Download A stoichiometric exception to the competitive exclusion principle.

A stoichiometric exception to the competitive
exclusion principle.
Irakli Loladze∗, Yang Kuang
Department of Mathematics
Arizona State University
Tempe AZ 85287-1804
[email protected], [email protected]
James J. Elser and William F. Fagan
Department of Biology
Arizona State University
Tempe AZ 85287-1501
[email protected], [email protected]
September 26, 2001
Abstract
The competitive exclusion principle (CEP) states that no equilibrium
is possible if n species exploit fewer than n resources. This principle does
not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homeogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been
proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can
help in understanding the stable maintenance of biodiverse communities.
Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements (carbon, nitrogen, phosphorus) and that the relative relative proportions of such elements vary within and among species.
We incorporate stoichiometric principles into a standard predator-prey
model to analyze competition between two consumers on one autotrophic
resource. The model tracks two essential elements, carbon and phosphorus, in each species. We show that a stable equilibrium is possible with
two competitors on this single biotic resource. At this equilibrium both
consumers are limited by the phosphorus content of the resource. This
suggests that within a stoichiometric framework, the CEP may be violated because one biotic resource may support multiple competitors at a
stable equilibrium.
∗ Current address: Department of Ecology and Evolutionary Biology, Princeton University,
Princeton, NJ 08544-1033
1
1
Introduction
The competitive exclusion principle, one of the oldest and most intriguing paradigms in community ecology, states that at most n species can coexist on n
resources (Volterra, 1926; Gause, 1934; Hardin, 1960; MacArthur & Levins,
1964; Levin, 1970). The principle relates to one of the central problems in
ecology: finding mechanisms that maintain vast biodiversity. Since the time
Hardin (1959, 1960) coined the term “the competitive exclusion principle,” subsequent advances in theoretical ecology (by no means a complete list) have
identified various mechanisms that promote coexistence and demonstrated that
the above formulation of the principle is too restrictive. First, if resources
are not explicitly modelled, competitive exclusion is not mandatory. Strobeck
(1973) showed that n species can coexist at a locally stable equilibrium in a
homogeneous (spatially and temporally) classical Lotka-Volterra model with no
resources. In such a model, however, competition is expressed in an ad hoc
fashion via constant competition coefficients. In explicit resource models, exploitative competition among consumers is realized, as in real life, directly via
the consumption of shared resources. Second, even for models that explicitly
consider resources, the coexistence of n species on fewer than n resources is possible via internally produced limit cycles or chaotic fluctuations (Armstrong &
McGehee, 1980; Huisman & Weissing, 1999). Third, other factors such as spatial
(Tilman, 1994; Richards et al., 1999) or temporal (Butler et al., 1985; Lenas &
Pavlou 1995) heterogeneity, light fluctuations (Litchman & Klausmeier, 2001),
predation (Armstrong, 1994; Leibold, 1996), disturbance (Hastings, 1980), and
interference between consumers (Vance, 1984, 1985) provide mechanisms for
coexistence.
In view of these findings, the competitive exclusion principle can be restated
as following: a stable equilibrium is impossible in a homogeneous system with
n species exploiting fewer than n resources. For the rest of this paper, by
the competitive exclusion principle (CEP), we will mean this formulation. It
assumes exploitative competition; that is, a situation in which consumers do not
interfere directly but affect each other via the consumption of shared resources.
It also assumes spatial and temporal homogeneity. Despite these restrictions,
the CEP is extremely robust to various types of models: whether resources are
abiotic such as chemical elements or biotic such as prey species (Armstrong and
McGehee, 1980), essential or substitutable (Leon & Tumpson, 1975; Tilman,
1982; Grover, 1997) the CEP holds firmly. All these models share the same
general structure: one set of equations describes resource dynamics and the
second set describes the dynamics of consumers. The CEP follows as a rigorous
mathematical result under an assumption that consumers and resources are in
a predator-prey relationship. In other words, the partial derivative of consumer
abundance with respect to resource abundance is positive or zero, while the
partial derivative of resource abundance with respect to the consumer is negative
or zero. (One notable exception to this rule is Levin’s (1970) extension of
the CEP, where resources were replaced with more abstract “limiting factors.”
Levin’s formulation requires only that species-specific growth rates be linear
2
functions of the limiting factors, an assumption that may not hold in reality.
Under this assumption, neither a stable equilibrium nor a stable cycle is possible
with n species on fewer than n limiting factors.)
As it frequently happens in mathematical biology, the verbal version of a
rigorous mathematical statement is often stated without some of its underlying
assumptions. Such omission of assumptions can and often leads to widespread
misconceptions by forcing the argument beyond the realm of its original formulation (e.g. the CEP or May’s complexity-stability argument). While this
should not negate the versatility and effectiveness of mathematical applications
in ecology, it highlights the importance of explicit statement of the precise conditions under which the statement holds.
Whether resources are biotic or abiotic, similar assumptions have been made
about consumer-resource interactions (Armstrong & McGehee, 1980). Hence, a
conclusion follows that the CEP holds irrespective of the type of resources (biotic
or abiotic) one models. However, any biotic resource (i.e. prey item), as all life,
consists of multiple abiotic resources that are essential to its consumer. For
example, one may consider a biotic resource as a mixture of macro-compounds
such as carbohydrates, proteins, and lipids, which in turn are made of chemical
elements such as carbon, nitrogen, phosphorus and so on. If a mathematical
model incorporates such properties of biotic resources, will the CEP remain true
as a rigorous mathematical result? It is the purpose of this paper to construct
a simple plausible model to address this issue. A question arises regarding the
