Conditions for a Species to Gain Advantage from the Presence of Competitors
Author(s): Lewi Stone and Alan Roberts
Reviewed work(s):
Source: Ecology, Vol. 72, No. 6 (Dec., 1991), pp. 1964-1972
Published by: Ecological Society of America
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Ecology. 72(6), 1991, pp. 1964-1972
© 1991 by the Ecological Society of America
CONDITIONS
FOR A SPECIES TO GAIN ADVANTAGE
THE PRESENCE OF COMPETITORS1
FROM
LEWISTONE
Australian Environmental Studies, Griffith University, Nathan, Queensland 4111, Australia
ALAN ROBERTS
Graduate School of Environmental Science, Monash University, Clayton, Victoria 3168, Australia
Abstract. The interaction between two species is usually assigned as though they were
in isolation from all other species. Here we use a (known) method that determines species
interactions more realistically, within the framework of the community to which they
belong. This "inverse" method evaluates all the effects that one species experiences from
another, both direct and indirect.
We use this method to study the classical (though highly controversial) "competition
community," where each species is considered (in the "isolated pair" approach) to suffer
from the presence of every other. The model we use takes account of the fluctuations in
interaction coefficients that one must expect in the real world, both from one species pair
to another, and as the effect of ambient environmental variations.
Remarkably, the "inverse" method finds that generally a high proportion (20-40%) of
the interactions must be beneficial, or "advantageous," when not lifted out of the community context in which they actually occur. The contrary case, called here "hypercompetitive," in which each species suffers from every other species, can occur only if the
environment is nearly constant, and the species closely akin to each other, with both of
these conditions holding and persisting to a degree that must be considered implausible.
The results of the model remain valid, even after incorporating a number of major
structural modifications, thus indicating robustness in its predictions. We survey the available field data and show that they are in good general agreement with the conclusions
reached, on the high proportion of interactions which must be "Advantageous in a Community Context" (ACC).
Key words: community effects; competition; environmental impacts; indirect interaction.
be an element of paradox in the classical theory of
competitive equilibrium communities, which might
sometimes be describing communities in which a large
proportion of interactions are in fact beneficial (or "advantageous").
For competition communities, only three interacting
competitors are required for indirect advantaging to
occur. The direct effect of Yj (species j) on Yi (species
i) is given by the coefficient -oai (with aij necessarily
>0). Now, 52 affects f51 in two ways:
(1) by the negative direct interaction a12, denoting
how ,91 suffers from the presence of 52 (because, for
example, they compete for a common resource); and
(2) by the (less evident) indirect interaction mediated
through Y'3. Via the two negative interactions measured by aO3 and 0a32, 92 exerts a positive effect on ,1.
Following Boucher et al. (1982), we might describe this
sort of advantaging as due to a species' "enemy's enemy."
It is important to realize that an interaction can be
termed direct or indirect depending on the observer's
reference frame, so that ambiguity is possible (see
Abrams 1987). For example, f1 and 5°2 were viewed
1
8
revised
received
'Manuscript
August 1990;
February above as direct competitors, but their interaction could
well be mediated through a common resource and thus,
1991; accepted8 February1991.
INTRODUCTION
The theory of competition, with its associated notion
of competitive equilibrium, has been a major if controversial theme in community ecology (Diamond 1978,
Simberloff 1982, 1984). MacArthur (1972) expounded
the theory in depth, and developed a model of "diffuse
competition" to explain fundamental processes in
communities of multiple competitors. In his scheme,
which treats competitive interactions as if their effects
are always additive, each species suffers from the presence of every other.
However, it is now appreciated that, from a community-wide perspective, these so-called competitive
interactions may result in some species actually benefitting from the presence of others. For instance, two
consumers that are strong competitors in isolation may,
when put in the context of a community, have a mutualistic association. For this mutualism to become
evident, one needs to take into account not only the
"direct" competitive interactions between a pair of
species, but also their "indirect" interactions as mediated through other species. Consequently, there may
WHEN DOES COMPETITIONBENEFITA SPECIES?
December 1991
in a sense, be indirect. For this reason, Vandermeer et
al. (1985) have argued that any case of resource competition could be plausibly viewed as indirect.
For the configuration of three competitors above,
the positive indirect advantages can outweigh the losses
Y1 suffers from the direct competition of 99,. Levine
(1976) shows that if a13 a32 > ao12, the overall net effect
of 5°2 on ,99 will be positive. Such an interaction will
here be termed "Advantageous in a Community Context," or ACC. (The term "facilitative" can be found
in the literature to describe this effect, but we prefer
"advantageous" as simpler and more expressive.)
Cases in which a species is thus "advantaged" by a
competitor have been noted in the literature (and are
discussed further below). But how widespread is the
phenomenon? In a system where all the direct interactions are competitive, what should we expect as the
community-context norm, i.e., a significant fraction of
advantageous interactions, or their near if not total
absence? And can we say anything about the conditions
that favor this advantaging and those that exclude it?
