Ecological Specialization and Trade Affect the Outcome of Negotiations in Mutualism.
Author(s): Emily Grman, Todd M. P. Robinson, Christopher A. Klausmeier
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Source: The American Naturalist, Vol. 179, No. 5 (May 2012), pp. 567-581
Published by: The University of Chicago Press for The American Society of Naturalists
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vol. 179, no. 5
the american naturalist
may 2012
Ecological Specialization and Trade Affect the Outcome of
Negotiations in Mutualism
Emily Grman,* Todd M. P. Robinson, and Christopher A. Klausmeier
Kellogg Biological Station and Department of Plant Biology, Michigan State University, Hickory Corners, Michigan 49060
Submitted July 14, 2011; Accepted January 8, 2012; Electronically published March 30, 2012
Online enhancements: appendix, zip file.
abstract: By definition, mutualisms involve the exchange of goods
or services between partners. It has been shown that mutualism can
grade into parasitism, but even when exchange is mutually beneficial,
a conflict of interest remains because each partner benefits from
reaping more benefits at a lower cost. Metaphorically, the partners
negotiate the conditions of trade, the outcome of which will determine the net benefit to each partner. Each partner can adjust its
allocation to self-provisioning while negotiating the ratio at which
benefits are exchanged. To understand how these two features of
trade affect mutualisms, we used the example of the plant–arbuscular
mycorrhizal mutualism and modeled uptake and trade of two resources, phosphorus and carbon. In most contexts, the fungus specialized on phosphorus uptake while the plant took up both phosphorus and carbon. However, when phosphorus was abundant and
light was scarce, the plant specialized, taking up only carbon and
relying on trade for phosphorus. Resource availability was the most
important factor determining specialization and the outcome of negotiation and trade, but other aspects of the context were also important. These results suggest experiments to link these two key
features of trade with environmental conditions to determine the
outcome of mutualism.
Keywords: Nash bargaining solution, trade-based mutualism, plantmycorrhizal interactions, context-dependent species interactions,
colimitation, biological market model.
Introduction
Mutualisms are ubiquitous, with most species on the
planet participating in at least one mutualism (Bronstein
2001). Furthermore, mutualisms can be an important
force structuring communities (e.g., Klironomos et al.
2011). However, mutualisms are only one conditional outcome of species interactions. Interactions that can result
in positive outcomes for the participating species may
sometimes result in neutral (or even negative) outcomes,
depending on the ecological context in which the inter* Corresponding author; e-mail: [email protected].
Am. Nat. 2012. Vol. 179, pp. 567–581. 䉷 2012 by The University of Chicago.
0003-0147/2012/17905-53192$15.00. All rights reserved.
DOI: 10.1086/665006
action occurs (Bronstein 1994). The traits of interacting
species, species abundances, and environmental conditions
can all affect how beneficial a mutualism is (Lekberg and
Koide 2005; Hoeksema et al. 2010). Many ecologists have
recognized the “context dependency” of mutualisms, but
it has been difficult to understand which factors determine
the outcome. Context dependency thus complicates our
understanding of how mutualisms structure communities.
Explaining the context-dependent outcome of species interactions continues to be one of the most pressing questions facing ecologists (Agrawal et al. 2007).
Although the outcome of mutualism can vary, all mutualisms share at least one feature: trade. One partner in
a mutualism receives some good or service from the other
and, in exchange, provides its partner some good or service
in what can be considered a biological market (Noë and
Hammerstein 1995). Bronstein (2001) identified three major classes of mutualisms (protection, transportation, and
nutritional mutualisms), and trade occurs in each class.
However, it may be easiest to quantify trade in nutritional
mutualisms, which involve reciprocal fluxes of resources
such as sugar, nitrogen, or phosphorus. Understanding the
fluxes into and out of each organism may help elucidate
variation in the outcome of mutualisms.
Allocation to direct uptake and the exchange ratio are
two key features of trade that can cause variation in the
outcome of mutualisms. Organisms may influence the
amount traded by adjusting their allocation to direct uptake
of the resources exchanged. An organism might “invest” in
its partner by increasing uptake of the resource it trades
away. For example, plants supporting belowground nutritional mutualists such as rhizobia or mycorrhizal fungi
might increase carbon uptake (photosynthesis) in order to
increase carbon supply to their partner (Miller et al. 2002).
However, optimal allocation to either trade or independent
uptake will depend on how efficiently each partner can
provide the resource. When each partner is relatively more
efficient at taking up the resource it trades away than the
partner receiving the resource, conditions of relative ad-
568 The American Naturalist
vantage exist (Ricardo 1817; Schwartz and Hoeksema 1998).
Under these conditions, it is thought that optimal allocation
should lead to partners specializing on uptake of the resource they trade away (Schwartz and Hoeksema 1998). For
plants and belowground mutualists, this would mean that
the plant should specialize on carbon uptake, while the
belowground mutualist should specialize on nutrient provisioning. However, empirical data indicate that plants do
not always allocate carbon to belowground mutualists
(Johnson 2010), suggesting that there may be variation in
the optimal strategy. To understand the outcome of mutualisms, we must understand the partners’ optimal allocation strategies and deviations from those strategies.
The second key feature of trade is the exchange ratio.
The gains obtained from trade depend on the cost (what
is exchanged away) and the benefit (what is received in
return). In other words, the ratio at which resources are
exchanged may affect whether a mutualism is beneficial
or not (Schwartz and Hoeksema 1998). Under conditions
of relative advantage, it is possible to identify a range of
favorable exchange ratios where resource acquisition is
“cheaper” for both partners through trade (Schwartz and
Hoeksema 1998). If trade at a favorable exchange ratio is
enforced, the mutualism will be evolutionarily stable (McGill 2005).
However, even among mutually beneficial exchange ratios, variation could cause differences in the degree to
which partners benefit from the mutualism. Each partner
would gain the most by minimizing its cost and maximizing its benefit (Schwartz and Hoeksema 1998; Bronstein 2001), creating a conflict of interest between the partners. Although this conflict of interest could lead to
selection for one partner to “cheat” and obtain the benefit
of trade without paying the cost (West et al. 2002), Bronstein (2001) points out that this need not be the case for
all mutualisms. Furthermore, many mutualisms typically
result in positive outcomes for both partners (Karst et al.
