Theoretical Population Biology TP1283
theoretical population biology 50, 281297 (1996)
article no. 0032
The Stability and Persistence of Mutualisms Embedded
in Community Interactions
Michael S. Ringel, Helen H. Hu, and Garrett Anderson
Department of Ecology and Evolutionary Biology, Princeton University,
Princeton, New Jersey
and
Michael S. Ringel
Department of Biology, Imperial College at Silwood Park,
Ascot, Berks SL5 7PY, United Kingdom
Received December 1, 1994
In this paper we argue that two-species models of mutualism may be oversimplifications of the real world that lead to erroneous predictions. We present a fourspecies model of a pollination mutualism embedded in other types of community
interactions. Conclusions derived from two-species models about the destabilizing
effect of mutualisms are misleading when applied to the present scenario; although
the mutualisms are locally destabilizing, the effect is more than canceled by an
increased chance of feasibility. The crucial difference is the interaction of the
mutualists with other species in a larger web. Furthermore, community persistence
(without unrealistic population explosion), arguably a superior ecological criterion,
is greatly enhanced by the presence of mutualisms. Therefore, we predict that
mutualisms should be common in the real world, a prediction matching empirial
findings and in contrast to the predictions from local stability analysis of basic twospecies models. This method of stabilizing a mutualism appears superior in some
ways to the often-used method of introducing density dependence in the strength of
the mutualism, because it permits obligate mutualisms to exist even at low densities,
again matching empirical findings. Lastly, this study is an example of how complex
model assemblages can behave qualitatively differently from analogous simpler
ones. 1996 Academic Press, Inc.
1. INTRODUCTION
Mutualisms are widespread in nature, occuring in almost every region of
the globe, and in almost every ecosystem (Lewis, 1982; Heithaus et al.,
1980). Unfortunately, mutualisms have not yet received the theoretical
attention they deserve, with far less attention being paid to them than to
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Copyright 1996 by Academic Press, Inc.
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predatorprey and competition interactions (Boucher et al., 1982; Murray,
1989; Cohen, 1993; Cohen et al., 1993). Furthermore, the little work that
has been done has focused almost entirely on two-species systems (Kumar
and Freedman, 1989; Zaghrout, 1991). Although small model assemblages
are more mathematically tractable, their analysis may yield different results
from that of larger assemblages. Since real-world food webs are far more
complex than such simple assemblages, the results from these basic analysis
may be invalid when applied to the real world. Instead, they should serve
as control examples, against which more complex model assemblages can
be measured (Hassell, 1978).
The basic two-species model of mutualism predicts that mutualisms
should be destabilizing (May, 1973), a finding which is not corroborated
by real-world evidence (Lewis, 1982; Heithaus et al., 1980; Roubik, 1992;
Thompson, 1982). In contrast, in this analysis of mutualisms embedded
within other community interactions, mutualisms can have a stabilizing
influence, especially with respect to assemblage persistence. As a corollary,
it is demonstrated that the stability of larger model assemblages can be
quitte different from that of analogous but smaller model assemblages.
2. OUTLINE OF PAPER
We begin by introducing the structure of the model assemblage, which
depicts a pollination interaction. This is followed by the derivation of a
basic two-species model of mutualism, which can be easily generalized to
a four-species model. The assemblage is then analyzed, first in a way
analogous to the analysis of the two species model, but also with a more
detailed look at both local stability and global persistence. Lastly, a comparison is made of this method of stabilizing mutualism with other
methods, with special reference to the problem of obligate mutualisms.
3. STRUCTURE OF THE ASSEMBLAGE
The model analyzed here depicts a plant-pollinator mutualism, shown
schematically in Fig. 1. Some standard assumptions are implicit in the
model:
v There is intraspecific resource competition in the plant species.
v The pollinator is a generalist, feeding at both species of plants
shown, as well as at other plants not explicitly shown (Faegri and
van der Pijl, 1971). There is effectively no intraspecific competition in the
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283
Fig. 1. A schematic of the model community. Black indicates a positive effect, gray a
negative one. We assume that there is intraspecific resource competition in the plant species,
that the pollinator is a generalist regulated by predation, and that the top predator has
sources of food not explicitly shown. Three community structures are analyzed: one with
mutualisms between the pollinator and both plants, one with no mutualism (pure nectar
theft), and one with both pollination and nectar theft. The last condition is the one illustrated.