We study these questions below, by analyzing randomly constructed models of competition communities. In doing so, we find that it is the total absence of
advantageous interaction that is indicated as rare, and
that we should expect 20-40% of the interactions to
be net-beneficial. The "uniform" model, in which all
species compete with equal strength, is thus a quite
misleading guide here. Some bodies of available field
data are then surveyed, and shown to be consistent
with the findings.
DERIVING COMMUNITYEFFECTSBY THE
INVERSEMETHOD
We consider a set of M interacting species, and take
the equations describing their growth and their direct
interactions in the form
dNk/dt = fk(NI, N2 . . ., NM, Ckl, Ck2 . . , CkRk),
k = 1 to M.
(1)
(For an outline of what this form implies, and the
empirical support for it, see, e.g., Yodzis 1988.)
Here the species counts Nk may be taken to measure
either population or biomass; for definiteness, we take
them here as population densities. In addition to these
state variables, the history of the species 9k depends
also on the Rk parameters Ck,, describing species behavior (inter- and intra-species interactions, birth rates,
migration flows, etc.).
We assume a feasible equilibrium achieved at the
species
counts Nl*, N2*, ...,
NM*. They satisfy the
equations
fk(Nl*,
N2*,
..
., NM*, Ckl, Ck2 .
k = 1 to M.
, CkRk) = 0,
(2)
Here, as elsewhere below, the asterisk indicates that
the attached quantity is to be evaluated at equilibrium.
The stability of this equilibrium (see, e.g., May 1973:
1965
51) depends on the eigenvalues of the "community
matrix"A, where
A, = (f,f/aNj)*.
(3)
The entry Ai,gives the "direct" effect on the ithspecies
,9i (around equilibrium) of a single /j individual. But,
as discussed above, 9j's actual effect on Yi, in the
context of the whole community, can be quite different.
To ascertain it, we note that the number of Yi individuals the system can carry is given by the equilibrium
count NAi,and we ask: how will this carrying capacity
change if more Cj individuals enter the system? If this
,9i count then increases (decreases), it seems reasonable
to interpret this as meaning that Yi is benefitting (suffering) from Yj.
To find this response, we select any parameter whose
increase implies a faster growth for Yj, i.e., one that
satisfies the condition
af/aCp> 0.
(4)
Differentiating Eq. 2 with respect to C,p (on which the
Ni are implicitly dependent), and using Eq. 3, we obtain
Ajk
k
2k
oNi*/C,,
A,ik N*/Cp
= -9f/9C,,,
= 0,
i=
(5a)
1 to M, i = j.
Solving Eq. 5 for the unknown derivatives:
= -(A)- ' ijf,/p.
Ni*/aC,
(5b)
(6)
The left side here tells how the equilibrium number
of the ,9i responds to an infinitesimal increase in the
9j's. This is the indicator that we seek, of the "interaction in a community context"; since Cjp has been
chosen so that the second factor on the right side will
be positive, Eq. 6 gives this indicator as just the sign
of the (i, j)th entry of the inverse of -A (Stone 1988).
The following remark is made to avoid possible misunderstandings. Only an infinitesimal increase in the
39jcan be considered, because the interaction between
Y, and 5j that interests us is the one existing at the
current equilibrium. It would thus defeat our purpose
if the change in Yj numbers produced a significantly
different equilibrium state. We stress this because, with
some choices of C,p, the change in 9j/ numbers could
be interpreted as due to an actual physical process,
perhaps even one that is experimentally controllable,
as, for instance, when Cjphappens to be an immigration
rate. But, for our present purposes, any such process
would have to be envisaged as infinitesimal.
This is worth pointing out since an expression altogether similar to Eq. 6 can occur in the treatment of
"press perturbations"; see, for instance, Yodzis 1988:
Eq. 7.15. But the context and goal are then quite different from ours; their aim is to handle and analyze
actual press experiments, in which new equilibria are
brought about by, for example, the steady introduction
Ecology, Vol. 72, No. 6
LEWISTONE AND ALAN ROBERTS
1966
from the constantly changing intensity and focus of
In general, competition
competitive interactions...
will be for specific limiting resources, which change
through time and space, as well as through a species'
life history stages." After noting that genetic changes
and even differences in age may drastically alter competitive interactions, Huston concludes: "Constant
competitive coefficients may well be meaningless except in uniform stable environments which never exist
in nature."
The design of the model used here incorporates this
consideration, and allows species interactions to vary
MODELLING COMPETITION
over time. It studies the totality of possible interaction
The Eqs. 1 describe a very general set of interacting patterns, that might describe a particular ecosystem at
different stages in its life history, given that its interspecies; if they are to model only systems of pure competition, containing no other types of species interac- actions fluctuate. The approach here is similar to that
tion, we must impose some constraints. To ensure that of Cohen and Newman (1988) and their "changing
community matrix." The final picture obtained from
each pair of species, if isolated from the rest, would
interact competitively near the community equilibriexamining this "ensemble" of community patterns turns
out to be quite different from that found by analyzing
derivatives
um, it is sufficient to require that the partial
a single, unchanging system, with defining parameters
on the right of Eq. 3 should be negative. To incorporate
intraspecific competition, this must also hold for the that remain constant for all time (see Stone [1988] for
case j = i. Thus, to ensure that Eqs. 1-3 describe a a more detailed discussion).