2008; Chamberlain and Holland 2009; Hoeksema et al.
2010), suggesting that partners somehow negotiate a mutually beneficial exchange ratio. Thus, even in the absence
of antagonism, variation in the exchange ratio is important. To determine how the partners would solve this conflict of interest to negotiate a single mutually beneficial
exchange ratio, Akçay and Roughgarden (2007a) built a
mechanistic model of trade between rhizobia and legumes.
The negotiated exchange ratio turned out to be identical
to the Nash bargaining solution (Nash 1950, 1953), an
outcome commonly found in cooperative game theory
(Akçay and Roughgarden 2007a). However, they did not
explore the context dependency of this negotiated exchange ratio, whether it would interact with the partners’
allocation strategies, or whether it would explain variation
in the outcome of the mutualism.
We asked what determines these two key features of
trade—allocation and the exchange ratio—and how they
predict the degree to which two partners gain from a mutualistic interaction. We then varied species traits and resource availability to explore the relationship’s context dependency. Uniting and building on the key results of Akçay
and Roughgarden (2007a) and Schwartz and Hoeksema
(1998), we modeled a classic example of a nutritional mutualism, plants and mycorrhizal fungi. In this interaction,
fungi develop a network of hyphae in the soil and plant
roots. Plants transfer photosynthetically fixed carbon to
the fungi in exchange for benefits, primarily soil nutrients
(Smith and Read 1997). Plants are capable of taking up
nutrients directly, but mycorrhizal fungi could be more
efficient because, per unit length, hyphae require less carbon to build than roots (Smith and Read 1997). Therefore
plant allocation to mycorrhizae may be a better investment
than allocation toward direct uptake of nutrients. Plants
and fungi differ in their relative requirements for carbon
and nutrients, which could also affect the benefit of trade
(Hoeksema and Schwartz 2003). Empirically, plant benefit
from fungi is highly variable and incompletely understood.
A meta-analysis of hundreds of studies explained only
23%–41% of variation in plant benefit, despite including
at least eight important predictor variables such as species
identity, species traits, and soil fertility (Hoeksema et al.
2010). This suggests that not all important predictor variables could be included in that analysis. In particular,
context-dependent variation in the fluxes of carbon and
soil nutrients is insufficiently understood.
We modeled trade between a plant and a fungus able
to take up a soil nutrient and carbon in conditions of
relative advantage (Schwartz and Hoeksema 1998) and
negotiate an exchange ratio according to the Nash bargaining solution (Akçay and Roughgarden 2007a) on a
very short (behavioral) timescale. Antagonistic outcomes
are not an option in this framework, but the model does
predict variation in the magnitude of benefit the partners
experience. Although we discuss the model in the context
of a specific example (plants and arbuscular mycorrhizal
fungi), it is conceptually applicable to all mutualisms because all involve trade. This model allows us to answer
three ecological questions: (1) Will both partners specialize
on uptake of the resource traded away? (2) What exchange
ratio will they negotiate? (3) Can these two features of
trade help explain why key predictors fail to fully explain
variation in the outcome of mycorrhizal mutualisms? We
parameterized the model to explore realistic ranges of critical ecological factors, specifically the traits of the partners
(resource uptake efficiency), community structure (relative
abundances), and the availability of resources thought to
be particularly important to this mutualism (light and a
soil nutrient).
Ecological Specialization and Trade 569
The Model
We model plant growth rate per unit biomass (gP) as the
lesser of the growth allowed by two limiting resources, carbon (C) and a soil nutrient (N for nutrient). We model the
plant as a population consisting of a single clonal individual.
Plant acquisition of available nutrient is the sum of direct
uptake and nutrient gained in trade. The nutrient taken up
is evenly distributed throughout the plant’s biomass. The
rate of direct uptake per unit biomass is fNP, which can be
thought of as a function of available nutrient in the soil
and allocation of effort to direct uptake. The amount of the
nutrient obtained through trade is determined by X, the
total flux of carbon exchange (summed across all plant biomass), and by T, the C : N exchange ratio that determines
how much carbon the plant trades away for a given amount
of the nutrient. The amount of growth allowed by the plant’s
nutrient income is the product of its nutrient acquisition
and the yield of biomass per unit nutrient (YNP). Similarly,
the net rate of plant acquisition of carbon is the difference
between photosynthetically driven carbon assimilation (fCP)
and carbon lost in trade, which is converted into carbonlimited growth rate by YCP.
[ (
g P p min YNP f NP ⫹
) (
)]
X
X
, YCP fCP ⫺ .
P#T
P
(1)
We model growth of a single, clonally spreading individual fungus (gF) similarly, except that the fungus gains carbon
and loses nutrient as a result of trade. Note that the fungus
is capable of direct carbon uptake, so this model is appropriate for both arbuscular (fCF p 0) and ectomycorrhizal
fungi (fCF 1 0), as well as other nutritional mutualisms such
as lichens, corals, and legumes and rhizobia.
[ (
g F p min YNF f NF ⫺
) (
)]
X
X
, YCF fCF ⫹ .
F#T
F
(2)
Henceforth we measure species biomass in terms of carbon, so we set the yield coefficients YCP p YCF p 1 without loss of generality; in this case YNP and YNF represent the
C : N stoichiometry of the plant and fungus, respectively.
We assume that the plant and the fungus can modify
the amount of nutrient and carbon taken up by adjusting
allocation to resource uptake over short behavioral timescales. At this timescale, allocation occurs on a physiological scale, for example, through altering phosphorus uptake
or carbon fixation machinery. Slower allocation adjustment, for example, through changes in root : shoot ratio,
results in qualitatively similar results (appendix, “Reallocation of Uptake Machinery Slower than Negotiation,”
available online). We model allocation as the proportion
of effort dedicated to obtaining one resource at the expense
of the other. We assume a linear trade-off (Klausmeier et
al. 2007), so that plant allocation for nutrient uptake is
ANP and allocation for carbon uptake is 1 ⫺ A NP; fungus
allocation for carbon uptake is ACF, and allocation for
nutrient uptake is 1 ⫺ A CF. Therefore, f NP p A NP f NP
,
fCP p (1 ⫺ A NP)fCP, f NF p (1 ⫺ A CF)f NF, and fCF p A CF fCF .
