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vol. 158, no. 5
the american naturalist
november 2001
The Effect of Density-Independent Mortality on the
Coexistence of Exploitative Competitors
for Renewing Resources
Peter A. Abrams*
Department of Zoology, University of Toronto, 25 Harbord Street,
Toronto, Ontario M5S 3G5, Canada
Submitted September 6, 2000; Accepted May 24, 2001
abstract: Many ecologists believe that higher mortality imposed
on competing species increases the probability that they will coexist.
This belief has persisted in spite of many theoretical counterarguments. However, few of those counterarguments have been based on
models having explicit representation of the resources for which
competition is occurring. This article analyzes a series of consumerresource models of competition for nutritionally substitutable renewable resources and determines the range of relative resource requirements that allow coexistence. In most cases, if consumers are
initially efficient at reducing resource densities, increasing densityindependent mortality widens the range of resource requirements of
the consumers that allow coexistence, provided the increase in mortality is not too great. The coexistence-promoting effects of mortality
occur because a very efficient consumer species usually reduces the
diversity of the set of resources it consumes. This lessens the extent
to which resource utilization differences between consumer species
can be expressed. Mortality, in this case, increases the diversity of
resource types, widening the conditions for coexistence. However,
sufficiently high mortality will usually reduce the range of parameters
allowing coexistence, in agreement with much previous theory. The
results presented here also predict maximal diversity at intermediate
levels of productivity. Previous empirical studies and theory are reviewed in light of the theory developed here.
Keywords: competition, coexistence, diversity, harshness, intermediate
disturbance hypothesis, mortality, predation, productivity.
It has frequently been claimed that imposing mortality on
competitors will make it easier for them to coexist (e.g.,
Harper 1969; MacArthur 1972; Pianka 1972; Connell 1975,
1980; Gurevitch et al. 2000; see critical discussion in Chesson
* E-mail: [email protected].
Am. Nat. 2001. Vol. 158, pp. 459–470. 䉷 2001 by The University of Chicago.
0003-0147/2001/15805-0001$03.00. All rights reserved.
and Huntley 1997). A corollary is that predation by a nonselective predator on competing prey species will also make
it easier for them to coexist. Although there are several
mechanisms that have been shown to produce this outcome,
all of these mechanisms are of limited generality; they apply
to only a fraction of the systems where this popular belief
has been invoked. In particular, there has been no demonstration that increased density-independent mortality of
competing consumers should facilitate their coexistence
when they are exploitative competitors for renewing resources. In fact, a variety of theoretical works have argued
that increased mortality, in general, will not act to promote
the coexistence of such competitors (Cramer and May 1971;
Van Valen 1974; Abrams 1977; Yodzis 1977, 1986, 1989;
Holt 1985). This article will analyze coexistence in several
simple models of consumer-resource systems and will suggest that mortality can, in fact, facilitate the coexistence of
exploitative competitors (consumptive competitors sensu
Schoener 1983) under a broad range of conditions.
I begin by using the Lotka-Volterra model of two-species
competition to illustrate a measure of the “ease of coexistence.” When applied to the Lotka-Volterra model, this
measure predicts that coexistence becomes more difficult
with increased density-independent mortality. The bulk of
the article is then devoted to determining whether this
conclusion holds for several different consumer-resource
models of competition and to showing why these models
often predict that mortality will facilitate coexistence of
efficient consumers. The article also reviews related theory,
suggests experimental tests, and discusses how the effects
of predator-caused mortality may differ from the densityindependent mortality considered here.
A Simple Analysis Based on LotkaVolterra Competition
The Lotka-Volterra equations provide a useful baseline for
judging the effects of mortality in more mechanistic models. Because of the prevalence of this model in competition
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The American Naturalist
theory, it is also important to assess whether the relationship between mortality and competition that it predicts is
representative of a broader class of models.
Before proceeding, it is necessary to establish a definition of “ease of coexistence.” Here, ease of coexistence will
be measured by the breadth of the range of conditions
(parameter values) that permit coexistence. Armstrong
(1976) termed this the “coexistence bandwidth.” This section analyzes coexistence bandwidth for all of the parameters that can affect coexistence under two-species LotkaVolterra competition.