Since all pairwise species interactions near equilibsystem of pure competition, we add the requirement
are described by the community matrix A in Eq.
rium
i, j = 1 to M.
(8)
A,i < 0,
3, fluctuations in their strength can be modelled by
We first examine the "uniform competition" model
appropriate changes to the entries Aij. A large ensemble
of competitive communities may be specified, with -c
direct
in
which
the
"uniform
the
model"),
(or simply
interaction between a pair of species always has the as the average over this ensemble of any particular
same strength -c (with c > 0). This corresponds to entry (regarded, in this approach, as a time-average
over the life of a single system). Thus we may, if we
the "simplex" model discussed in detail by May (1973:
other
with
wish, regard any particular community as the current
interacts
in
which
every
"every species
162)
response to ambient conditions by what was originally
species in a manner which is completely symmetrical."
a "uniform model" of interaction strength (-c).
We adopt also the normalization (see, e.g., Roberts
To incorporate stochasticity, we let the community
1974, Pomerantz and Gilpin 1979) in which all species
matrix A have the form
are self regulated at unity. The interaction coefficients
are thus:
A = Ao - B,
of particular biota. However, as emphasized above, it
is the community-context interactions at the current
equilibrium that are the focus here.
Eq. 7.22, for example, in Yodzis 1988, expresses the
effect of a parameter change as a time series of successively more complex interaction loops. While this
development holds obvious interest in the case of a
finite press perturbation, it is not relevant to the present
exercise (where all the phenomena of community interaction that we seek to describe are already present
in the first [time-independent] term of Eq. 7.22).
Ai = -c,
i = 1 to M, i 4 j,
(9a)
and
A = -1.
(9b)
For this uniform model, it is shown in Appendix 1
that all the entries (-A -)ij (i # j) have the same magnitude and are of negative sign, that is, the interactions
in a community context are, like the direct interactions,
all competitive and all of equal strength. Thus the uniform model depicts a community in which each species
suffers from the presence of every other. We call such
a community hypercompetitive.
THE STOCHASTIC ENSEMBLE MODEL
There are good reasons to be wary before taking the
uniform model, or indeed any model in which the interactions never change, as a guide to competitive interactions in nature. As Huston (1979: 82) has noted,
competition has an "elusive quality ... (which) results
where Ao describes the uniform model (see Eq. 9), and
B imposes small perturbations. The entries (B)ij = b,,
are chosen independently from a distribution uniform
in (-d, d), with
(b,) = 0,
var(bj) =
The community matrix
A =-1,
2 = d2/3 (i
A
j),
b, - 0. (10)
now has elements
Aij= -(c
+ b),
i = j.
(11)
While this is a procedure familiar enough in the literature, it can benefit from closer examination than it
usually receives. The doubtful points include: the
method used to generate a uniform variate, the assumption of independent b' s, and the assumption that
an attainable equilibrium (i.e., a feasible one, with all
the populations positive, see Roberts 1974) can indeed
give rise to the A'S as the corresponding community
matrices.
These points need to be examined, and we have done
so. Since the conclusions emerge, we believe, essen-
December 1991
WHEN DOES COMPETITION BENEFIT A SPECIES?
tially intact from this examination, the discussion of
these questions is relegated to Appendix 2.
We now need to ask: what ranges of the parameters
involved should be chosen, as likely to have ecological
relevance? Here a number of considerations enter.
First, to ensure that it is communities of pure competition that are being modelled, we require that Aij <
0. This is achieved by choosing the bij to lie in the
interval (-cO, +c0) where 0 is a number between 0
and 1, so that the A,j are always negative and lie in the
interval [-c(l + 0), -c(l - 0)]. This gives a = d/V\
= c0/3.
We must also decide what range of the mean value
c should be chosen. We first note that the uniform
model is unstable for c > 1, and take as our upper
limit IA,1 = c(l + 0) < 1, on the assumption (which
numerical calculations, incidentally, have confirmed)
that the stochastic model will also have few feasible,
stable solutions otherwise.
To fix a lower limit for the range of c, we note that
values that are too small could not be regarded as modelling a system in which interaction between species
was significant, since each species would then be influenced more by its own intraspecific competition than
by that from other species. We therefore take for our
smallest c a value of 1/(M - 1), at which the sum of
the mean interspecific coefficients is equal to the intraspecific (self-regulation) term.
We collect here these various constraints. To ensure
all the direct interactions are negative:
0 < 0 < 1. (12a)
b,j confined to (-cO, c0),
To avoid instability:
c(l + 0) < 1.
(12b)
To make interspecific competition significant:
c > 1/(M-
1).
(12c)
In this model, environmental fluctuations make the
interaction strengths vary about the community's mean
strength of competition. Thus two communities may
have the same number of species and the same average
interaction strength c, but the one undergoing stronger
perturbation will show a greater variation in its interaction coefficients. Hence the stochastic model expresses increasing disturbance by increasing 0 and
therefore a2(=02c2/3),
the variance of the bij.