appropriate “chemical scale” for modeling. The overwhelming complexity of
organic compounds and their intricate metabolic pathways leave us no hope to
follow them with a simple model. Fortunately, there is a powerful approach
advocated by Alfred Lotka (1925), an originator of the famous Lotka-Volterra
equations. Lotka, as a physical chemist, stressed that for biological systems,
mass conservation with respect to all elements must hold: “for any self-contained
system, we shall have an equation of the form
X
X1 + X2 + ... + Xn =
X = A = const.
(1)
expressing the constancy of the total mass of the system; and a similar equation
holds separately for every chemical element (except in those rare cases in which
radioactive or other atomic disintegration occurs)”.
Equation (1) is a part of the foundation of “stoichiometry”, a term that
Lotka borrowed from chemistry and redefined for mathematical biology as a
“branch of the science which concerns itself with material transformations, with
the relations between the masses of the components.” Unfortunately, this approach has been largely neglected in the subsequent development of theoretical
and mathematical population dynamics. This oversight of Lotka’s ideas is especially regrettable because the underlying assumptions of stoichiometry are
among those few laws that can be firmly stated about biological systems. Incorporating such assumptions in mathematical models can be crucial for generating
sound biological insights. In recent years, Lotka’s stoichiometric vision has been
revived in such works as Reiners (1986), Sterner (1990), Andersen (1997), Hes3
sen (1997), Elser et al. 2000, Sterner & Elser (2001).
Here we state five facts that, on our opinion, underline the foundation of
stoichiometry:
- biological systems, despite their overwhelming complexity, are not able to
create or destroy chemical elements, nor they are able to convert one element
into any other;
- there are several chemical elements required by all life;
- the proportions of these elements vary within and among species;
- all such proportions have upper and lower bounds;
- the amount of any element in any ecosystem is finite.
Lotka’s equation (1) follows from these facts. Although Lotka favored a
stoichiometric approach, his population dynamics equations did not incorporate
stoichiometry.
We formulate a simple model that captures the critical elements of stoichiometry. Our motivation is to find out if and when the CEP remains true
in a model where consumers compete for a biotic resource in a stoichiometric
setting. In the next section we construct the model. In section 3, we analyze
it qualitatively. Numerical and graphical analysis of the model with Monod
(or Michaelis-Menten) functional responses are in section 4. We will demonstrate that, in a stoichiometrically explicit model, two consumers exploiting one
prey can coexist at a stable equilibrium. In the discussion, we will address the
implications of these findings for our understanding of real ecosystems.
2
Model
We model two consumers exploiting one biotic resource in a system with no
spatial heterogeneity or external variability. The difference between our approach and the models mentioned above is that we do not assume, a priori,
the existence of conventional predator-prey interactions (that is, those in which
increased prey abundance never hurts predator growth). Instead, our model
relies mainly on the stoichiometric principles listed in the previous section and
the basic law of mass conservation.
To have some concrete system in mind, we can think of the biotic resource
as a single species of alga, while the consumers are two distinct zooplankton
species, all placed in a well-mixed system open only to light and air. (One can
imagine a continuously stirred culture in an open-top clear chamber.) In the
model construction, we follow a course outlined in Loladze et al. (2000), which
deals with one consumer-one biotic resource interactions.
Let us start with a conventional model, which describes a system of two
consumers feeding on one biotic resource, here assumed to be a photoautotroph
at the base of the food web (e.g. phytoplanktonic algae):
4
³
x´
dx
= rx 1 −
− f1 (x)y1 − f2 (x)y2 ,
dt
K
dy1
= e1 f1 (x)y1 − d1 y1 ,
dt
dy2
= e2 f2 (x)y2 − d2 y2 .
dt
(2a)
(2b)
(2c)
Here,
x, y1 and y2 are the densities of the biotic resource and two consumers
respectively (in milligrams of carbon per liter, mg C/l),
r is the intrinsic growth rate of the resource (day−1 ),
d1 and d2 are the specific loss rates of the consumers that include rates of
respiration and death (day−1 ).
f1 (x) and f2 (x) are the consumers’ ingestion rates, which we assume to
follow Holling type II functional response. In other words, fi (x) is a bounded
smooth function that satisfies the following assumptions:
fi (0) = 0, fi0 (x) > 0 and fi00 (x) < 0 for x ≥ 0, i = 1, 2.
(3)
e1 and e2 are constant growth efficiencies (conversion rates or yield constants) of converting ingested resource into consumer biomasses. The second
law of thermodynamics requires that e1 and e2 be less than 1.
K represents a constant carrying capacity that we relate to light in the
following way: suppose that we fix light intensity at a certain value, then let
the resource (which is a photoautotroph) grow with no consumers but with
ample nutrients. The resource density will increase until self-shading ultimately
stabilizes it at some value, K. Thus, we might model the influence of higher
light intensity as having the effect of raising K, all else being equal (Loladze et
al., 2000). More mechanistic descriptions of light limited growth are possible
(e.g. Huisman & Weissing, 1994), but these would significantly complicate our
model and are not attempted here.
Model (2) views the biotic resource, x, as a chemically homogeneous substance. Hence, this resource can not support two consumers at fixed densities.
As we pointed out in the Introduction, however, any biotic resource is not chemically homogeneous, but is made of multiple chemical elements. In particular,
no life on Earth exists without carbon or phosphorus (we can add nitrogen, sulfur, potassium and other elements as well). The requirement for these elements
was established early in evolution. For example, every nucleotide requires one
atom of phosphorus. Millions of nucleotides linked together make DNA and
RNA in every cell. Hence, at least some amount of phosphorus must be present
in every known organism. The bulk of the dry weight of most organisms is carbon comprises, and phosphorus to carbon ratio, P:C, varies within and among
species (e.g. Elser et al., 2000). The cellular physiology of autotrophs allows