The Lotka-Volterra model may be written as follows:
dN1
p r1N1 ⫺ k 1(N1 ⫹ a12 N2 )N1 ⫺ D1fN1,
dt
(1a)
dN2
p r2 N2 ⫺ k 2(N2 ⫹ a 21N1)N2 ⫺ D2 fN2 .
dt
(1b)
This form separates density-independent components of
population growth for species i into the maximum per
capita growth rate, ri, and a per capita mortality term, Dif.
The mortality term is factored into a general factor, f, and
a species-specific factor, Di. The parameter ki measures the
per capita effect of species i on its own per capita growth
rate. The carrying capacity of species i is (ri ⫺ Di f)/k i. The
competition coefficient, aij, has its usual meaning as the
ratio of per capita competitive effect of j on i relative to
that of i on i. Exploitative competition generally results in
a12 a 21 ! 1, so that condition will be assumed here.
Coexistence bandwidth in the two-species LotkaVolterra model can be measured by the ranges of ri, ki,
or aij that permit coexistence. These bandwidths can be
derived from the well-known coexistence conditions,
a12 ! K1/K 2 ! 1/a 21, where Ki is the carrying capacity of
species i. The coexistence bandwidth of ri is given by
ri max ⫺ ri min p
(rj k i ⫺ Dj k i f)(1 ⫺ aij aji)
k j aji
(ri ⫺ Di f)(1 ⫺ aij aji)
.
rj ⫺ Dj f
aij !
k j (ri ⫺ Di f)
k i(rj ⫺ Dj f)
.
(4)
The right-hand side of inequality (4) differs from that of
inequality (3) by a constant, which implies that the constraint on one competition coefficient is relaxed while the
other is made more stringent by increased mortality (unless r1D2 p r2 D1, in which case mortality has no effect on
coexistence). If we consider the results for all three parameters, the decrease in the bandwidth for r with mortality is likely to imply that mortality decreases the total
volume of parameter space allowing coexistence.
The Lotka-Volterra model has been criticized for its lack
of competitive mechanism (Abrams 1975, 1977; Schoener
1976). Analyses of consumer-resource models have shown
that competition coefficients usually change as consumer
populations change (Schoener 1976; Abrams 1977, 1980,
1983, 1987), contrary to the underlying assumption of the
Lotka-Volterra model. By examining models with explicit
representation of resources as well as consumers and by
examining a range of functional forms for model components, it is possible to get a more general answer to the
question, How does mortality affect the ease of coexistence
of competing consumers? The analysis here will focus on
a set of models with two consumers and either two or
three resources. Within that framework, a variety of different assumptions will be made about the form of the
consumer functional response, the form of the resource
growth, and the presence or absence of resources that are
used exclusively by each consumer.
Consumer-Resource Models of Competition
.
(2)
This range must decrease with overall mortality, f, because
of the requirement that a12 a 21 ! 1. The bandwidth for ki
is given by
k i max ⫺ k i min p
The limits on the competition coefficients determined by
coexistence conditions are only maximum values and are
given by
(3)
From equation (3), it follows that the effect of overall
mortality, f, is to decrease the bandwidth for ki if ri Dj !
rj Di and to increase the bandwidth when ri Dj 1 rj Di. This
implies that mortality has no effect on the bandwidth for
either ki when r1D2 p r2 D1 but otherwise increases the k
bandwidth for one species while decreasing it for the other.