The results of this paper hold qualitatively true for other community shapes of 3 to 7 species.
pollinator; instead, it is regulated by predation (see Hairston, et al., 1960,
Strong, et al., 1984, Fretwell, 1987, DeAngelis, 1992, and many others).
v A predator species preys on the pollinator, but also has sources of
food not explicitly shown. There is intraspecific competition in the predator
for these other resources (again, see Hairston, et al., 1960, etc.).
The results described in this paper are not dependent upon the exact
layout of this model community, holding qualitatively true for randomly
constructed communities of three to seven species (Ringel, unpublished
data).
However, the validity of food-web models in general has been called into
question by Polis (1991) and others, with the recognition that some of the
basic predictions from food-web theory do not match the empirical findings
from well-documented real-world food webs. These real-world webs are far
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more complex and species-rich than almost all model communities, including the simple model analyzed here. Ideally, we would analyze the stability
and persistence of a more realistic web. But there is a dearth of such food
webs in the literature (Cohen, et al., 1993, Polis, 1991), and even the best
available webs intentionally omit mutualisms and other non-feeding interactions (Cohen, et al., 1993). Randomly constructed communities of a
realistic size (e.g.: thousands of species even in Polis's species-poor desert)
would essentially never be stable. Thus, we use a very simple model in our
analysis. It is by no means presented as a definitive proof that embedding
mutualisms within a community of other interactions will stabilize the
community. But it is an important first step towards looking at mutualisms
in the context of a community of species with both mutualisms and other
kinds of interactions.
We analyze the community in three different combinations: without
mutualism, with only one pollinator-plant mutualism (as depicted in Fig.
1), and with two pollinator-plant mutualisms. It is reasonable for a
pollinator to have a mutualistic interaction with one plant species, but a
predator-prey interaction with the other. While few pollinators behave as
nectar robbers (making a hole to access nectar) (Inouye, 1980, Mainera
and Martinez del Rio, 1985), the theft of nectar is very common (Inouye,
1980). It can result, for example, when a small bee visits a large flower and
fails to contact the stamens or stigma (Inouye, 1980), or when a foraging
trip is limited to a single flower, which would fail to pollinate an obligate
outcrosser (Heinrich and Raven, 1972).
4. A BASIC TWO-SPECIES MODEL OF MUTUALISM
A basic two-species model shown in Fig. 2, similar to that examined by
May (1973, 1981) and others, is of the form
F 1 =dN 1 dt=N 1(b 1 +a 11 N 1 +a 12 N 2 )
(4.1)
F 2 =dN 2 dt=N 2(b 2 +a 22 N 2 +a 21 n 1 ),
(4.2)
where N i is the size of the population of the ith species, b i is the intrinsic
growth rate of the ith species, and a ij is the per capita effect of the jth
species on the ith. The self effects (a ij when i= j) are negative, while the
interactions (a ij when i{j) are positive in the case of mutualism.
Admittedly, a chief draw-back of such LotkaVolterra equations is that
they are not mechanistic and lack biological detail. However, close to
equilibrium such models behave qualitatively similarly to any model of the
form dN i dt=N i F i (N i } } } N m ) given that F i is continuous, that F i >0
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Fig. 2. A simple schematic of a two-species mutualism. With LotkaVolterra dynamics
this model is locally stable only if intraspecific damping is greater than the interspecific
benefits.
where N i <K i , and that F i >0 where N i >K i , so that conclusions drawn
from LotkaVolterra models about local stability are not significantly different from those derived from more mechanistic models (Goh, 1979). Still,
it is important to keep in mind that the formulation of the models may
miss important details, especially when no longer close to equilibrium. One
should therefore temper the interpretation of any LotkaVolterra models
with these caveats.
The stability of a two-species interaction can be evaluated in the usual
way by constructing the Jacobian matrix and analyzing the eigenvalues,
using the RouthHurwitz criteria. Given a feasible equilibrium (i.e.: no
N*<0), the condition for stability in this interaction is simply
a 11 a 22 >a 12 a 21 .