them to exhibit highly variable phosphorus content. For example, the green
alga Scenedesmus acutus can have cellular P:C ratios (by mass) that range
5
from 1.6 · 10−3 to 13 · 10−3 , almost an order of magnitude. On the other hand,
in animals P:C varies much less within a species. For example, the zooplankton
Daphnia’s P:C stays within relatively narrow bounds around 31 · 10−3 by mass,
decreasing only by around 30% when food P:C is more than 20-times lower than
Daphnia’s P:C (Sterner & Hessen, 1994; DeMott, 1998). For simplicity, we will
model only two elements, phosphorus and carbon. Though this is a gross simplification of reality, it is a qualitative improvement of the conventional models
like model (2).
We express the above considerations in the following assumptions:
(A1): phosphorus to carbon ratio (P:C) in the resource varies, but never
falls below a minimum q (mg P/mg C); the two consumers maintain a constant
P:C ratio, s1 and s2 (mg P/mg C), respectively.
(A2): the system is closed for phosphorus, with a total of P milligrams of
phosphorus per liter (mg P/l), which is divided into two pools: phosphorus in
the consumers and the rest as phosphorus potentially available for the biotic
resource.
From these two assumptions it follows that phosphorus available for the resource at any given time is (P − s1 y1 − s2 y2 ) (mg P)/l. This amount determines
the “phosphorus carrying capacity” of the resource: (P − s1 y1 − s2 y2 ) /q (mg
C)/l. In the spirit of Liebig’s Law of the Minimum, the combination of light
and phosphorus limits the carrying capacity of the resource to
µ
¶
P − s1 y1 − s2 y2
min K,
.
(4)
q
To determine the P:C ratio of the resource at any given time, we follow
Andersen (1997) by assuming that the biotic resource can absorb all potentially
available phosphorus. Then its P:C ratio in (mg P : mg C) is
(P − s1 y1 − s2 y2 ) /x.
(5)
This assumption is not far from reality in freshwater systems, where algae can
absorb almost all available phosphorus, bringing free phosphate levels in water below detection. However, it may not be suitable for terrestrial systems,
where soil particles are a major stock of phosphorus, or for modeling competition among multiple biotic resources, where free phosphorus uptake affects
competition outcome.
While resource stoichiometry varies according to (5), the i-th consumer maintains its constant P:C, si . If resource P:C is more than si , the i-th consumer is
able to maximally use the energy (carbon) content of consumed food with efficiency ei , and excretes any excess phosphorus it ingests. If P:C of the resource is
below si , the i-th consumer wastes the excess of ingested carbon. This waste is
assumed to be proportional to the ratio of resource P:C to i-th consumer’s P:C
(Andersen, 1997). Hence, the growth efficiency in carbon terms, ei , is reduced.
The following minimum function reflects growth efficiency under both good and
6
bad resource quality conditions:
¶
µ
(P − s1 y1 − s2 y2 ) /x
ei min 1,
.
si
(6)
The considerations of (4) and (6) are captured in the following model:
µ
¶
dx
x
= rx 1 −
− f1 (x)y1 − f2 (x)y2
dt
min(K, (P − s1 y1 − s2 y2 ) /q)
µ
¶
(P − s1 y1 − s2 y2 ) /x
dy1
f1 (x)y1 − d1 y1
= e1 min 1,
dt
s1
¶
µ
(P − s1 y1 − s2 y2 ) /x
dy2
f2 (x)y2 − d2 y2
= e2 min 1,
dt
s2
(7a)
(7b)
(7c)
Note, that the growth rate of the biotic resource in the absence of consumers
µ
¶
dx
x
= rx 1 −
dt
min(K, (P − s1 y1 − s2 y2 ) /q)
can be equivalently represented as the minimum of a logistic equation and a
biomass version of Droop’s (1974) model:
µ ³
µ
¶¶
x´
x
dx
= min rx 1 −
, rx 1 −
.
(8)
dt
K
(P − s1 y1 − s2 y2 ) /q
3
Mathematical Analysis
In this section, through rigorous mathematical derivations we obtain some analytical results on the solutions and aspects of positive equilibria of model (7).
The interpretations of these results are presented in subsequent sections.
Model (2) almost never has a positive equilibrium (that is, an equilibrium
with all three species present). It is easy to see why this is the case. If such
an equilibrium exists, then the resource density, x∗ , at such an equilibrium
should simultaneously satisfy the following two equations to keep consumers’
net growth rates equal to zero:
f1 (x∗ ) =
d1
d2
and f2 (x∗ ) = ,
e1
e2
(9)
which is almost impossible (the set of parameter values satisfying both equations
is of measure zero). Moreover, model (2) belongs to the class of models studied
by Armstrong & McGehee (1980), where resources are regarded as biotic. These
authors have rigorously shown that an attracting equilibrium is impossible (not
just almost impossible) with n species on fewer than n resources. Nevertheless,
we are going to show that such an equilibrium exists in our stoichiometrically
explicit model (7).
7
First, we point out that model (7) is well defined as x → 0. The conditions
(3) assure that for fi (x) /x, the following holds:
µ
¶0
fi (x)
fi (x)
0
< 0 for x > 0, i = 1, 2.
(10)
lim
= f (0) < ∞ and
x→0
x
x
This means that yi0 (t) is well defined as x → 0, since for i = 1, 2,
µ
µ
¶
¶
P − s1 y1 − s2 y2
(P − s1 y1 − s2 y2 )fi (x)
min 1,
fi (x) = min fi (x),
.
x
x
The condition (10) ensures that system (7) is well defined and the functions at
the right hand sides are locally Lipschitzian, which ensures that solutions to
initial values are unique.
Stoichiometric considerations provide natural bounds on densities of all species.
We have the clear upper bound defined by phosphorus: the sum of phosphorus
in the resource and the consumers cannot exceed P - the total phosphorus in the
system. The following theorem provides an analytical result on the boundedness
and invariance of solutions of model (7); its proof can be found in the appendix
.
Theorem 1 Let k = min(K, P/q). Then, solutions with initial conditions in
∆ ≡ {(x, y1 , y2 ) : 0 < x < k, 0 < y1 , 0 < y2 , qx + s1 y + s2 y < P }
(11)
remain there for all forward times.
˘ we denote the region ∆ with its entire boundary except of biologically
By ∆,
unrealistic edge γ = {(x, y1 , y2 ) : x = 0, P = s1 y1 + s2 y2 }. (At this edge
consumers contain all phosphorus, thus there is no any resource in the system.)
That is,
˘ = (∂∆\γ) ∪ ∆.
∆
To simplify our analysis, we rewrite the system (7) in the following form:
x0 = xF (x, y1 , y2 )
y10 = y1 G1 (x, y1 , y2 )
y20 = y2 G2 (x, y1 , y2 ),
(12a)
(12b)
(12c)
where
µ
F (x, y1 , y2 ) = r 1 −
x
min(K, (P − s1 y1 − s2 y2 )/q)
¶
−
2
X
fi (x)
i=1
x
¶
µ
(P − s1 y1 − s2 y2 ) /x
fi (x) − di
Gi (x, y) = ei min 1,
si
µ
¶
P − s1 y1 − s2 y2 fi (x)
= ei min fi (x),
− di , i = 1, 2.
si
x
8
yi ,
(13a)
(13b)
Condition (10) ensures that all partial derivatives of F and Gi exist almost
˘ :
everywhere on ∆
¶0
2 µ
X
∂F
fi (x)
r
yi
(14)
=−
−
∂x
min(K, (P − s1 y1 − s2 y2 )/q) i=1
x
For i, j = 1, 2, we have