We consider nonlinear trade-offs in the appendix, “Extension to Nonlinear Trade-offs.” Efficiencies of direct re
source uptake rates (f NF
, f NP
, fCF , and fCP
) are functions of
fixed traits of the organisms and of resource available in
the environment. Because we are interested in situations
of relative advantage, we assume the plant is better at
taking up carbon than nutrient (fCP
) and that the
1 f NP
fungus is better at taking up nutrient than carbon (f NF
1
fCF). Notation is summarized in table 1.
Plant and fungal growth rates are context dependent in
our model. Specifically, they depend on plant and fungal
abundances (P and F), organismal stoichiometry (YNP and
YNF), and carbon and nutrient uptake rates (f NF
, f NP
, fCF ,
and fCP). We assume that the interactions modeled here
(allocation decisions and exchange ratio negotiations) take
place on much faster timescales than changes in species
abundances or nutrient availabilities, which we treat as
fixed parameters rather than dynamic variables. Note that
although the plant and the fungus take up carbon and
nutrient through very different mechanisms (e.g., the plant
fixes carbon through photosynthesis in the leaves and takes
up the nutrient through the roots, while the fungus takes
up the soil nutrient and potentially carbon through extraradical hyphae), these differences do not matter because
we do not model resource depletion. In addition to these
species-specific or environmental contexts, plant and fungus growth also depend on the amount of resources obtained through trade, which could also depend on those
specific contexts.
Model Analysis
To understand how the two key features of trade, allocation
and the exchange ratio, affect the outcome of trade, we
analyze the effect of environmental parameters on three
interrelated factors: the total amount of carbon exchanged
(X), allocation to uptake of the resource each species is better
at obtaining (ANP, ACF), and the exchange ratio (T). Modeling the outcome of trade between the plant and the mycorrhizal fungus requires that we solve these three problems
simultaneously, but we will describe them sequentially to
build up the problem. First, we will examine what determines the volume of trade (X) for fixed allocation (ANP,
ACF) and exchange ratio (T). Second, given X, we then allow
the plant and the fungus to optimize ANP and ACF to maximize their growth rates. Finally, we allow the plant and the
fungus to negotiate T while simultaneously determining X
and optimizing ANP and ACF. All three problems are solved
570 The American Naturalist
Table 1: List of parameters in the model, their biological interpretations, and estimates of values with references
Parameter
ANP
YNP
fNP
VCP
KCP
ACF
YNF
fNF
VCF
KCF
P
F
N
C
X
T
Interpretation (units)
Range of values
References
Plant allocation to nutrient uptake ([g C root] .3–.8
Craine et al. 2002; Mahaney et al. 2008;
[g C plant]⫺1)
Grman 2011
Plant carbon yield per unit nutrient ([g plant C 150–500
Elser et al. 2000; Niklas and Cobb 2006; Jonas
m⫺2] [g plant N⫺1 m⫺2])
and Joern 2008
Plant nutrient uptake rate ([g N m⫺2][s⫺1] [g
3.4–600 # 10⫺10 Sanders and Tinker 1973; Smith 1982; Mcroot C m⫺2]⫺1)
Gonigle and Fitter 1988; Miller et al. 1995;
Craine et al. 2002
Plant max carbon uptake (photosynthesis) rate 9.7–67.2 # 10⫺7 Awada et al. 2003; Lee et al. 2003; Allred et al.
([g C fixed m⫺2][s⫺1][g shoot C m⫺2]⫺1)
2010
Plant carbon half-saturation constant ([g C]
200–400
Awada et al. 2003; Allred et al. 2010
[mmol m⫺2 s⫺1])
Fungal allocation to carbon uptake ([g C intra- .33–.9
Miller et al. 1987
radical][g C fungus]⫺1)
Fungus carbon yield per unit nutrient ([g fun- 30–100
Cleveland and Liptzin 2007
gal C m⫺2][g fungal N⫺1 m⫺2])
Fungus nutrient uptake rate ([g N m⫺2][s⫺1][g
1.2–124 # 10⫺6 Jakobsen et al. 1992; Miller et al. 1995; Schweiextraradical C m⫺2]⫺1)
ger and Jakobsen 1999; Smith et al. 2000,
2004
Fungus max carbon uptake rate ([g C
0
m⫺2][s⫺1][g intraradical C m⫺2]⫺1)
Fungus carbon half-saturation constant ([g
NA
C][mmol m⫺2 s⫺1])
Plant biomass ([g plant C] m⫺2)
686–2,128.5
Craine et al. 2002; Mahaney et al. 2008
Fungal biomass ([g fungus C] m⫺2)
.006–45
Miller et al. 1995; Grman 2011
Nutrient available in the environment ([g N][g 10⫺8–0.0005
Grman 2011
soil]⫺1)
Carbon (light) available in the environment
100–1,600
Grman 2011
(mmol m⫺2 s⫺1)
Amount of carbon exchanged ([g C] m⫺2 s⫺1)
Exchange (trade) ratio of carbon for nutrient
.004–.007
Pearson and Jakobsen 1993; Kiers et al. 2011
([g C][g N]⫺1)
Note: See appendix, “Calculations of Parameter Values,” available online, for details of calculations. We use phosphorus as the soil nutrient (N) and
assume 50% of biomass is carbon (C). Where possible, values for plant parameters are taken from Andropogon gerardii (big bluestem), Bromus inermis
(smooth brome), and Elymus repens (quackgrass).
simultaneously and organisms can adjust the volume of
carbon exchanged, allocation, and the exchange ratio over
very short, behavioral (physiological) timescales. Details of
the analysis are presented in the appendix, “Model Analysis,” and a zip file with Mathematica code is provided as
supplementary material, available online.
What Determines X, the Total Amount of
Resources Exchanged?
Clearly, the amount of carbon and nutrient traded will
determine how beneficial the interaction is for both partners. In general, one partner or the other will limit the
amount traded by failing to produce enough resource to
match its partner’s production. There are three possible
cases: plant-limited, fungus-limited, and colimited trade.