(4.3)
In other words, that effects must be greater than interaction effects. Because
this is a rather strict condition for stability, it has been suggested that
mutualisms are, in general, destabilizing, and should be less likely in the
real world than other types of interactions (May, 1973). Numerous
investigators have shown, however, that some additions to this two-species
model can help to stabilize the interaction. For example, the introduction
of handling times or other constraints on the benefits derived from the
mutualism (May, 1981, Vandermeer and Boucher, 1978, Addicott, 1986),
both intrinsic (e.g.: Wright, 1989, Thompson, 1982, Soberon and Martinez
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del Rio, 1981) and extrinsic (e.g.: Dean, 1983) to the interaction can stabilize a two-species assemblage. Other plausible stabilizing factors include
temporal and spatial variability, the influence of genetics when both species
are polymorphic (Levin and Udovic, 1977), allowing species to affect each
other in different ways (influencing K or r; e.g.: Addicott, 1981, Pierce and
Young, 1986), or incorporating the damping effect of a third species, either
a predator or a competitor (Heithaus, et al., 1980, Freedman, et al., 1987,
Freedman and Rai, 1987, 1988). The present analysis is an extension of
these three-species scenarios. Of particular note, however, is that the
stabilizing influence of the web can go beyond that of a predator or competitor damping the mutualism, to include subtler influences that are less
intuitive. For instance, the presence of additional prey for one of the
mutualists stabilizes the model.
5. THE MODEL
The assemblage shown in Fig. 1 can be described by four coupled differential equations that represent a generalization of the two-species
LotkaVolterra model. Consider the system of equations
4
dN i
=N b i + : a ij N j ,
dt
j=1
\
+
i=1, 2, 3, 4,
(5.1)
where again N i is the size of the population of the ith species, b i is the
intrinsic growth rate of the ith species, and a ij is the per capita effect of the
jth species on the ith. Self effects (a ij , i= j) and predator effects are
Fig. 3. The signs of the community matrix, consisting of the per capita effects of species
j on species i. Self effects and predator effects are negative, while prey effects and mutualisms
are positive. Terms involving two species that do not directly interact are zero. The terms
marked indeterminate (V) are positive for mutualism, negative for nectar theft.
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negative, while prey effects and mutualisms are positive. Terms involving two
species that do not directly interact are zero (these account for about half of all
combinations). Figure 3 gives a diagram of the signs of these terms.
Analytical Results
The eigenvalues of this system are complicated, the RouthHurwitz
criteria being
1. F 1 =a 11 +a 22 +a 44 <0
(5.2)
2. F 2 =a 13 a 31 +a 23 a 32 +a 34 a 43 &a 11 a 22 &a 11 a 44 &a 22 a 44 <0
(5.3)
3. F 3 = &a 13 a 31(a 22 +a 44 )&a 23 a 32(a 11 +a 44 )&a 34 a 43(a 11 +a 22 )
+a 11 a 22 a 44 <0
(5.4)
4. F 4 =a 11 a 22 a 34 a 43 +a 11 a 44 a 23 a 32 +a 22a 44 a 13 a 31 <0
(5.5)
5. F 5 =F 1 F 2 +F 4 >0
(5.6)
given a feasible equilibrium. As with any model community of more than
two species, it is difficult to disentangle these stability conditions. In this
case any of the conditions other than F 1 can be the crucial one that determines stability. A simple analysis of the signs of the a ij terms shows that
introducing a mutualism between the pollinator and one or both plants is
nearly always destabilizing, as expected. This result, however, is misleading,
for it does not include the effects of mutualism on feasibility. The explicit
feasibility criteria in this model are even more convoluted than the stability
criteria, but they can be easily analyzed using simulations.
TABLE I
Feasibility and Local Stability in Each Model Community
No mutualism
One mutualism
One strong mutualism
Two mutualisms
Two strong mutualisms
Feasible
Stable
Feasible and stable
259
179
191
1,966
2,818
10,000
9,545
9,199
7,229
5,977
259
153
138
273
1,070
t-test
***
***
***
Note. Mutualisms decrease the likelihood of stability of a point equilibrium, but the consideration of local stability in isolation of feasibility (i.e., all population densities >0) is
biologically meaningless. When looking at the stability of only feasible equilibria a different
pattern emerges; mutualisms can be strongly stabilizing. Two-tailed Student's t-test of feasible
and stable equilibria vs no mutualism: one mutualism (t=5.28), one strong mutualism
(t=6.14), two mutualisms (t=0.62), two strong mutualisms (t=23.64).