fi (x)

 −
<0
if K < P −s1 yq1 −s2 y2
∂F
x
=
,
rqsi x
fi (x)
P −s1 y1 −s2 y2
∂yi 
 −
−
<
0
if
K
>
q
x
(P − s1 y1 − s2 y2 )2

if P −s1 yx1 −s2 y2 > si
e f 0 (x) > 0
µ
¶0
∂Gi  i i
,
=
P − s1 y1 − s2 y2 fi (x)
 ei
∂x
< 0 if P −s1 yx1 −s2 y2 < si
si
x

if P −s1 yx1 −s2 y2 > si
∂Gi  0
=
,
s f (x)
 −ei j i
∂yj
< 0 if P −s1 yx1 −s2 y2 < si
si x
(15)
(16)
(17)
To determine the type of species interactions at equilibria and the stability
properties of equilibria, it is convenient to examine the Jacobian of system (12):


∂F
∂F
∂F
F+
x
x
x


∂x
∂y1
∂y2



 ∂G
∂G1
∂G1


1
G1 +
y1
y
y1
J(x, y1 , y2 ) ≡ (aij )3×3 ≡ 
 (18)


∂x
∂y1
∂y2



 ∂G2
∂G2
∂G2
y2
G2 +
y2
y2
∂x
∂y1
∂y2
If the following system has a solution
F (x, y1 , y2 ) = G1 (x, y1 , y2 ) = G2 (x, y1 , y2 ) = 0,
(19)
then system (12) has an internal (positive) equilibrium. We will show that for
biologically realistic parameter sets this condition can be satisfied. Let us denote
such an equilibrium as (x∗ , y1∗ , y2∗ ). Then the Jacobian at (x∗ , y1∗ , y2∗ ) is

 ∗
x
0 0
(20)
J(x∗ , y1∗ , y2∗ ) =  0 y1∗ 0  E(x∗ , y1∗ , y2∗ ),
0
0 y2∗
where





E(x∗ , y1∗ , y2∗ ) = 



∂F
∂x
∂G1
∂x
∂G2
∂x
9
∂F
∂y1
∂G1
∂y1
∂G2
∂y1
∂F
∂y2
∂G1
∂y2
∂G2
∂y2









(21)
is the ecosystem matrix of our system. Here, ij-th term in the matrix measures
the effect of j-th species on i-th species growth rate.
If
(P − s1 y1∗ − s2 y2∗ ) /x∗ > si ,
i = 1, 2
(22)
then the quality of the resource is good for both consumers, and, as (15-17)
suggest, the signs in the ecosystem matrix are


+/− − −
 +
0 0 ,
(23)
+
0 0
meaning that a conventional predator-prey type interaction exists between the
competitors and the resource. In this case, fi (x∗ ) = di /ei , i = 1, 2, an almost
impossible situation.
If
(P − s1 y1∗ − s2 y2∗ ) /x∗ < si ,
i = 1, 2
(24)
then the quality of the resource is bad for both consumers, and (15-17) indicate
that the signs of the ecosystem matrix are