If the plant limits trade, the plant has no surplus carbon,
and we can solve for the amount of carbon exchanged
(eq. [A3], available online) and surplus fungal nutrient
(eq. [A4]; appendix, “What Determines X, the Amount of
Resources Exchanged?”). Similarly, if the fungus limits
trade, the fungus has no surplus nutrient, and we can solve
for the amount of carbon exchanged (eq. [A6]) and surplus plant carbon (eq. [A7]). The borderline case of colimited trade occurs when both the plant and the fungus
have no surplus resource. Which case occurs depends on
population sizes, resource uptake rates, stoichiometries,
and the exchange ratio (eqq. [A5], [A8]).
How do the Plant and the Fungus Adjust Their
Allocation, ACF and ANP?
Should a nutrient-limited plant increase direct nutrient
uptake (increase ANP), or should it increase its surplus
Ecological Specialization and Trade 571
carbon for trade with a carbon-limited fungus (decrease
ANP)? We again consider the cases of plant-, fungal-, or
colimited trade. In the plant-limited case, the derivative
of plant growth rate (eq. [A13]) with respect to ANP is
negative when T ! fCP
(appendix, “How Do the Plant
/f NP
and the Fungus Adjust Their Allocation to Carbon or Nutrient Uptake, ACF and ANP?”). This means that the price
(T) the plant pays for nutrient by trading carbon is less
than the opportunity cost of allocating more effort to taking up nutrient directly. Therefore, the plant benefits by
decreasing its own uptake of nutrient (decreasing ANP),
increasing carbon uptake, and allocating more carbon to
trade (fig. A1), as long as the fungus can match trade. If
trade is plant limited for all ANP, then the optimal
A*NP p 0. However, when the fungus limits trade, the derivative of plant growth rate (eq. [A14]) with respect to
p 0), so the plant should inANP is positive (unless f NP
crease its allocation to nutrient uptake until it is colimited.
The fungal optimal strategy is similar, because the fungal
growth rate function is symmetrical with the plant growth
rate function. Therefore, if fCF /f NF
, the opti! T ! fCP
/f NP
mal allocation strategy for both partners is when one or
both of the partners is completely allocated toward uptake
of the resource traded away (eqq. [A16], [A19], [A22]; fig.
A1). The optimal allocation strategy is best for both partners, even though each partner acts selfishly to maximize
its growth rate given the allocation strategy of its partner.
How Will the Plant and the Fungus Negotiate the
Exchange Ratio T?
Above, we showed that if exchange ratios are favorable to
both partners (fCF /fNF
), selfish allocation de! T ! fCP
/fNP
cisions by one partner consistently benefit the other partner and there is no conflict. However, there is conflict in
the third problem that the plant and the fungus must
simultaneously solve: negotiating the C : N exchange ratio
T. Each partner gains more from trade when T is closer
to the ratio at which the other partner takes up carbon
and the nutrient directly (fig. 1). Plant gain from trade is
0 when T is equal to the plant C : N uptake ratio (fCP
/
f NP, the isolation cost ratio in Schwartz and Hoeksema
1998). Plant gain from trade increases when nutrient becomes cheaper (relative to carbon) through trade than it
is through direct uptake. In contrast, the fungus benefits
from trade whenever T is greater than the fungal C : N
uptake ratio (fCF /f NF
), with gain from trade increasing as
T increases.
How do the plant and the fungus settle on a single,
mutually acceptable T when they have competing interests?
One solution to this bargaining problem is the Nash bargaining solution (NBS; Nash 1953), which is the strategy
that maximizes the product of the gains to both players
Figure 1: Plant (solid line) and fungal (dashed line) gain from trade
(g*P ⫺ g*P, no trade and gF* ⫺ gF,* no trade) as a function of the C : nutrient exchange ratio (T). Trade is beneficial for both partners only when the
exchange ratio is between the ratio at which the fungus directly takes
up carbon and the nutrient (fCF /fNF
) and the ratio at which the plant
directly takes up carbon and the nutrient (fCP /fNP
). Parameters are
chosen to simulate the case of colimited trade (YNP p 1, YCP p 1,
YNF p 1, YCF p 1, fCP p 3, fNP
p 1, fNF
p 3, fCF p 1, F p 1, P p
1).
(the Nash product). The NBS was originally derived on
the basis of an axiomatic approach (Nash 1953) but has
been shown to be the optimal solution of a model of
strategic bargaining (Binmore et al. 1986). The NBS has
been applied to a wide variety of social and economic
situations (van Damme 1986; Border and Segal 1997) but
only recently introduced to ecology (Roughgarden et al.
2006; Akçay and Roughgarden 2007a). As an assumption
of an alternative theory of evolution to sexual selection,
the NBS has been controversial (McNamara et al. 2006;
Roughgarden et al. 2006; Akçay and Roughgarden 2007b).
However, Akçay and Roughgarden (2007a) demonstrated
that a negotiation model based on offers and rejections or
acceptances of nutrient fluxes between two mututalists
leads to the NBS. In that model, each partner acts individualistically to increase its own gain from trade. We assume that such flux-based negotiations take place between
a plant and a mycorrhizal fungus, and therefore adopt the
NBS to solve the third problem of how the plant and the
fungus will negotiate the exchange ratio T (see appendix,
“How Will the Plant and the Fungus Negotiate the Exchange Ratio T?”). The NBS assumes only that partners
can sense the resource they receive from trade, a reasonable
assumption in plant-mycorrhizal interactions (Bücking
and Shachar-Hill 2005; Javot et al. 2007).
The Nash product P (eq. [A24]) is the product of the
gains to the two partners. Because fitness depends on
whether the plant or the fungus limits trade, the Nash
572 The American Naturalist
product can be composed as the minimum of two unimodal curves, and the NBS is the maximum of the minimum of these two curves (fig. A2). There are three candidate points for this maximum: the maximum of
plant-limited Nash product (eq. [A25a]), the maximum
of fungus-limited Nash product (eq. [A25b]), or the intersection of the two curves (eq. [A25c]; Abrams 1987).
The NBS, T*, is the T of these three candidates that leads
to the largest P. This comparison also determines whether
trade is plant-, fungus-, or colimited and therefore determines the optimal allocation strategies of the partners.
When trade is ultimately plant limited the plant specializes
on carbon uptake at the NBS (fig. A2A); when trade is
ultimately fungus limited the fungus specializes on nutrient uptake at the NBS (fig. A2C); when trade is colimited
both partners allocate optimally to take up only the resource they trade away (fig. A2B).