ADVANTAGEOUSINTERACTIONSIN A STOCHASTIC
COMPETITIONCOMMUNITY
We have seen (in the Introduction) that, even if 59
directly competes with 5i when the two species are
alone together, it is possible for the net interaction to
be Advantageous in a Community Context (ACC), i.e.,
A
< 0,
while (A-1),j > 0.
1967
Percentageof interactionsthat are Advantageous
in a Community Context (ACC), arranged in ascending
order for the stochastic competition model. Each of the
horizontal rows of the table gives results for a model competition community in which interactions are uniformly
distributed between -c(1 - O) and -c(1 + 0).
TABLE 1.
Interaction
strength, c
0
Spread
a/c (%)
%ACC
GLV (%
feasible)
0.100
0.100
0.200
0.100
0.300
0.200
0.100
0.100
0.100
0.200
0.100
0.300
0.100
0.200
0.400
0.500
5.8
11.5
5.8
17.3
5.8
11.5
23.1
28.9
0.0
0.0
0.0
1.3
1.7
7.8
10.1
18.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
0.200
0.100
0.200
0.300
0.200
0.300
0.300
0.300
0.300
0.600
0.400
0.300
0.500
0.400
0.500
0.600
17.3
34.6
23.1
17.3
28.9
23.1
28.9
34.7
21.1
23.5
28.3
31.9
32.8
100.0
100.0
100.0
97.1
97.1
62.9
24.3
7.4
36.5
39.0
41.7
We can find how common this phenomenon is, by
examining an ensemble of competitive interaction matrices and then determining the frequency of advantageous "community effects." The parameter ranges
must of course obey the constraints (Ineq. 12) above,
if the work is to have ecological relevance.
We generated matrices of size 21 x 21 (each representing a 21-species community) so as to satisfy both
Eq. 1 1 and also the constraints (Eq. 12). We chose mean
interaction strengths as successively c = 0.1, 0.2, and
0.3, corresponding to a ratio (sum of interspecifics/
intraspecific) of 2, 4, and 6, respectively. The values
of 0 were stepped in increments of 0.1 from 0.1 to 0.6.
For each couple (c, 0) 200 sample matrices were
generated. Each matrix contained 420 (directly) competitive interactions; a representative sample of the
results is displayed in Table 1. The ratio (standard
deviation)/(mean value) is used to measure the disorder in the interaction coefficients, and termed their
"spread." (The last column will be explained in Appendix 2.)
The results have been arranged in ascending order
of the percentage of advantageous interactions (%ACC),
which make up from 0 to >40% of the total possible,
depending upon the magnitudes of c and 0. We see
that hypercompetitive communities, in which each
species suffers from the presence of every other, are
quite rare. Only those cases above the horizontal dotted
line have % ACC < 20%; studying them, we note that
they correspond to either an unrealistically small spread
of disorder (< 20%) or a very small degree of interspecific competition (c = 0.1, meaning that the biota suffer
10 times more from their own species, on the average,
than from another). Further comment on these results
is postponed to the Conclusions.
Ecology, Vol. 72, No. 6
LEWI STONE AND ALAN ROBERTS
1968
2. Percentageof interactionsthat are Advantageous we thus reduce heterogeneity, by making the direct
interactions no longer independent, the number of advantageous interactions likewise decreases.
In a further major step, we changed the model to
simulate M species each characterized by a different
mean strength of interaction. The interaction coefficients were now
TABLE
in a Community Context (ACC), under different model
conditions, arranged in ascending order: (a) results from
original stochastic model; (b) incorporating a negative correlation between interaction coefficients Ai and A,i; (c) each
species <9 given its own mean strength of interaction c, so
that (A,j) = -Ci; (d) as (c), but incorporating negative correlations between interaction coefficients.
% of advantageous interactions
(a)
(c)
(b)
(d)
Spread
Spread
a/c (%)
M = 4 species, mean competition strength c = 0.5
11.5
0.8
0.7
0.3
0
23.1
14.3
16.5
12.3
14.7
34.6
23.5
21.8
24.0
26.3
46.2
28.3
33.8
25.3
29.2
M = 5 species, mean competition strength c = 0.38
16.7
2.2
3.0
1.2
2.6
33.3
19.2
21.1
19.2
20.6
50.0
26.6
30.9
26.9
29.1
66.7
32.8
35.7
31.5
36.7
83.3
34.1
40.2
33.1
40.0
ROBUSTNESS OF THE MODEL'S PREDICTIONS
Thus, in this model, the extent to which the competition coefficients vary is important in determining
the number of interactions that are Advantageous in a
Community Context. But we should see how far these
results depend on the particular details of the model
chosen, that is, how robust they are.
The analysis so far has concentrated on 21-species
systems. However, further simulations have shown that
ACC interactions can appear just as prominently in
smaller communities. The left-hand column of Table
2 shows the results when four- and five-species communities are simulated. We see that up to 40% of interactions can still be Advantageous in a Community
Context.