+/− − −
 −
− − .
(25)
−
− −
This means that all species compete with each other and the consumers endure
self-limitation.
In this case,
A ≡ P − s1 y1∗ − s2 y2∗ =
d1 s1 x∗
d2 s2 x∗
=
e1 f1 (x∗ )
e2 f2 (x∗ )
and
µ
B ≡ rx∗ 1 −
x∗
min(K, (d2 s2 x∗ )/(qe2 f2 (x∗ ))
¶
= f1 (x∗ )y1∗ + f2 (x∗ )y2∗ .
This case is highly likely to happen with the value of x∗ determined by
d1 s1 x∗
d2 s2 x∗
=
,
e1 f1 (x∗ )
e2 f2 (x∗ )
(26)
and the values of y1∗ and y2∗ given by
y1∗ =
f2 (x∗ )P − f2 (x∗ )A − s2 B
,
s1 f2 (x∗ ) − s2 f1 (x∗ )
y2∗ =
f1 (x∗ )A + s1 B − f1 (x∗ )P
. (27)
s1 f2 (x∗ ) − s2 f1 (x∗ )
It is straightforward to find that condition (24) is equivalent to
di < ei fi (x∗ ),
10
i = 1, 2,
(28)
which is to say that if the resource at density, x∗ , would have been of good quality, then the growth rate of each consumer would have exceeded its death rate.
At this equilibrium both consumers are limited by the quality (i.e. phosphorus
content) of the resource.
An intermediate scenario can also occur, where the quality is bad for one
consumer, but good for the other. The coexistence of all species at a stable
positive equilibrium is possible as well in this case (parameter values for this
case are listed in Table 1 and K=0.65).
Observe that (7) has E0 ≡ (0, 0, 0) and Ek ≡ (min(K, P/q), 0, 0) as its only
axial equilibria. Clearly, E0 is always a saddle. It is easy to show that when
any other steady state exists, then Ek is also a saddle. For general functions
fi (x) satisfying (3), there can be many other nonnegative equilibria on the
boundary surfaces (x − yi surface, i = 1, 2) of ∆ (Loladze et al., 2000). The
stability of these equilibria and positive equilibria (if any) can be routinely studied via Routh-Hurwitz criteria (Edelstein-Keshet (1988), p. 234) when specific
functions fi (x), i = 1, 2 are given. Persistent properties of the system can be
obtained by applying existing theory (e.g. Thieme (1993) and the references
cited there).
4
Numerical and Graphical Analysis
To illustrate some of the above analytical findings, we run simulations on model
(7) with both fi (x), i = 1, 2 as Monod (or Michaelis-Menten) functions of the
form
ci x
fi (x) =
,
i = 1, 2.
(29)
ai + x
With these functions, if a positive equilibrium exists with bad resource quality (conditions (24)), then equation (26) yields the following resource density at
the equilibrium:
d2 s2 a2 e1 c1 − d1 s1 a1 e2 c2
.
(30)
x∗ =
d1 s1 e2 c2 − d2 s2 e1 c1
We provide practical graphical interpretations for some of the analytical and
numerical results below.
4.1
Numerical Analysis
Table 1 lists biologically realistic parameter values that are based on literature
sources (Urabe & Sterner, 1996; Andersen, 1997; Elser & Urabe, 1999).
These values indicate that the second consumer requires more phosphorus
(s2 > s1 ) but, as a trade-off, has a higher growth rate (e2 f2 (x) > e1 f1 (x) for
x > 0) when resource quality is good. Such a trade-off is consistent with “the
growth rate hypothesis” (Elser et al., 2000) and has been empirically supported
in studies of zooplankton (Main et al., 1997)
First, we analyze the system when light intensity is low, so that self-shading
of the resource bounds its density at K = 0.35 (mg C)/l. Under such low
11
P
r
K
c1
c2
a1
a2
e1
e2
d1
d2
s1
s2
q
P
Description
intrinsic growth rate of the resource
resource carrying capacity determined by light
maximal ingestion rate of the 1st consumer
maximal ingestion rate of the 2nd consumer
half-saturation constant of the 1st consumer
half-saturation constant of the 2nd consumer
maximal conversion rate of the 1st consumer
maximal conversion rate of the 2nd consumer
loss rate of the 1st consumer
loss rate of the 2nd consumer
constant P:C of the 1st consumer
constant P:C of the 2st consumer
minimal possible P:C of the resource
total phosphorus in the system
V
0.93
0.35 − 1.0
0.7
0.8
0.3
0.2
0.72
0.76
0.23
0.2
0.032
0.05
0.004
0.03
Units
day−1
(mg C)/l
day−1
day−1
(mg C)/l
(mg C)/l
day−1
day−1
(mgP)/(mg C)
(mgP)/(mg C)
(mgP)/(mg C)
(mg P)/l
Table 1: The parameters (P) of the model and their values (V) used for numerical simulations.
light conditions, resource quality is good (due to low algal biomass relative to
phosphorus) and both consumers are limited by resource quantity (carbon).
Model (7) behaves like conventional resource based models. In other words, the
CEP holds and the second consumer (whose break-even resource concentration
is lower) wins. The only attracting equilibrium is
(x, y1 , y2 ) = (0.098, 0, 0.23),
(31)
where we rounded these values (in mg C/l) and all values hereafter to a thousandth.
Next, we analyze the system under high light intensity, so that self-shading
of the resource occurs at a higher level, with K = 1.0 (mg C)/l. Under such
high light conditions, resource quality can be bad for both consumers. The
CEP does not hold anymore, because the system has the following positive
equilibrium with all species present:
(x, y1 , y2 ) = (0.59, 0.26, 0.17) .
(32)
All eigenvalues at this equilibrium are negative, so the equilibrium is locally
asymptotically stable. It can be shown that this is the only positive equilibrium,
all boundary equilibria are unstable and the system appears to be persistent.
(That is, all species, if present, are expected to coexist. This implies that
the invasions of either consumer species will be successful.) Moreover, all our
numerical simulations (Fig. 4) indicate that trajectories with all positive initial
conditions in ∆ converge to this equilibrium.
12
4.2
Graphical analysis
Since system (7) involves three differential equations, its phase space is three
dimensional. To facilitate the visualization of the dynamics, let us recall a
two dimensional case: a single consumer on one biotic resource. This case was
rigorously analyzed in Loladze et al. (2000) using the method of nullclines.
The nullcline of a species on the phase plane (the coordinate plane with axes
representing species densities) is the set of points where the species’s growth rate
is zero . If nullclines of all species intersect at some common point, then at such
a point the growth rate of every species is zero, making this point an equilibrium.
The stability type of such an equilibrium can sometimes be determined by the
way nullclines intersect. As we see in Fig.1, the resource nullcline is hump
shaped, a usual feature of many predator-prey models. The consumer nullcline,
however, has an unusual shape of a right triangle (in conventional consumerresource models, it is a vertical line or an increasing curve). Here, multiple
equilibria are possible because two nullclines with such shapes can intersect
more than once. The vertical segment of the consumer nullcline lies in a region
where the quantity of the resource limits the consumer and where the system
behaves like a conventional predator-prey model. The slanted segment of the
consumer nullcline lies in a region where the resource quality limits the consumer
and qualitatively novel dynamics arise.
Introducing the second consumer adds a third dimension to Fig. 1 and
nullclines on the phase plane become nullsurfaces in the phase space. For example, system (7), with biologically realistic parameter values as in Table 1 and
K = 0.35, has nullsurfaces as shown in Fig. 2a. (It can be shown that for Monod
functional responses, consumer species nullsurfaces always take the shape of two
i
folding planes: a portion of the plane vertical to x-axis, x = ciaeii d−d
, i = 1, 2, and
i
di si
a portion of the plane slanted to x-axis, s1 y1 +s2 y2 + ci ei x = P − dci si ei ai i , i = 1, 2.)
For K = 0.35, the resource’s nullsurface lies entirely in the region where the consumers are limited by the quantity of the resource. The reason for this is that
a low value of K corresponds to low light intensity which keeps the resource
density low (in mg C/l), which in turn assures high P:C in the resource. In this
case, the system is no different from standard resource based models, the CEP
holds and the only attracting equilibrium is (31). Fig. 2b shows the slice of
the phase space in Fig. 2a cut out by y1 = 0 plane. The second consumer has
a lower break-even concentration (the vertical segment is closer to the origin)
and, as resource competition theory predicts, this consumer wins.
The picture changes drastically when we increase K to 1.0 and keep all other
parameter values unchanged. This K value corresponds to high light intensity,
which lowers P:C ratio of the resource to such levels that consume growth becomes phosphorus-limited. Fig. 3 shows that all three nullsurfaces intersect at
one point, equilibrium (32). Its negative eigenvalues prove that this equilibrium
is locally attracting. For conventional model (2), which ignores stoichiometry,
consumers’ nullsurfaces are parallel planes, so they cannot intersect and an equilibrium is impossible, which provides a graphical interpretation of condition (9).
To simplify Fig. 3, we slice the phase space through equilibrium (32) with plane
13
y1 = 0.26 (mg C)/l (slicing with plane y2 = 0.17 yields a qualitatively similar
picture). This slice is shown in Fig. 4. Drawing an analogy from Fig. 1, we
see that in Fig. 4 both consumers are limited by resource quality. Indeed, at
equilibrium (32) inequalities (24) hold, implying that both consumers are limited by the same element - phosphorus. Hence, both consumers coexist on one
biotic resource and are limited by the same chemical element!