***# p<0.0001.
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Simulation ResultsLocal Stability
It is clear that as the number of species in the assemblage increases the
conditions for stability usually become more complicated. Even in this
system of four species the conditions are algebraically messy, with F 4 often
but not always being the relevant criterion. Therefore, in addition to the
above analysis a computer simulation was used to analyze local stability.
We used a root-finding routine to solve for the roots of the characteristic
polynomial. We followed Pimm and Lawton (1977) in setting the range of
parameters in the model. The per capita effects of prey on predators (e.g.:
a 31 , a 43 ) were randomly assigned from a uniform distribution over the
interval (0,0.1]. The effects of predators on prey varied over the interval
(0, &10]. Lastly, self terms (a 11 , a 22 , a 44 ) varied from (0, &1]. The b i 's
(the intrinsic rates of natural increase) were set equal to 1 for the K-selected
species, and 2 for the bee, which has a higher reproductive rate.
10,000 trials were generated for each of the following five assemblages:
no mutualism, one mutualism (a 23 # (0, 1]), one strong mutualism
Fig. 4. Histograms of return times for (a) non-mutualist, (b) one-mutualist, and (c) twomutualist assemblages. Small return time indicates strong local stability, in that perturbations
of an equilibrium will be short-lived. Communities (a) and (b) are roughly equivalent in
resilience to perturbation, while community (c) (two-mutualisms) is much more stable both
in the absolute number of stable equilibria and in the resilience of those equilibria. See also
Table I.
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STABILITY OF MUTUALISM
Fig. 4 (continued)
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(a 23 # (0, 2]), two mutualisms (a 23 and a 13 # (0, 1]), and two strong
mutualisms (a 23 and a 13 # (0, 2]). The ranges for these parameters are
somewhat arbitrary but they are probably a reasonable approximation to
the relative magnitudes of the different types of interactions (Pimm and
Lawton, 1977).
Simulations showed that communities with mutualisms had a lesser
chance of stability, as we have already seen. But the consideration of local
stability in isolation of feasibility is meaninglessin biological terms it is the
consideration of equilibria involving populations of negative individuals.
When looking at both stability and feasibility a different pattern emerges:
mutualisms can be strongly stabilizing. Of the 10,000 non-mutualist assemblages created, 259 had a feasible, stable equilibrium. 153 and 138 of the
one-mutualist and one-strong-mutualist assemblages had a feasible, stable
equilibrium. Lastly, 273 an 1,070 of the two-mutualists and two-strongmutualists assemblages were feasible and stable. These results are summarized in Table I.
A more exact test is not just qualitative stability, but a measure of how
stable an equilibrium is, its resilience to perturbation. A long time returning to equilibrium following a perturbation implies an increased probability
of extinction (Pimm and Lawton, 1977). Return time is given by
Return timer &1Re(* max), given all *<0,
(5.7)
where Re(* max) is the real part of the largest eigenvalue (Pimm and Lawton,
1977). The distribution of return times was essentially identical for both the
non-mutualist and the one-mutualist assemblages, (see Figs. 4a and 4b),
but the assemblage with two mutualisms was much more resilient to perturbation (see Fig. 4c).
Simulation ResultsPersistence
Although the local stability of an equilibrium is important, a crucial
aspect of the assemblage is the persistence of all its populations. Persistence
will differ from local stability in that it captures slowly dying transients,
stable cycles and otherwise bounded behavior, while losing some stable
equilibria due to the influence of initial conditions. Thus, local stability
neither implies persistence nor is a necessary condition for persistence
(Travis and Post, 1979, Cull, 1986, Law and Blackford, 1992). In fact,
systems that are locally unstable yet globally persistent may be quite common (Hastings and Wolin, 1989).