It is common for the interactions between species to
show a fairly obvious mutual dependence. For example, a strong competitor can have large effects on a
particular species but itself experience only small effects from that species. It seemed useful to determine
whether such correlations can significantly affect the
results. To model this phenomenon, the interaction
coefficients Aoiand Aj,were given negative correlations.
Taking, as before
A, = -(c + b,)
for i > j;
bji)
for i < j.
we set
Ai=
-(c-
The second column of Table 2 shows some results
from this type of simulation. It is evident that ACC
interactions are still often found in the 20-40% range.
However, a comparison with the first column of Table
2 reveals that the introduction of negative correlations
into the coefficients A, slightly lowers the frequency of
ACC interactions. This general feature is found over a
wide range of parameter space. It appears that when
A = -(c,
+ bii)
where the ci were randomly chosen from a distribution
uniform in the interval (0.8c, 1.2c), so that
(A)
= -c,.
This new model incorporated additional heterogeneity.
As the simulation results of Table 2 (column 3) show,
this increased heterogeneity carries over into a slightly
increased frequency of ACC interactions.
Column 4 of Table 2 shows the effect when negative
correlations in the interaction coefficients are added to
this last model. Again we note that a decrease in heterogeneity gives a reduction in ACC interactions.
All these various simulations go to confirm that the
original predictions of the ensemble model have some
degree of robustness. Important structural changes of
the model seem to change the frequency of ACC interactions by only a slight amount, rarely > 5%. All of
them confirm the apparent rule: "More heterogeneity
implies that more competitors are helpful."
COMPARISONWITH FIELD DATA
To be sure that the findings above are more than
mere mathematical artifacts of the ensemble model,
we now survey field data for various real systems that
are believed to be purely competitive.
MacArthur (19 5 8) gave his classic data on competing
warbler species,
and later pointed
out that "...
an
increase in blackburnia warblers should, by itself, make
invasion easier for the bay-breasted" (MacArthur 1968).
This advantaging occurred despite the competitive direct link between the two species.
Lawlor (1979) examined Cody's data for eight avian
communities. He found that, of the entries in the community-effects matrix, between 30 and 40% were advantageous. Davidson (1980, 1985) studied granivorous ants in the Chihuahuan Desert near Rodeo, New
Mexico, over a 5-yr period. She constructed the direct
interaction matrix, based on the dietary overlaps of six
ant species, and made careful allowance for any interference competition. Performing manipulatory experiments and then examining correlations between
populations, Davidson confirmed that her communityeffects matrix indeed gave realistic predictions. From
her results one sees that 11 of the 30 off-diagonal coefficients within the community-effects matrix are of
positive sign, showing that 34% of interactions were
advantageous.
Lane (1975) examined four zooplankton communities in which, she argued, interspecific competition
was a predominant community force. Using the GLV
TABLE 3.
1969
WHEN DOES COMPETITION BENEFIT A SPECIES?
December 1991
The percentageof interactionsthat were Advantageousin a CommunityContext(%ACC)are tabulatedfor Lane's
(1975) interaction matrices.
Lake
Gull
Michigan
Cruise
Species
%ACC
1
8
43
2
3
0
3
8
36
4
8
50
5
9
42
model, Lane calculated 11 interaction matrices from
data on four lakes at various times. She derived the ai
from equations (based on resource turnovers) developed by Richard Levins. After inverting the interaction
matrices provided by Lane, we obtained the community-effects matrices and tabulated the percentage of
interactions that were Advantageous in a Community
Context (Table 3). The only case lacking such interactions was the small community (only three species)
of Cruise 2 on Lake Michigan.
Literature reviews of field data also tend to confirm
the prevalence of advantageous indirect interactions.
For example, Connell (1983) surveyed 72 studies of
interspecific competition; 14, or 19%, showed advantageous interactions, usually of an indirect nature, while
competition itself was successfully demonstrated by
only 23, or 32% (see Vandermeer et al. 1985).
To sum up, then: to the extent that the field studies
examined above correctly determine competition coefficients (this note of caution is needed; see, e.g., Lawlor 1979: 356), they give a reasonably consistent picture
in the large proportion they display of interactions that
are Advantageous in a Community Context. This high
proportion verifies what the stochastic model has suggested, i.e., that ACC interactions are quite common
in systems usually designated as competitive.
DETERMININGCOMMUNITYEFFECTSBY
OTHER METHODS
Levine (1976), Bender et al. (1984), and Davidson
(1980, 1985) have all made use of the inverse interaction matrix to elicit information concerning indirect
effects, though not proceeding to the generalization presented here. Perhaps Levins (1973, 1975) developed
the most comprehensive theory of the subject with his
loop analysis, which has been used extensively (e.g.,
Lane 1975, 1985, Briand and McCauley 1978, Puccia
and Levins 1985). When analyzing a community by
Levins' method, it is necessary to enumerate the number of feedback loops (or indirect pathways) of various
"lengths" and "levels," embedded in the community
matrix. Once they are found, the community effects
can be determined.
Because loop analysis assigns only the values + 1,
-1, or 0 to an interaction coefficient, the task of determining community effects is greatly simplified.