In our study, we consider two essential chemical elements, carbon and phosphorus, in the composition of a biotic resource. For resource competition for two
nutrients, Leon & Tumpson (1975) introduced nullcline analysis on the plane
with axes representing nutrient availability. Tilman (1982) greatly expanded this
approach in developing resource-competition theory. When essential nutrients
are considered in this theory, each consumer nullcline is L shaped. Neither the
vertical legs of these two L’s can intersect each other nor can the horizontal
ones. Hence, the stable coexistence of two consumers on the same nutrient is
impossible. For system (7), such nullclines on the carbon-phosphorus plane have
the shape of tilted V, where the angle of the tilt is species specific as seen on Fig.
5. The resulting intersection of two V-nullclines can yield a stable equilibrium
where two consumers are limited by the content of the same essential element
in the prey (Fig. 3, Fig.4 and Fig.5).
5
Discussion
We showed that an attracting equilibrium is possible for two consumers on one
biotic resource. This result disagrees with the competitive exclusion principle
(CEP) that an attracting equilibrium is impossible in a homogeneous system
with n species exploiting fewer than n resources (whether resources are biotic
or abiotic). Since the CEP is a rigorous mathematical result (Armstrong &
McGehee 1980) and one of the seemingly solid pillars in ecological theory, one
may ask why such a result is possible at all. Our model, unlike conventional
resource based models, does not assume a (+,-) relationship between consumers
and resources nor does it have linear growth assumptions as in Levin’s (1970)
“limiting factor” model. Instead, it relies on general stoichiometric principles.
In particular, we use the fact that all species share common essential chemical
elements, but in different proportions. Consumer species differ in their chemical
constitution, both from each other and from the resources they consume. Model
(7) captures this by considering different consumer P:C ratios, while allowing
the P:C ratio of the autotroph to vary, as it does in nature. It is a fact that
any biotic resource contains several chemical elements that are essential to all
of its consumers (e.g. carbon, nitrogen, phosphorus, sulfur, and calcium.) This
suggests that biotic resources can be viewed and can be modelled as channels
through which multiple nutrients flow to their consumers.
All three species in our system, and in fact, all species in any ecosystem,
share a common need for carbon and phosphorus. Such shared requirements for
essential elements connect population level properties to those on the level of the
entire ecosystem, and create feedbacks between these two levels of organizational
14
complexity that lead to dynamics that may seem paradoxical or impossible from
conventional points of view. Stoichiometric considerations provide biologically
meaningful bounds on population densities: every organism requires at least
some minimum amount of an essential chemical element. Therefore, the total
amount of this element puts a ceiling on overall biomass in the system. We
rigorously showed that in Theorem 1 by proving that all dynamics in the system
is confined to a region defined by the interplay between ecosystem properties
(total phosphorus and light intensity) and species-specific stoichiometries (P:C
ratio of the consumers and the resource). That is why, the carrying capacity of
the resource is not static, as it is usually assumed in the logistic equation, but
instead is a dynamic quantity affected by consumer densities, energy inputs and
phosphorus in the system (see eq. 4).
In addition to imposing natural bounds on the dynamics, the consideration
of multiple elements and their ratios reveals rich dynamics within these bounds.
When the quality of the resource is good, the sheer quantity of the resource limits
consumers. Their dynamics are completely predicted by resource competition
theory: the consumer with the lowest break-even concentration wins (see eq.
(9). The signs of ecosystem matrix (23) indicate the usual (+,-) interaction
between consumers and the resource. However, when resource quality is bad, a
novel attracting equilibrium can arise. For example, realistic parameter values
listed in Table 1 (with K = 1) yield a locally attracting equilibrium (32). At
this equilibrium, not only do two consumers coexist on one biotic resource,
but both are limited by the same chemical element, phosphorus, which is in low
concentration in the resource. The ecosystem matrix (25) takes an unusual form
for consumer-resource models: negative signs indicate that the common need
for limited essential element fosters competition between the consumers and
the resource as well as between the consumers. Such an outcome is impossible
in conventional resource-based models because the (+,-) relationship between
consumers and resources is mandated by the model assumptions.
In the Graphical Analysis section, we analyzed the model using nullsurfaces
and nullclines on the coordinate system with each axis representing species densities. However, in resource competition theory (Leon & Tumpson 1975; Tilman,
1982; Grover, 1997), nullclines are drawn on the plane with each axis representing nutrient concentrations. For essential nutrients each consumer nullcline is
L shaped. This shape preclude the possibility of an equilibrium with two competitors being limited by the same nutrient. For system (7), however, nullclines
on the carbon-phosphorus plane have the shape of a tilted V, where angle of the
tilt is species specific. The intersection of two Vs with different inclinations, as
seen in Fig. 5, results in an equilibrium where two competitors are limited by
the content of the same chemical element (phosphorus) in the prey.
The key observation obtained here is that stoichiometric considerations provide and expand the possibility for coexistence in the form of a stable equilibrium
- the most restrictive form of coexistence. Limit cycles, chaotic fluctuations, and
persistence more generally, provide other forms for coexistence. How stoichiometry affects these other forms of coexistence remains to be examined. This
question brings up Hutchinson’s (1961) famous “paradox of the plankton”: how
15
so many phytoplankton species can coexist on relatively few resources. Huisman & Weissing (1999, 2000, 2001), using numerical simulations, showed that
coexistence of phytoplankton can be achieved through chaotic fluctuations: for
example, by twelve species on five resources. They used a conventional resourcebased model with fixed chemical composition of all species. But in reality, phytoplankton can exhibit highly variable chemical composition (Elser & Urabe,
1999) suggesting that stoichiometric considerations may provide a fresh and
alternative avenue to the resolution of “paradox of the plankton.”
We should caution, however, that stoichiometric considerations provide limited possibilities for coexistence at an equilibrium. For example, equilibrium
is impossible when we add a third competitor to system (7) but continue to
track only two elements, carbon and phosphorus. One can use this limitation
to argue that our result does not disagree with the CEP, because still at most
two consumers can coexist on two chemical elements (abiotic resources) that are
just packaged in one biotic resource. Abrams (1988), discussing the problem of
counting resources in the context of the CEP, stated that “it would clearly be
inappropriate to define resources in such a way that the theory [of the CEP]
is simply false”. Given our result then, if one wishes to save the CEP, one
should not count a biotic resource as a single resource. The idea that one biotic
resource can represent more than one resource is not novel. For example, different parts of plants can provide distinct resources to herbivores (Tilman, 1982,
Abrams, 1988). What is novel in our approach is its reliance not on a particular
species-specific interpretation of the biotic resource but on the undisputable fact
that all biotic resources are always a package of multiple chemical elements
to all their consumers. This fact, which is independent of the species considered, invalidates the CEP with respect to biotic resources, unless one views a
single biotic resource as multiple abiotic resources (chemical elements). It is
intriguing, however, that a stable equilibrium can exist when two consumers
are simultaneously limited by one essential element in the prey (in our specific
case, phosphorus). In fact, at such an equilibrium, the two consumers and the
resource are all competitors (as the signs in the ecosystem matrix (25) suggest),
and one can argue that three competitors coexist on just two essential elements.
Next, we discuss possible extensions of stoichiometric model (7). Though
phosphorus is of special interest because it appears tightly linked to growth
rates of individual organisms through ribosomal function and protein synthesis
(Elser et al., 1996), other elements (such as N, S, or Ca) may also have important
but unappreciated effects on consumer-resource interactions (Sterner & Elser,
2001; Williams & Frausto da Silva, 2001). To explore how such multi-element
stoichiometry may affect coexistence and other ecological dynamics, one may
analyze the following version of system (7) that models k consumers exploiting
16
one biotic resource consisting of n essential chemical elements:














x




x0 (t) = bx 1 −

Ã
Ã
!
!


k
k



X
X





min(K,
N
−
s
y
,
...,
N
−
s
y
)
/q
/q

1
1i
i
1
n
ni
i
n



i=1
i=1


m

X
−
fi (x)yi


i=1 

Ã
Ã
!
!


k
k

X
X




N1 −
Nn −
s1i yi
sni yi 








i=1
i=1
0

y
(t)
=
e
min
1,
,
...,
 fi (x)yi − di yi


i
i




xs
xs
1
n







(33)
where Ni is the total amount of i-th nutrient in the system, qi is the resource’s
minimal i-th nutrient content, sij is the j-th consumer’s constant (homeostatic)
i-th nutrient content and the other parameters are as in the model (7).
Our experience with model (7) suggests that the ultimate dynamics of model
(33) will depend on both model parameters and initial population densities.
Straightforward algebraic study and numerical work indicate that stable positive
steady states are possible for some set of parameters if k < n. This suggests that
within a multi-element stoichiometric framework, the CEP may be thoroughly
violated. It also suggests a new metric by which to judge the maintenance of
biodiversity in competitive systems. In particular, if one biotic resource provides
n essential chemical elements for competing consumers, it may support up to n
different competitors at a stable equilibrium.
The formulation of a plausible and tractable mathematical version of model
(33) with more than one biotic resource is challenging. This difficulty stems from
the competition among resource species, which forces one either to consider
free pools of essential elements (i.e., not bound in any biomass) and model
their uptake, or to find some other way to distribute each essential element
among all resource species. A mathematically easier case can be imagined in a
patchy habitat, where specialist consumers in each patch exploit a distinct biotic
resource that provides n essential chemical elements. Then, m distinct biotic
resources can support up to m × n consumer species at a stable equilibrium.
In our model (7), we assumed no spatial heterogeneity and no external fluctuations in the system. Instead, chemical heterogeneity within and among species
alone provides mechanisms for coexistence. Can puzzling aspects of biodiversity
on Earth be, at least partially, explained by such chemical heterogeneity? We
note that the worsening quality of the resource weakens consumer-resource interactions by limiting carbon flow into consumer biomass. As we have shown, such
weakening promotes coexistence, a result that agrees with “the weak interaction
strength” hypothesis (McCann et al., 1998). This suggests that stoichiometric
considerations can play an important role in the biodiversity-stability debate
17
(McCann, 2000).
There are other areas of population dynamics and ecology where stoichiometry may provide new insights. For example, chemostat theory is one of the
most thoroughly studied parts of population dynamics and one where nutrients
play a crucial role (e.g. Hsu et al., 1981; Smith & Waltman, 1995; Li et al.,
2000). This theory is particularly suitable for extensions that encompass stoichiometric principles. Another interesting avenue is a coupling of stoichiometric
effects with spatial ones. Spatial models are central to ecology (e.g. Durrett &
Levin, 1994; Lewis, 1994); how the interplay between multiple patches and the
ratios of multiple nutrients results in observed ecological patterns remains to be
thoroughly evaluated, but some work in this area emerged recently. Work by
Fagan and Bishop (2000) on Mount St. Helens suggests that nutritional quality
of invading plants may depend on their spatial position, leading to differential
effects on herbivores and on the dynamics of primary succession. Codeco &
Grover (2001) provided another starting point by considering the effects of various P:C supply ratios on competition between algae and bacteria in a gradostat
(interconnected multiple chemostats).
Understanding how the balance of chemical elements affect population and
ecosystem dynamics becomes ever more important as human activities profoundly change global cycles of carbon and phosphorus as well as other elements
essential for all life, like nitrogen and sulfur. In model (7) with just two chemical elements, variation in stoichiometry of one prey species qualitatively alters
food web dynamics. It is plausible that similar effects take place in diverse
ecosystems where multiple elements and many species are at play.
6
Acknowledgements
We thank Chris Klausmeier for insightful comments. This research has been
partially supported by NSF grant DMS-0077790 and IBN-9977047. I.L. also
has been partially supported by NSF grant DEB-0083566.
7
Appendix
Here is the proof of the Theorem 1 on boundedness and invariance of solutions
of model (7).
Proof. Let us prove by contradiction, i.e. assume that there is time t∗ > 0,
s.t. a trajectory with initial conditions in ∆ touches or crosses the boundary
∂∆ for the first time. Since the boundary consists of at most five surfaces (four
if K > P/q), we divide the crossing into five possible cases: x(t∗ ) = 0, x(t∗ ) =
k = min(K, P/q), y1 (t∗ ) = 0, y2 (t∗ ) = 0 and qx(t∗ ) + s1 y1 (t∗ ) + s2 y2 (t∗ ) = P.