Having solved the volume of resource exchanged, how
the plant and the fungus optimally allocate to uptake of
carbon and nutrient and at what ratio they exchange carbon for nutrient, we will now explore how these variables
depend on organismal traits (stoichiometry and uptake
efficiencies), community structure (relative abundances),
and environmental conditions (resource availability). We
will also explore how the two key features of trade, allocation and the exchange ratio, affect plant and fungal gain
from trade. We define gain from trade as the difference
between growth rate with trade and growth rate without
trade.
Effect of Environmental and Species Parameters on the
Identity of the Specialist Partner
A key result of our model is that environmental and species
characteristics determine which partner will specialize in
uptake of the resource traded away by determining which
partner limits trade. In general, the flows of the soil nutrient N and carbon C into negotiations and the demand
for those resources determine which partner comes up
short. This partner limits trade, failing to provide enough
of the resource it trades away to satisfy both its own requirements and its partner’s requirements. This occurs
even though the limiting partner adjusts allocation to specialize on uptake of the resource it trades away and eliminates uptake of the other resource. The nonlimiting partner, on the other hand, is a generalist and takes up both
resources directly from the environment. For example,
when the fungus limits trade, the fungus cannot supply
enough nutrient even when it acts as a nutrient specialist,
eliminating direct carbon uptake and allocating toward
maximal nutrient uptake. Its plant partner therefore acts
as a generalist, allocating to uptake of both nutrient and
carbon (for its own use and to trade away). In the case
of colimitation, both partners specialize on uptake of the
resource traded away.
What Determines Which Partner Will Specialize?
To determine when each partner should specialize, we parameterized the model with values from the literature on
the symbiosis between temperate grasses and arbuscular
mycorrhizal fungi (table 1). We assume that each partner
exists in a homogenous, clonally spreading population
where resources taken up are instantaneously evenly distributed across the population. We study soil phosphorus
as the soil nutrient, modeling uptake as a type II functional
response. We assume that the fungus cannot directly take
up carbon from the environment (fCF p 0; Smith and
Read 1997) and model plant carbon uptake as photosynthesis, which is a function of light availability. We run the
model with all 1,024 possible combinations of the extremes
of empirically derived parameters (table 1) and record the
identity of the specialist (limiting) partner. Fungal specialization on phosphorus uptake occurs in 89% of combinations of environmental conditions and species traits,
10% of combinations result in plant specialization on carbon uptake, and 1% result in the borderline case where
the plant specializes on carbon uptake and the fungus
specializes on phosphorus uptake.
Using a classification tree (library tree in R, ver. 2.12.1),
we determine which parameters are most important in
determining the identity of the specialist (limiting) partner
(fig. 2). Given the range of empirically measured parameter
values, the availability of soil phosphorus is the most important determinant of the identity of the specialist, accounting for 22% of the total deviance. If phosphorus is
low, the fungus always specializes on phosphorus uptake
and the plant is a generalist, taking up both carbon and
phosphorus directly. If phosphorus is high, the identity of
the specialist is determined by fungus maximum phosphorus uptake rate. If the fungus has low capacity for
phosphorus uptake, the fungus is the specialist partner in
254 of 256 cases.
If phosphorus is high and fungus maximum uptake rate
is high, fungal abundance becomes important in determining the identity of the specialist (fig. 2). When the
fungus is small, the plant is the specialist in only 4 of 128
cases. All of these cases are associated with very low carbon
inputs into the symbiosis (low plant maximum photosynthetic rate, small plant biomass, low light availability, and
high plant C : phosphorus ratio). On the other hand, when
the fungus is large, the plant specializes on carbon uptake
in most cases (96/128). This tendency is especially pronounced when carbon inputs into the symbiosis are low.
Over all the empirically derived ranges of the parameters
we investigate, the three most important (phosphorus
Ecological Specialization and Trade 573
Figure 2: Classification tree indicating which parameters control the identity of the specialist (limiting) partner. We examine all possible
combinations of the extremes of measured parameter values (table 1) for maximum photosynthetic rate (VCP), photosynthetic rate halfsaturation constant (KCP), carbon : phosphorus stoichiometries of the fungus (YNF) and plant (YNP), biomasses of the plant (P) and fungus
(F), light availability, and soil phosphorus availability. We use the extremes of measured phosphorus uptake efficiencies (fNP
and fNF
from
table 1) as maximum uptake rates (VNP and VNF, respectively). There are 210 p 1,024 combinations of parameter values in total. We estimate
a single value for the phosphorus uptake half-saturation constants (KNP and KNF p 0.0002 ). Pie sizes indicate the number of parameter
combinations in the branch; the largest pie contains 512 combinations of parameter values and the smallest pie contains 8. The color of
the pie indicates the identity of the specialist partner: black indicates that the plant specializes on uptake of carbon (ANP p 0 ), light gray
indicates that the fungus specializes on phosphorus uptake (ACF p 0 ), and the intermediate gray indicates parameter combinations where
both partners specialize on the resource they trade away.
availability, fungal maximum phosphorus uptake rate, and
fungal abundance) account for 67% of the variation in the
identity of the specialist. Less important parameters determine the identity of the specialist partner for some combinations of those three parameters (figs. A3, A4). These
less important parameters include plant biomass, light
availability, and plant species traits such as the maximum
rate of photosynthesis and carbon : phosphorus stoichiometry. The remaining parameters, fungal carbon : phosphorus stoichiometry, plant maximum phosphorus uptake
rate, and plant photosynthetic half-saturation constant, do
not appear in the tree. These unimportant parameters occasionally cause variation within the terminal nodes of the
tree but explain only 12% of total deviance in the identity
of the specialist. Because our model is deterministic, this
variation in the terminal nodes is not error, so we did not
perform cross-validation or calculate a misclassification
table (Urban 2002; see also Crawley 2007). Instead, we
interpret these parameters as unimportant in determining
the identity of the specialist partner, perhaps because the
ranges of empirically derived parameters are too small to
have much effect.
574 The American Naturalist
What Are the Consequences of Changes in the
Identity of the Specialist Partner?