The global persistence of the model was tested using a fourth-order
classical RungeKutta routine that numerically solves the system of differential equations already outlined. The performances of mutualist and
non-mutualist assemblages were compared according to the ability of their
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constituent members to coexist. Dynamic simulations were run from t=0
to 40, which allows most transients to die out, with an integration timestep of 0.02. Extinction or unrealistic explosion resulted in a failed trial,
while persistence of all four species with reasonable population levels
resulted in a succesful one. The four individual components of the initial
population vector were varied systematically from 0.1 to 1, in increments
of 0.1, where 1 is the carrying capacity in the absence of other species. This
resulted in 10,000 initial population vectors. For each such vector we ran
10 trials, randomly assigning new values to the interaction terms (aij's) in
each trial. We used exactly the same protocol for determining a ij 's and b i 's
as in the analysis of local stability. In total, 100,000 trials were run for each
of the following assemblages: an assemblage entirely without mutualisms,
an assemblage with one mutualism (a 23 # (0, 1]), an assemblage with one
strong mutualism (a 23 # (0, 2]), an assemblage with two mutualisms (a 23
and a 13 # (0, 1]), and an assemblage with two strong mutualisms (a 23 and
a 13 # (0, 2]). In the assemblage lacking mutualism, only 831 trials resulted
in persistence, whereas 1025 and 923 trials resulted in persistence in the
assemblage with one mutualism and a strong mutualism, respectively. Fully
2594 and 1547 trials resulted in persistence in the assemblages with two
mutualisms (see Table II and Fig. 5).
TABLE II
Persistence in Each Model Community
No mutualism
One mutualism
One strong mutualism
Two mutualisms
Two strong mutualisms
Persistence
t-test
831
1025
923
2594
1547
***
*
***
***
Note. Mutualisms strongly increase community persistence (the maintenance of all species at positive population
densities without unrealistic population explosion). Persistence differs from a local stability analysis in that it captures slowly dying transients, stable cycles, and otherwise
bounded behavior, while losing some stable equilibria due
to the influence of initial conditions. Here persistence is out
of 100,000 trials (compared with 10,000 trials for local
stability in Table I). Two-tailed Student's t-test of persistence vs no mutualism: one mutualism (t=4.52), one
strong mutualism (t=2.21), two mutualisms (t=30.46),
two strong mutualisms (t=14.78).
*# p<0.05.
***# p<0.0001.
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Fig. 5. Comparison of persistence in each model community. Persistence of community is
the maintenance of all species at positive population densities, regardless of the underlying
dynamics. Mutualisms greatly increase the chances of community persistence. See also
Table II.
6. OBLIGATE MUTUALISM
Embedding a mutualistic interaction within other types of community
interactions is only one of several possible ways of stabilizing
LotkaVolterra models of mutualism. A frequently used approach (e.g.:
Vandermeer and Boucher, 1978, Addicott, 1986, Wright, 1989, Thompson,
1982, Soberon and Martinez del Rio, 1981, Dean, 1983) has been to incorporate density-dependent limitations on the strength of the mutualism. The
condition for stability of a facultative mutualism without such limitations
is given by Eq. (4.3) as we have already seen. When these conditions are
not met both populations explode unrealistically. By introducing densitydependent limitations on the strength of the mutualism, however, a stable
equilibrium is guaranteed; there is no orgy of mutual benefaction.
These same LotkaVolterra models can be easily applied to obligate
mutualisms by making the intrinsic rate(s) of increase negative. Without
density-dependent limitations on the strength of the mutualism, however, a
two-species obligate mutualism can never be stable. Obligate mutualism
with limitations on the strength on the mutualism can produce a feasible
stable equilibrium, but at best the system has multiple stable states, such
that below a certain threshold population the system will deterministically
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collapse to extinction (May, 1981). Such an interaction could only persist
in a constant environment (May, 1981). The greater abundance of obligate
mutualisms in tropical environments has sometimes been considered
evidence for this.
However, the finding that obligate mutualisms are more common in the
tropics is not, by itself, evidence that obligate mutualisms need constant
environments in order to persist. Boucher, et al. (1982) point out that there
are phylogenetic considerations, such that the tropical mutualisms are not
independent examples. Furthermore, even without this caveat we should
expect the absolute number of obligate mutualisms to increase towards the
equator, regardless of environmental stability, since species richness,
biomass, and productivity also increase towards the equator (Boucher, et
al., 1982).
In fact, obligate mutualisms are rather common in the temperate zone,
and in pioneer communities where populations necessarily start at low
densities and should, by the above models, go to extinction (Heithaus,
et al., 1980, Keeler, 1985, Hutson, et al., 1985). Thus, introducing densitydependent limitations on the strength of a mutualism in a two-species system
seems to fail in its predictions for obligate mutualism.