However, this restriction on the coefficients becomes
a severe limitation. Levins (1975) warns that it is quite
Cranberry
6
8
29
7
8
45
8
5
20
9
5
20
George
10
7
24
11
7
24
possible to get ambiguous or even wrong results when
the magnitudes of interactions are neglected in this
way. Even with this simplification, loop analysis becomes tedious for the study of large communities, and
a computer (with a specialized program) is needed to
enumerate the loops, of each particular "length" and
"level," within the interaction matrix. In contrast, the
inverse method above requires only the generally available computer software used for inverting matrices. It
has the added advantage that interaction intensities are
taken fully into consideration.
If one allows for the way that the interaction matrix
has been simplified, it turns out that loop analysis is
almost identical with the matrix inversion method. An
examination of the underlying mathematics (Levins
1973: 131, Eq. 23) reveals that, in using loop analysis
procedure to obtain community effects, we do nothing
more than compute the inverse of the interaction matrix. (Calculating loops is, after all, the process required
in order to expand as determinants the cofactors needed to invert a matrix.)
Lawlor (1979) presented a technique that was designed to analyze indirect interactions in an M-species
competitive community. He calculated the net per capita effect on the equilibrium density of Y99,when the
equilibrium of 5/'2 is manipulated.
It is not hard to show that, apart from a self-regulation term, his technique for obtaining community
effects is equivalent to the matrix inversion method
described here, although in a rather more complicated
form. The path linking the two results is sketched in
Appendix 3.
CONCLUSIONS
The role and effects of indirect interactions have
received far less attention than that given to the postulated "direct" interactions. This seems curious, when
we remember that these latter can be justified only by
an act of abstraction, in which the species pair is considered in isolation from its actual context, the community. A striking difference between this abstract relationship and the real one has been noted above, for
competition system models.
The findings displayed in Table 1 can be put in a
different and more suggestive way as follows. Recall
first that, when the number of species M and mean
interaction strength c are fixed, the "spread" parameter
1970
LEWISTONE AND ALAN ROBERTS
Ecology, Vol. 72, No. 6
(a/c) is a measure of the variability in the model. This
ACKNOWLEDGMENTS
variability comprises that between different species pairs
We wish to thank Chris Wallace, of the Monash University
at a given time, and also (in the "life history" inter- Computer Science Department, for drawing our attention to
pretation of the random ensemble) the fluctuations in the possible dangers in random number generators.
environmental conditions from one time to another.
LITERATURECITED
We can thus say that, in the very general model used
Abrams, P. A. 1987. On classifying interactions between
above for a 21 -species competition system, each spepopulations. Oecologia (Berlin) 73:272-281.
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tice. Ecology 65:1-13.
the most that an interaction coefficient can ever vary
Boucher, D. H., S. James, and K. H. Keeler. 1982. The
by is < 12%. In more disturbed (and, we suggest, more
ecology of mutualism. Annual Review of Ecology and Sysrealistic) environments, the system cannot be "hypertematics 13:315-347.
competitive" like this; between 20 and 40% of the Briand, F., and E. McCauley. 1978. Cybernetic mechanisms
in lake plankton systems: how to control undesirable algae.
effects that one species has on another must be beneNature 273:228-230.
ficial. These results were shown to have some robustCohen, J. E., and C. M. Newman. 1988. Dynamic basis of
ness upon variation in the number of competing species
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and in various aspects of the model's structure.
Connell, J. H. 1983. On the prevalence and relative importance of interspecific competition: evidence from field
Moreover, even at a given moment, that is, for a
experiments. American Naturalist 122:661-696.
fixed set of interaction coefficients, these coefficients
D. W. 1980. Some consequences of diffuse comDavidson,
can differ between themselves by < 12%, if all interacpetition in a desert ant community. American Naturalist
tions are to be harmful. This would seem to be another
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1985. An experimental study of diffuse competition
implausible uniformity, this time, one holding between
in harvester ants. American Naturalist 125:500-506.
species pairs; in their internal interactions, they have
Diamond, J. M. 1978. Niche shifts and the rediscovery of
to resemble each other with an unbelievable closeness.
interspecific competition. American Scientist 66:323-331.
Let us recall the "uniform model" that was found Huston, M. 1979. A general hypothesis of species diversity.
to be hypercompetitive. It is now clear that it is not as
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special a case as it might have appeared; it is simply Lane, P. A. 1975. The dynamics of aquatic systems: a comparative study of the structure of four zooplankton comthe extreme member of the group of cases that repremunities. Ecological Monographs 45:307-336.
sent environments not varying much in time, popu1985. A food web approach to mutualism in lake
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themselves. This whole group can, we suggest, reprebiology of mutualism. Croom Helm, London, England.
sent communities found rarely if ever in the real, in- Lawlor, L. R. 1979. Direct and indirect effects of n-species
competition. Oecologia (Berlin) 43:355-364.
constant world.
Levine, S. H. 1976. Competitive interactions in ecosystems.