To better understand the causes and consequences of
changes in the identity of the specialist partner in the
arbuscular mycorrhizal symbiosis, we examine a few combinations of parameters in detail. The availability of resources, especially phosphorus, affects the identity of the
specialist partner (fig. 2). When light (carbon) availability
is low (fig. 3A, 3C, 3E), a habitat with low soil phosphorus
would have a specialist fungus and a generalist plant. In
these conditions, fungal uptake of scarce phosphorus cannot satisfy fungus and plant requirements, so the plant
allocates to direct phosphorus uptake as well as carbon
uptake. Increases in soil phosphorus within this fungusspecialist region would increase the fungus’s ability to trade
away phosphorus, increasing the total flux of carbon and
increasing both partners’ gains from trade but having no
effect on the exchange ratio. A habitat with higher soil
phosphorus, on the other hand, would allow the fungus
to provide enough phosphorus and would challenge the
plant to take up enough carbon, leading to both partners
specializing on uptake of the resource they trade away
(colimited trade). When both partners specialize, increases
in soil phosphorus drive up the exchange ratio (fig. 3A).
Figure 3: Effect of light and phosphorus availability on the total amount of carbon (C) exchanged per day (X), the C : nutrient exchange
ratio T (A, B), plant gain from trade (C, D), fungal gain from trade (E, F), and the identity of the specializing partner who limits trade
(shaded bars in A, B; F indicates fungal specialist, B indicates both partners specialize, and “Plant specialist” indicates plant specialization).
Parameters are set at YNP p 150, YCP p 1, YNF p 30, YCF p 1, fCF p 0, KCP p 200, VCP p 9.7 # 10⫺7, VNP p 3.4 # 10⫺10, KNP p 0.0002,
VNF p 1.2 # 10⫺4, KNF p 0.0002, P p 2,129, and F p 0.36. For low light availability (A, C, E), C p 100 ; for high light availability (B, D,
F), C p 1,600.
Ecological Specialization and Trade 575
This occurs because, when both partners specialize, only
the fungus can take up the phosphorus that would increase
both partners’ growth rates; this power allows the fungus
to negotiate very high exchange ratios that favor the fungus
(fig. 1). These higher exchange ratios increase the total
amount of carbon the plant trades away (fig. 3A), increasing fungal gain from trade but decreasing plant gain from
trade (fig. 3C, 3E). Further increases in soil phosphorus
and consequent increases in the exchange ratio and the
flux of carbon traded away mean that the plant can no
longer supply enough carbon to satisfy both partners. At
this level of phosphorus, the plant must continue to specialize on carbon uptake, but the fungus’s strategy of specialization on phosphorus brings in more phosphorus than
the plant can pay for. However, because the fungus is
incapable of taking up carbon (fCF p 0; Smith and Read
1997), changes in fungal allocation do not change fungal
growth rate in this region of the generalist fungus. Therefore, in the case of plant limitation, we assume the fungus
continues to function as a specialist, even though there is
selection for it to be a generalist. At high carbon (light)
availability, the pattern is similar (fig. 3B, 3D, 3F), but
both partners specialize at the highest phosphorus availability examined. Therefore, changes in the identity of the
specialist (limiting) partner have important effects on how
carbon flux between partners, the exchange ratio, and
plant and fungal gain from trade respond to increases in
phosphorus availability.
Fungus phosphorus uptake is also an important determinant of the identity of the specialist (limiting) partner
(fig. 2). Examining this parameter more closely, we find
that increases in fungus phosphorus uptake efficiency can
switch the identity of the specialist from the fungus, to
both partners, to only the plant (fig. 4A). A fungus with
very low phosphorus uptake efficiency is unable to take
up enough phosphorus to satisfy plant requirements even
though it specializes on phosphorus uptake. The plant
associating with this fungus is a generalist, allocating to
both carbon and phosphorus uptake. As long as the fungus
is the specialist, a fungus with higher phosphorus uptake
would be able to increase the total amount of carbon
exchanged between partners; this increased trade volume
would increase both partners’ gains from trade. However,
a fungus with even higher phosphorus uptake efficiency
could satisfy plant phosphorus requirements, allowing the
plant to specialize on carbon uptake (fig. 4A), increasing
the carbon exchanged even further because trade would
be the plant’s only phosphorus source. In this region, the
exchange ratio increases, again because the fungus’ monopoly on phosphorus uptake allows it to negotiate an
exchange ratio favorable to the fungus (fig. 1). This shift
toward a much higher exchange ratio, accompanied by
larger quantities of carbon exchanged, combines to result
Figure 4: Effects of fungus phosphorus uptake efficiency (A–C) on
the total amount of carbon (C) exchanged per day (X), the amount
of C traded per unit nutrient (C : nutrient exchange ratio, T; A),
plant gain from trade (B), fungal gain from trade (C), and the identity
of the specializing partner who limits trade (shaded bars in A; F
indicates fungal specialist, B indicates both partners specialize, and
“Plant specialist” indicates plant specialization). Parameters are set
at YNP p 500, YCP p 1, YNF p 100, YCF p 1, fCF p 0, VCP p 9.7 #
10⫺7, fNP
p 3.4 # 10⫺10, KCP p 400, P p 2,129, F p 0.36, C p
1,600.
in higher fungal gain from trade and lower plant gain from
trade (fig. 4B, 4C). A fungus with still higher phosphorus
uptake efficiency can provide more than enough phosphorus for both its own requirements and plant phosphorus requirements, again resulting in a specialist plant
and a fungus constrained to act as a specialist (because
fCF p 0) even though there is selection for it to be a generalist. Again, changes in the identity of the specialist partner affect the impacts of shifts in fungal species traits on
the outcome of trade.
The biomass of each partner also contributes to determining whether the plant, the fungus, or both specialize
on uptake of the resource traded away (fig. 5). A plant
576 The American Naturalist
Figure 5: Effect of varying abundances of the plant (A, C, E) and the fungus (B, D, F) on the total amount of C exchanged per day (X),
the amount of C exchanged per unit nutrient (C : nutrient exchange ratio, T; A, B), plant gain from trade (C, D), fungal gain from trade
(E, F), and the identity of the specializing partner who limits trade (shaded bars in A, B; F indicates fungal specialist, “Both” indicates both
partners specialize, and “Plant” indicates plant specialization). Parameters are set at YNP p 150 , YCP p 1 , YNF p 30, YCF p 1, fCF p 0,
KCP p 200, VCP p 6.72 # 10⫺6, fNP
p 3.4 # 10⫺10, fNF
p 1.24 # 10⫺4, and C p 100. Fungal biomass is large when varying plant abundance
(F p 0.36 in A, C, E); plant biomass is small when varying fungal abundance (P p 686 in B, D, F).
with more biomass is better able to take up enough carbon
to satisfy itself and its partner than a smaller plant, leaving
a larger plant free to allocate to direct phosphorus uptake
as well as direct carbon uptake (fig. 5A, 5C, 5E). To understand the dynamics of trade during plant invasion of
a habitat occupied only by the fungus, we investigated
plant biomasses below the lowest equilibrium biomass
listed in table 1 (686 g plant C m⫺2). Not surprisingly, the
plant specializes on carbon uptake during invasion of new
habitat. However, even above the lowest equilibrium plant
biomass, the plant might still be the specialist partner (fig.