One solution is to model the interacting species as metapopulations with
diffusion. Populations with average densities below the critical value can
Fig. 6. Persistence in communities with obligate mutualisms. An obligate mutualism is
one in which the partners decay to extinction without each other (here both species are taken
to be obligate). Communities with obligate mutualisms achieve both local stability and persistence even without density-dependent limitations on the strength of mutualisms, in contrast
to two-species models. Thus, the model suggests that obligate mutualisms need not be constrained to high population densities, a prediction in accordance with empirical findings. See
text for a more detailed explanation.
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then still persist, so long as some foci of high population density exist
(Hutson, et al., 1985). Another is to introduce the effects of other species
(Freedman and Rai, 1987), as been done in this paper.
In simulations of obligate mutualism in which the plant is obligately
dependent on the bee for pollination (b 2 <0, otherwise following the same
protocol as before), local stability and persistence can be as great in
obligate mutualisms as in facultative ones, depending primarily on the
decay rate of the obligate species (b 2 ). With b 2 =&0.01 local stability and
feasibility were found in 121 out of 10,000 trials, while persistence was
found in 936 of 100,000 trials. Both local stability and persistence were still
found with the extremely high decay rate of b 2 =&1, but less often: 7 out
of 10,000 and 85 out of 100,000 trials, respectively (see Fig. 6).
7. DISCUSSION AND CONCLUSIONS
The model analyzed here is several orders of magnitude less complex
than real-world interaction webs. But it differs from simple two-species
models by beginning to explicitly incorporate the effects of a larger web on
mutualism, and vice versa. Even the slight difference of adding two species
can have profound effects on the predictions the model makes. These
results are qualitatively similar for a series of randomly constructed communities of three to seven species (Ringel, unpublished data).
Communities with mutualisms do have a lesser chance of a locally stable
point equilibrium, but this is a spurious result. Many of the equilibria
examined are biologically meaningless, involving negative population densities. When corrected so that only the stability of feasible equilibria is
considered a different pattern emerges: mutualisms can very often be
stabilizing. A more exact analysis of the quantitative strength of stability
(or resilience, measured by time taken to return to equilibrium following a
disturbance) corroborates and strengthens these findings. Furthermore,
mutualisms greatly enhance the chances of community persistence. Persistence differs from local stability in that it captures slowly dying transients, stable cycles and otherwise bounded behavior, while losing some
stable equilibria due to the influence of initial conditions; in many senses
it is an equally valid ecological criterion (Travis and Post, 1979, Cull, 1986,
Law and Blackford, 1992). Taken as a whole, these results suggest that
mutualisms may be stabilizing in real-world communities, not as simple
two-species models indicate.
This method of stabilizing a mutualism appears superior in some ways
to the often-used method of introducing density-dependence in the per
capita strength of a mutualism. The latter method makes the prediction
that obligate mutualisms suffer from on Allee effect, and can only be
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feasible and stable at high population densities (May, 1981). Since obligate
mutualisms are prevalent in many pioneer communities at low densities
(Heithaus, et al., 1980, Keeler, 1985, Hutson, et al., 1985) this method
seems to fail in its predictions for obligate mutualisms. In contrast, the
present method allows obligate mutualisms to be feasible and stable,
depending primarily on the decay rate of the mutualists.
The take-home message is that the failure of some previous models in
their predictions about mutualism may be due to the simplification of a
complete community to a mere two-species interaction. In larger model
communities, assemblages with mutualisms can have greater local stability,
resilience and persistence than analogous assemblages without mutualisms.
As with all models, care must be taken in drawing conclusions about the
real world. However, it does appear that although mutualisms destabilize
a two-species model assemblage, they need not do so for larger, more
realistic assemblages.
ACKNOWLEDGMENTS
The autors are gratefully indebted to Charles Godfray, George Hurtt, John Lawton, Simon
Levin, and three anonymous reviewers for comments, and espesially to Diane Wagner, who
first proposed this project. MSR was supported by a Graduate Research Fellowship from the
U.S. National Science Foundation and an Overseas Research Student Award from the Committee of Vice-Chancellors and Principals of the Universities of the United Kingdom.
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