The significance of competition, in the distribution
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of species, has been much debated. Simberloff (1982), Levins, R. L. 1973. Discussion paper: the qualitative analin his important critique "The status of competition
ysis of partially specified systems. Annals of the New York
Academy of Sciences 231:123-138.
theory in ecology," raises a number of issues that our
1975. Evolution in communities near equilibrium.
model confirms to be of absolutely crucial concern here.
Pages 16-50 in M. L. Cody and J. M. Diamond, editors.
He argues that the traditional theory of interspecific
Ecology and evolution of communities. Belknap, Harvard
competition means little, unless important factors such
University, Boston, Massachusetts, USA.
as chance events, heterogeneous environments, and MacArthur, R. H. 1958. Population ecology of some warblers of northeastern coniferous forests. Ecology 39:599community-context interactions are taken into ac619.
count. The procedure used above goes what might be
. 1968. The theory of the niche. Pages 159-176 in R.
considered only a short distance in reckoning with such
C. Lewontin, editor. Population biology and evolution. Syracuse University Press, Syracuse, New York, USA.
factors, yet the results indicate how drastically they can
. 1972. Geographic ecology. Princeton University,
modify our perceptions of even the simplest compe& Row, New York, New York, USA.
Harper
tition system. If by competition system, or subsystem,
May, R. M. 1973. Stability and complexity in model ecois meant a hypercompetitive one with all interactions
systems. Princeton University Press, Princeton, New Jerharmful (and this view is not unknown), then the findsey, USA.
ings above set a task for those who advocate its im- Park, S. K., and K. W. Miller. 1988. Random number generators: good ones are hard to find. Communications of the
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exhibit
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Until this is done, the realism and even the self-concovariance and coexistence. Journal of Theoretical Biology
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Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T.
doubt.
Vetterling. 1987. Numerical recipes: the art of scientific
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gland.
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Puccia, C. J., and R. Levins. 1985. Qualitative modeling of
WHEN DOES COMPETITION BENEFIT A SPECIES?
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Roberts, A. 1974. The stability of a feasible random ecosystem. Nature 251:607-608.
1989. When will a large complex system be viable?
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1984. The great god of competition. The Sciences
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Stone, L. 1988. Some problems of community ecology: processes, patterns and species persistence in ecosystems. Dissertation. Monash University, Melbourne, Australia.
Vandermeer, J., B. Hazlett, and B. Rathcke. 1985. Indirect
facilitation and mutualism. Pages 326-343 in D. H. Boucher, editor. The biology of mutualism. Croom Helm, London, England.
Yodzis, P. 1988. The indeterminancy of ecological interactions as perceived through perturbation experiments.
Ecology 69:508-515.
APPE lNDIX 1
For the uniform model, the interaction matrix Ao is given
Hence, from Eq. 7, and c < 1, NA'(Ij)/dIj< 0, so that the
uniform model has no ACC interactions; it is hypercompetiby Eq. 9, with c < 1 (to ensure stability). Its negative inverse
' has as elements:
tive.
-A
(-A),
(1 -c)[1
(-,
+(m
-
)c]'
i
j,
+1 (m - 2)c
(1 (1
c)[1 + (m - 1)c]'
APPENDIX 2
Here the r, are the intrinsic increase rates, and the a,( (>0)
Here we consider the objections foreshadowed in the disare the competition coefficients, interspecific (i # j) and incussion of the stochastic ensemble model.
'
traspecific (i = j). The matrix whose (i, j)th entry is al has a
As the simplest to deal with, let us first note that a satisfor its inverse.
factory series of "pseudorandom" numbers will not necesThe "community matrix" A (Eq. 3) is given by (see, e.g.,
sarily be supplied by a procedure purporting to do so, whether
May 1973: 51)
it is intrinsic to a higher order language or offered in some
other way. Procedures of the "linear congruential" type, for
(2.2)
'Ni*.
A, = (df/dN)*
instance, can exhibit a high degree of sequential correlation
in the successive, allegedly "independent" values given for a From Eq. 2. lb, the N,* are (rational) functions of the a,j. Thus
uniform variate (see, e.g., Park and Miller 1988).
any two entries in A, since they must both be functions of the
This point is far from being a mere nicety; generating pro- set {a,X},cannot vary independently of each other. It is evident
cedures to be found in text books can be so inefficient that that a similar dependence will occur no matter what form the
they give as few as 125 distinct values, which they thereafter functions f have in Eq. 1.
A further difficulty lies in the assumption that, no matter
repeat forever (Park and Miller 1988). For the 21 x 21 matrices treated in this paper, this would mean that, far from what values are given to the entries of A by the "randomizing"
obtaining a succession of statistically independent matrices, process, A can still be regarded as a community (or "stability")
the 420 off-diagonal elements in one matrix alone would con- matrix, that has arisen from the standard first-order expansion
tain at least three repetitions of each entry with the set of of some realistic ecosystem model, about an equilibrium in
which all the population values are positive.
distinct entries being, moreover, the same in each matrix!
We might agree that some equations of the form of Eqs. 1
With this in mind, and programming in Turbo Pascal, we
replaced its RANDOM intrinsic function by the procedure and 2, containing specific functionsf' say, could in principle
be found to yield a given A as their community matrix. What
RAN3 given by Press et al. (1987: 199, 715).