5A). Similarly, a fungus invading a new habitat would
specialize (fig. 5B, 5D, 5F). However, in habitat where the
fungus is more abundant, the fungus might be able to take
up enough phosphorus to satisfy both plant and fungus
requirements. Therefore, with increases in fungus biomass,
trade would transition from fungus specialization, to both
partners specializing, to plant specialization. Again,
changes in the identity of the specialist partner affects
whether changes in parameters increase, decrease, or have
no effect on the exchange ratio and plant and fungal gain
from trade.
These results reveal two important points. First, the
identity of the specialist partner affects the impacts of
Ecological Specialization and Trade 577
changes in parameters on the outcome of the symbiosis.
In figures 3–5, sharp breakpoints in the curves occur wherever the identity of the specialist partner switches. Increases in species uptake efficiencies, biomass, or environmental characteristics can increase the total volume of
carbon exchanged, the exchange ratio, or plant and fungal
gain from trade while one partner specializes, but decrease
them (or have no effect) when the other partner specializes. Second, changes in plant and fungal gain from trade
in response to changes in parameter values are frequently
counter-intuitive. While the exchange ratio is always negotiated according to the Nash bargaining solution, some
exchange ratios are nearer the lower bound of the beneficial range of trade, an exchange ratio more favorable for
the plant (fig. 1). Other exchange ratios are nearer the
upper bound of the beneficial range of trade, benefitting
the fungus more than the plant. Understanding these results relies on knowledge of the identity of the specialist
partner and the exchange ratio.
Discussion
Specialization
In contrast to Schwartz and Hoeksema (1998), who suggested that specialization by both partners should be common, our results suggest that this is not the only possible
outcome: it is common for one of the partners to specialize
and for the other partner to act as a generalist, taking up
both resources directly from the environment. In the case
of diminishing returns on investment in uptake machinery,
it is possible for both species to experience selection for
uptake of both resources (appendix, “Extension to Nonlinear Trade-Offs”), but this does not occur for the empirically derived parameter values (table 1).
In the majority of cases in our simulations of plant
interactions with arbuscular mycorrhizal fungi, the fungus
is the specialist partner, while the plant takes up both
carbon and phosphorus. This finding is reassuring, because
it is backed by empirical observations that the plant photosynthesizes and takes up soil nutrients and that the fungus only takes up soil nutrients. Because the fungus is the
specialist partner in most environments it encounters, it
is tempting to speculate that it may have experienced selection for the loss of its direct carbon uptake machinery.
This hypothesis could explain the apparent evolution of
specialization in arbuscular mycorrhizal fungi: in other
words, it may explain why arbuscular mycorrhizal fungi
apparently cannot take up carbon from the environment.
However, our simulations reveal a few cases where the
plant is the specialist partner. This occurs even though we
simulate the arbuscular mycorrhizal case, where the fungus
is incapable of direct carbon uptake and is thus incapable
of acting as a generalist. In plant specialist situations where
the fungus can take up more phosphorus than both organisms can use, the surplus phosphorus is simply lost,
and fungus growth rate is flat with respect to fungal allocation. In another mutualism where both partners are
capable of acting as generalists (fCF 1 0), such as the ectomycorrhizal symbiosis (e.g., Vaario et al. 2011), the fungus would act as a generalist and increase its growth rate
by increasing independent carbon uptake.
Interestingly, there may be empirical support for the
idea that plants could specialize on carbon uptake and
eliminate direct phosphorus uptake when associating with
arbuscular mycorrhizas (Smith et al. 2009). In fact, suppression of plant direct phosphorus uptake may be proportional to the amount of phosphorus received from the
fungus (Burleigh et al. 2002).
Combined with empirical evidence suggesting that the
identity of the specialist partner in the arbuscular mycorrhizal symbiosis may actually vary, our finding that the
identity of the specialist partner affects the outcome of
trade is intriguing. It suggests that some of the context
dependency in plant-mycorrhizal interactions may indeed
be driven by variation in the identity of the specialist
partner.
Exchange Ratio
With a few notable exceptions (Hoeksema and Bruna 2000;
McGill 2005; Akçay and Roughgarden 2007a; Golubski
and Klausmeier 2010), relatively little theoretical work has
explored the implications of variation in the exchange ratio. However, the exchange of benefits lies at the heart of
any mutualism. Our results support this idea: the exchange
ratio affected how much each partner gained from trade,
even though exchange ratios are required to be mutually
beneficial. These changes sometimes coincide with the expectation that lower carbon : nutrient exchange ratios
would benefit the plant and that higher exchange ratios
would benefit the fungus. However, this was not always
the case. Changes in the identity of the specialist interact
with changes in the exchange ratio to determine gain from
trade.
We modeled negotiation using the Nash bargaining solution because it is appropriate for exchanges of resources
between single individuals of two guilds (Akçay and
Roughgarden 2007a). In nature, however, it is typical for
plants to associate with multiple species of mycorrhizal
fungus, and similarly for fungi to associate with many
plants (van der Heijden and Horton 2009). It is not clear
that the Nash bargaining solution will be appropriate to
predict the outcome of negotiation when either or both
partners can chose from among many potential partners.