Case 1: Assume x(t∗ ) = 0 (here we exclude the edge P = s1 y1 − s2 y2 lying
on x = 0 plane). Let Pmin = mint∈[0,t∗ ] (P − qx(t) − s1 y1 (t) − s2 y2 (t)) > 0,
18
ỹi = maxt∈[0,t∗ ] yi (t) and note that fi (x) ≤ fi0 (0)x. Then,
µ
x (t) = rx(t) 1 −
0
x(t)
min(K, (P − qx(t) − s1 y1 (t) − s2 y2 (t))/q)
!
à µ
¶ X
2
x(t)
fi0 (0)ỹi x(t) ≡ αx(t)
≥ r 1−
−
min(K, Pmin )
i=1
¶
−
2
X
fi (x(t))yi (t)
i=1
where α is some constant. Thus, x(t) ≥ x(0)eαt > 0, which implies that x(t∗ ) >
0.
Case 2: Assume x(t∗ ) = k. Then
µ
¶
µ
¶
x (t)
x (t)
0
x (t) ≤ rx (t) 1 −
≤ rx (t) 1 −
.
min(K, P/q)
k
The standard comparison argument yields that x(t) < k for all t ≥ 0.
Cases 3 and 4: Assume that y1 (t∗ ) = 0 or y2 (t∗ ) = 0. Since yi0 (t) ≥ −di yi (t)
it follows that yi (t) ≥ yi (0)e−dt > 0, i = 1, 2 for t ≥ 0.
Case 5 (includes the edge excluded in case 1) : Assume
qx(t∗ ) + s1 y1 (t∗ ) + s2 y2 (t∗ ) = P.
(34)
Since for all t ∈ [0, t∗ ), qx(t) + s1 y1 (t) + s2 y2 (t) < P, it follows that
qx0 (t∗ ) + s1 y10 (t∗ ) + s2 y20 (t∗ ) ≥ 0
(35)
First, let us show that x(t∗ ) 6= 0 (to avoid division by 0 later). If x(t∗ ) = 0,
then s1 y1 (t∗ ) + s2 y2 (t∗ ) = P.
However,
¶
2 µ
X
fi (x)
(s1 y1 + s2 y2 ) ≤
ei (P − s1 y1 + s2 y2 )
yi − di yi ≤
x
i=1
µ
¶
2
2
X
X
s1 y1 + s2 y2
(ei (P − s1 y1 + s2 y2 ) fi0 (0)yi ) =
(ei fi0 (0)yi ) P 1 −
,
P
i=1
i=1
0
and the standard comparison argument yields that s1 y1 (y) + s2 y2 (t) < P for all
t ≥ 0. Hence, x(t∗ ) 6= 0.
Observe that at t∗ , the producer reaches its carrying capacity allowed by
available phosphorus and has lowest quality, q, since (34) implies that
(P − s1 y1 (t∗ ) − s2 y2 (t∗ ))/q = x(t∗ ).
Substituting it to (7 a) yields:
µ
x0 (t∗ ) = rx (t∗ ) 1 −
¶ X
2
2
X
x(t∗ )
∗
∗
f
(x(t
))y
(t
)
≤
−
fi (x(t∗ ))yi (t∗ ).
−
i
i
min(K, x(t∗ ))
i=1
i=1
(36)
19
From (34) it also follows that (P − s1 y1 (t∗ ) − s2 y2 (t∗ ))/x(t∗ ) = q. Substituting
it to (7b, c) yields bounds on yi0 (t∗ ), i = 1, 2 :
¾
½
q
q
fi (x (t∗ ))yi (t∗ ) ≤ ei fi (x (t∗ ))yi (t∗ )
(37)
yi0 (t∗ ) = ei min 1,
si
si
Using (36), (37) and the fact that e < 1, we obtain the following:
qx0 (t∗ ) + s1 y10 (t∗ ) + s2 y20 (t∗ ) ≤
2
X
(−1 + ei )qfi (x (t∗ ))yi (t∗ ) < 0.
i=1
This contradicts (35) and completes the proof.
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8
Figure Legends
Figure 1. Stoichiometric properties confine dynamics to a trapezoid-shaped area
and divide the phase plane into two regions. In Region I, like in classical LotkaVolterra model, food quantity limits consumer growth. In shaded Region II food
quality (that is, the phosphorus content of the resource) constrains consumer
growth. Competition for the limiting chemical element between the consumer
and the resource alters their interactions from (+, -) in Region I to (-, -) in
Region II. This bends the consumer nullcline in Region II. This shape of the
consumer nullcline, with two x-intercepts, creates the potential for multiple
positive steady states. A solid circle denotes stable equilibrium, clear circles unstable equilibriums. (The figure is taken from Loladze et al. (2000).)
Figure 2. a) The phase space of model (7) with parameter values listed in
Table 1 and K = 0.3 mg C/l. The low value of K means low light intensity,
which assures good quality (high P:C) of the resource. Consumer dynamics are
governed by resource quantity and the competitive exclusion principle (CEP)
holds; that is, the second consumer, whose nullsurface has the lowest x-intercept,
wins. b) Slicing Fig.2a with plane y1 = 0 at the equilibrium shows its similarity
with Fig. 1.
Figure 3. Same as Fig. 2a, but with K=1.0 mg C/l. The high value of K
means high light intensity, which leads to bad quality (low P:C) of the resource.
The dynamics is governed by the resource quality and the CEP does not apply.
Both consumers are limited by phosphorus content of the resource and coexist
at an internal attracting equilibrium.
23
Figure 4. Slicing the phase space in Fig. 3 at the internal equilibrium (0.59,
0.26, 0.17) with plane y1=0.26. Unlike Fig. 2, where consumer-resource interactions are (+, -) at the boundary equilibrium, consumer-resource interactions
in Fig. 3 and Fig. 4 change to (-,-) at the internal equilibrium.
Figure 5. Analogous to resource competition theory, we draw consumer
nullclines on the carbon-phosphorus phase plane. Instead of the usual L shape,
the nullclines take the shape of tilted V, where the angle of the tilt is determined
by the stoichiometric properties of each species. This shape can lead to an
internal attracting equilibrium, where the two consumers coexist, even though
their growth is limited by the same chemical element in the prey.
24
y
II
Nullclines:
resource
consumer
I
0
Figure 1.
K
x
Slice produced by plane y1=0.26
0.35
trajectory
resource
1st consumer
2nd consumer
0.3
y2 (mg C/l)
0.25
0.2
.
0.15
0.1
0.05
0
Figure 4
0
0.2
0.4
0.6
x (mg C/l)
0.8
1
1.2
Nullclines on Carbon-Phosphorus plane
0.04
Total phosphorus, P (mg P/l)
0.035
0.03
0.025
0.02
1st consumer
2nd consumer
0.015
Figure 5
0
0.1
0.2
0.3
0.4
Carbon, x (mg C/l)
0.5
0.6
0.7
0.8