The next problem can be stated quite simply: although, in is much more doubtful is, first, whether the same functional
the literature, the entries of the community matrix A have formsf' could account for all the matrices in the sample, and,
frequently been allowed to vary independently of each other, second, whether these functions f' would be at all realistic
and acceptable, in the behavior they implied for species in
this is in fact impossible.
This can best be seen by an example, which will in any case the model.
It is hard to see how these objections can be met, without
be needed in the sequel. Consider the simplest nonlinear model of a community, the "Generalized Lotka-Volterra" (GLV) actually solving the equilibrium Eqs. 2 for each of the systems
or multispecies quadratic. For this model, the particular form composing a random sample. If the numbers satisfying Eq. 2
are positive, and therefore possible population values, i.e., if
of Eqs. 1 and 2, specialized to a competition community, is
the system is "feasible" (Roberts 1974), we indeed have an
equilibrium and can go on to examine its stability. Confining
=
=
i
1 to M,
(2.1a)
dN,/dt
Z aN,),
N(-r,i
our attention to this "feasible," stable subset, we can then
d
extract the properties of interest. These properties may or
=
may not be similar to the corresponding properties of the
M.
i=
to
1
(2.lb)
Nl*
2Z (a-'),/r,
whole random ensemble, whose membership generally inI
1972
Ecology, Vol. 72, No. 6
LEWI STONE AND ALAN ROBERTS
cludes the feasible and the unfeasible, the stable and the unstable.
It would be wrong to think that work that uses the procedure
criticized above, assuming the entries of A to be those of a
community matrix, and that they can vary independently,
must inevitably lead to wrong conclusions. The requirement
of feasibility, for example, singles out the ecologically relevant
subset of a random ensemble, but this subset may not differ
significantly from the whole ensemble in those properties that
happen to be the ones under study.
But we should not assume in advance that this will be true
for all properties, and especially for those involved in the
present study. It could well be that a feasible equilibrium, and
stability about it, occurs more frequently, or less frequently,
when there is a high proportion of advantageous interactions;
in either case, the results obtained from studying the whole
ensemble, feasibles and unfeasibles alike, could be quite misleading. Such points seemed to be worth examining.
Unfortunately, it does not seem possible to do so for a case
as general as that of Eqs. 1 and 2; it is necessary to specify
the form of the functions f, if the equilibrium equations are
to be solved. Accordingly, the GLV model described by Eq.
1.1 was examined.
The variations chosen were those described earlier (Eqs.
10-12, in text), but applied here to the coefficients ai rather
than to the community matrix entries Aj. Thus we let aii =
1, and put aoi = c + bj, with
var (b,j) = 02 (i: j),
(b,) = 0,
b,, = 0. (10a)
It is now possible for the b,j's to vary independently. The
community matrix is given by
(2.3)
A,ij= -N*aol .
As before, we examine the signs of the entries (-A- '),, but
only for those generated matrices that are feasible (all N,* >
0) and stable (the real parts of all eigenvalues of A being
negative).
The appropriate parameter range was easily determined.
Define the parameter y by
y2
=
(M-
1)2/(1
-
c).
(2.4)
Then it is known that feasible solutions of the competition
system, when represented by the GLV model, are almost all
confined to the region y < 1. Moreover, these feasible solutions are found empirically to be stable (locally at least), no
exceptions having been found in many thousands of random
systems solved numerically, extending over a wide range of
values of c, M, and y (see Roberts 1974, 1989, Stone 1988:
51 and Fig. 2c, p. 71). The range of values for c and t0 displayed
in Table 1 implies 7 < 0.7; we could thus assume that systems
found feasible were also stable.
We repeated our earlier calculations, but now interpreted
as modelling GLV systems, and discarding those found to be
unfeasible. The results were anticlimactic but reassuring: on
taking sampling fluctuation into account, the percent ACC in
Table 1, and the figures found under the requirement of feasibility, showed no significant difference. This means that, at
least for the GLV model, in respect to the property we are
studying, the feasible subset behaves essentially the same as
the whole ensemble. Thus the results presented in the main
body of the text survive this test.
APPENDIX 3
We now use the following result for the determinant of a
Using techniques similar to Schaffer (1981), we show that
partitioned matrix:
Lawlor's expression for indirect effects (his Eq. 4) is essentially
the same as obtained here from the "community-effects" marP Q
Is P - RS-'QI.
trixA.
, then
If r=T
Tr|
sj
[R
We write the inverse of A in terms of its cofactors. If we
form the matrix A,i, from A by deleting row j and column i,
then
Applying this to the two determinants in Eq. 3.1, partitioned
as indicated (here with p a 1 x 1 matrix) we obtain the numerator on the right side of Lawlor's Eq. 4. Thus, apart from
- A l _ 1+
_A
3.1
(A '
( )
the self-regulation term in his denominator, the two expres*3 1)
A
I)
sions for measuring total (direct plus indirect) interaction are
For convenience, shift the ith row and the jth column in A to
identical.
the leading position, and partition the resulting matrix in the
pattern
LM1[M
M1I1
1J'