerical examples illustrating the stabilizing effects of algal sinking (with velocity v p 0.1 m day⫺1) and reduced mineralization rate
(r) on shallow systems. Plots show the effects of decreasing mixed water column depth (z) on the asymptotic biomasses of algae (A; green lines)
and grazers (G; red lines), the algal nutrient quota (Q; blue lines), and the proportion of total nutrients locked up in the sediment (1/z · R s /R tot;
brown lines). Solid lines are stable equilibria. Dashed lines show upper and lower bounds of stable limit cycles. Light supply decreases from right
to left with decreasing z. Column A is for a relatively nutrient-poor system (R tot p 0.3 mM P) and has a very low mineralization rate (r p 0.02
day⫺1). This parameter combination corresponds to a vertical transect through figure 4A at R tot p 0.3 mM P. Column B is for a nutrient-enriched
system (R tot p 0.8 mM P) and has the default mineralization rate (r p 0.05 day⫺1). This parameter combination corresponds to a vertical transect
through figure 4B at R tot p 0.8 mM P. Remaining parameter values are as in table 1. Note that algal sinking leads to a buildup of nutrients in the
sediment (Rs), which withdraws, with decreasing depth, an increasing proportion of total nutrients Rtot (bottom row). In combination with increased
light supply, this nutrient depletion of the water column stabilizes system dynamics even at high levels of Rtot (note that the cycle amplitude of
grazers decreases at z ! 7.5 m in B). This effect is stronger if nutrient mineralization is slow (low r; note that 195% of Rtot may be stored in the
sediment at low z in A).
Paradoxes of Enrichment in Pelagic Producer-Grazer Systems
E189
APPENDIX D
Bifurcation Plots of the Full and Simplified Models for Nonsinking Algae
Figure D1: Two numerical examples illustrate that the full model (including nutrient recycling through a sedimented nutrient pool; A) behaves
very similarly to the simplified, two-dimensional model (with infinitely fast nutrient mineralization and nutrient uptake by algae; B) if algae do not
sink out of the water column (i.e., are neutrally buoyant or motile; v p 0 ). The two models do not behave exactly identically because, in the full
model (A), dead grazers sink to the bottom and contribute to a pool of sedimented nutrients. The major differences are that the stabilizing influence
on cycle amplitude operates over a larger range of nutrient supplies and that the region of the paradox of energy enrichment is larger in the full
than in the simplified model. Solid lines represent boundaries between system states. A p algae-only system (grazers extinct); AG p stable algaegrazer equilibrium; AG osc. p algae and grazers oscillate on a limit cycle. Numbers indicate regions of alternative system states. 1 p grazers extinct
or stable algae-grazer equilibrium; 2 p grazers extinct or algae and grazers oscillate on a limit cycle. Dashed red lines bound regions of the paradox
of energy enrichment from below; that is, along a gradient of enrichment with light (moving vertically from bottom to top through a plot), grazer
biomass increases to a maximum on the dashed line but decreases with further light enrichment. Dashed blue lines mark levels of light enrichment
beyond which further light enrichment decreases the amplitudes of persistent grazer oscillations. Parameters are as in table 1, except that v p 0.
Plot B is identical to figure 3C.
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