In particular, incorporating partner choice could substan-
578 The American Naturalist
tially change the dynamics of negotiation, possibly allowing the limiting partner to negotiate a more favorable exchange ratio. Other work has shown that partner choice
can affect the negotiated exchange ratio as well as the
benefits each partner gains from the association (Golubski
and Klausmeier 2010; Akçay and Simms 2011) and also
may determine community structure and diversity (Kummel and Salant 2006), so examining partner choice in negotiation is an important direction for future work. Furthermore, the Nash bargaining solution requires that
partners can enforce this negotiated exchange ratio (in
other words, cheating is not allowed) and that both partners benefit from the interaction and neither partner be
parasitized. Therefore, the NBS is not appropriate for
modeling the mutualism-parasitism continuum observed
in arbuscular mycorrhizal fungi (Johnson et al. 1997;
Hoeksema et al. 2010). Clearly, incorporating parasitism,
nonenforceable exchange ratios, or partner choice into a
model of negotiation would invalidate the use of the NBS
and could have profound effects on the outcome of trade,
both ecologically and evolutionarily (McGill 2005).
A few empirical studies have measured variation in the
exchange ratio in the arbuscular mycorrhizal symbiosis
and linked it to fungal species traits (Pearson and Jakobsen
1993) and resource availability (Kiers et al. 2011). However, too few studies have reported exchange ratios, and
none in conjunction with the identity of the specialist, so
it is too early to test our model’s predictions. The exchange
ratio may be even more difficult to measure in mutualisms
where behaviors such as pollination or protection are exchanged. The exchange ratio thus remains understudied,
but our results highlight its potential importance for understanding mutualisms.
Gain From Trade
The net effect of the arbuscular mycorrhizal symbiosis
ranges widely (Hoeksema et al. 2010). Several aspects of
the context are thought to affect the outcome, including
resource availability, the relative abundances of the partners, and the identity of the partners, probably because of
differences in traits involving resource uptake efficiency
(Hetrick et al. 1992; Allison and Goldberg 2002; Lekberg
and Koide 2005; Hoeksema et al. 2010). However, 59%–
77% of variation in plant growth remained unexplained
in a meta-analysis (Hoeksema et al. 2010). This surprisingly large amount of residual variation in the outcome
of the mutualism suggests that not all important predictors
have been examined. In particular, those investigators were
unable to include analyses of the uptake and trade of carbon and soil nutrients. In other words, trade itself (the
reciprocal flux of resources) has not been adequately
studied.
Our model suggests that two aspects of trade may be
important predictor variables. We found that both the
identity of the specialist and the exchange ratio help determine the degree to which the plant and the fungus gain
from trade. Variation in these two factors is driven by
changes in extrinsic forces such as environmental conditions and species traits. However, these two factors also
affect the impact of the extrinsic forces. In fact, the partners’ gains from trade could increase, decrease, or remain
constant with changes in resource availability, the partners’
resource uptake efficiencies, or the partners’ biomasses,
depending on the identity of the specialist and the exchange ratio. Without knowing the identity of the specialist
and the exchange ratio, this variation in the response of
plant and fungal benefit to environmental drivers would
appear to be highly variable and unpredictable, as it does
in the recent meta-analysis (Hoeksema et al. 2010). However, radioactive isotopes, genetic techniques, or other
methods may make identification of the specialist possible
(Smith et al. 2009). Measuring the exchange ratio should
also be possible (Kiers et al. 2011). By identifying the
specialist and quantifying variation in the exchange ratio
along gradients of key variables such as phosphorus availability, fungal phosphorus uptake efficiency, and fungal
biomass, we may be able to explain much of this residual
variation in the outcome of the mutualism.
Broader Theoretical Context
Previous models of trade have made important advances
in understanding the exchanges of costs and benefits in
mutualisms. These models have considered the conditions
under which trade could evolve (McGill 2005; de Mazancourt and Schwartz 2010) and could result in a range of
positive to negative impacts on populations based on resource availability (Neuhauser and Fargione 2004) or density-dependent functional responses to partners (Holland
et al. 2002). Another approach has more explicitly considered costs and benefits by modeling mutualisms as a
consumer-resource interaction (Holland and DeAngelis
2010). These and other models (Hoeksema and Bruna
2000; Okuyama and Holland 2008) have also asked
whether mutualism can affect community structure. However, none of these have investigated the impacts of behaviorally plastic allocation to trade as opposed to independent resource acquisition, clearly an important
determinant of the magnitude and therefore the impact
of trade on individuals and communities.
Although we have analyzed and interpreted our model
for the specific example of the plant-arbuscular mycorrhizal mutualism, many of the central concepts are applicable to other kinds of mutualisms. All mutualisms require the exchange of one benefit for another. Some of
Ecological Specialization and Trade 579
these benefits may be costly for partners to provide, opening up the possibility that there may be a conflict of interest
in determining the magnitude of provisioning (Bronstein
2001). Negotiation of the exchange ratio is therefore likely
to be a critical issue in determining the outcome of the
interaction. Furthermore, some mutualists may be capable
of adjusting allocation to provisioning their partner. Flexible allocation to provisioning raises questions about the
optimal allocation strategy, and whether either partner (or
both) will specialize on provisioning and becoming an
obligate mutualist. Our model is thus conceptually applicable to a wide range of mutualisms.
By uniting two previously disparate fields of theory, we
have gained new insight into two factors that should determine the outcome of mutualisms. Specifically, we have
incorporated aspects of a biological market (Schwartz and
Hoeksema 1998) and negotiation (Akçay and Roughgarden 2007a). By combining these two modeling approaches,
we are able to show that a mutualist’s decision to specialize
on uptake of a single resource can affect the exchange ratio
it negotiates with its partner. We also show that both of
these factors, allocation and the exchange ratio, can combine with other important ecological drivers to determine
the outcome of trade.
Acknowledgments
Development of the model and the manuscript benefited
from many comments and discussions with colleagues,
including T. Golubski, K. Gross, H. Reynolds, Y. ShacharHill, and two anonymous reviewers. Funding was provided
by a James S. McDonnell Foundation grant to C.A.K. This
material is based on work supported by the National Science Foundation (NSF) under grant 0909942. T.M.P.R. and
E.G. were supported by NSF GK-12 fellowships (DGE0538509); E.G. was also supported by an NSF Graduate
Research Fellowship. This is Kellogg Biological Station
contribution 1606.
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Associate Editor: Brian McGill
Editor: Judith L. Bronstein
“Everyone is familiar with the beautiful and delicate fabric made from the fibres spun by that crawling repulsive creature, the silk worm.”
From “The American Silk Worm” by L. Trouvelot (American Naturalist, 1867, 1:30–38).