MacArthur’s Consumer-Resource Model
Two Resources. MacArthur’s model assumes that resources
have logistic growth and consumers have linear functional
and numerical responses. I begin with this system in part
because it has been used (MacArthur 1972) as a justification for the Lotka-Volterra model and in part because
it is probably the most widely used consumer-resource
model of competition (Chesson 1990). However, MacArthur’s own (1970, 1972) analysis implicitly assumed that
resources were never excluded by apparent competition
(as pointed out by Hsu and Hubbell [1979] and Abrams
[1998]). Taking resource exclusion into account, the interspecific effects are only piecewise linear and do not
correspond to any single Lotka-Volterra model (Abrams
1998). This analysis begins with a scaled version (see
Mortality and Coexistence
Abrams 1998) of the two-consumer–two-resource MacArthur model having the following form:
dN1
p N1[CR1 ⫹ (1 ⫺ C)R 2 ⫺ d 1],
dt
(5a)
dN2
p N2[(1 ⫺ C)R1 ⫹ CR 2 ⫺ d 2 ],
dt
(5b)
dR1
p R1(1 ⫺ R1) ⫺ CR1N1 ⫺ (1 ⫺ C)R1N2 ,
dt
(5c)
dR 2
p R 2(1 ⫺ R 2 ) ⫺ CR 2 N2 ⫺ (1 ⫺ C)R 2 N1.
dt
(5d)
Time has been scaled relative to the common intrinsic
growth rate, r, of both resources, and resource densities
are scaled to the common carrying capacity, K. Consumer
densities were scaled relative to their summed per capita
consumption rates of the two resources and the consumer’s efficiencies of converting resources to offspring.
Both resources are assumed to have equal nutritional/energetic values to a given consumer. The consumers are
mirror images of each other in their relative abilities to
consume the two resources. This allows C to be used as
a measure of similarity in resource use (with C p 0 corresponding to no competition, and C p 0.5 corresponding to identical consumption). The main results obtained
do not depend on this symmetry assumption of “mirror
image” consumers, as is shown below.
The parameters di in equations (5) are the resource requirements for zero population growth, expressed as a fraction of the resource carrying capacity. The di determine the
maximum per capita growth rates of the two consumer
species, 1 ⫺ d i for species i. Thus 1 ⫺ d i in equations (5)
is analogous to ri in equations (1). The values of di reflect
three main biological features of the system: metabolic efficiency of consumer species i, which sets a lower limit to
d; mortality, which increases di; and resource carrying capacity, which decreases di. If d 1 p d 2 p d, the equilibrium
resource densities are equal to d when expressed as a fraction
of their carrying capacities. The consequences of imposing
higher density-independent mortality on both species can
be seen by raising both species’ resource requirements. The
consequences would be qualitatively the same if metabolic
requirements were raised or the resource carrying capacities
lowered. The question, How does density-independent mortality (higher d ) affect the parameter range allowing coexistence? can be answered using the same approach as
described above for the Lotka-Volterra model. If the range
of d1 allowing coexistence increases (decreases) with the
initial d, higher levels of mortality make coexistence easier
(harder). Because species numbering is arbitrary, results are
equivalent for d2. It is also possible to determine the max-
461
imum value of C that allows coexistence when d 1 ( d 2.
However, the results are similar to those based on the possible range of d1 and will not be presented here.
Appendix A presents some of the details of the analysis.
The results are shown in figure 1, which graphs the range
of d1 allowing coexistence as a function of d. Here, the
relationship consists of three linear segments; the two leftmost segments are increasing, and the rightmost segment
is decreasing. The range of d1 that allows coexistence approaches 0 as d approaches 0 or 1. This result implies that
coexistence is most likely at intermediate mortality rates.
The range of d1 allowing coexistence increases relatively rapidly from d p 0 to d p (C ⫺ 4C 2 ⫹ 4C 3)/[(1 ⫺ C)(1 ⫺
2C ⫹ 2C 3)]; it then increases more slowly until d p 1 ⫺
2C, at which point it starts to decrease (these results follow
from eqq. [A1]–[A3]). Thus, the increasing phase of this
relationship applies to the broadest range of resource requirements, d, when C is very small (i.e., when there is little
overlap in resource use).
If the range of d allowing coexistence is integrated over
all possible values of C, the same qualitative result emerges.
The parameter C only determines the magnitudes of the
slopes of the line segments in figure 1 but not their signs.
Averaged over all values of C, the range of d1 that allows
coexistence increases at low d and decreases at high d. The
reason for the difficulty of coexistence when d is small is
that one of the two resources is usually excluded (or depressed to very low densities) due to apparent competition
when the two consumers have sufficiently unequal requirements (Abrams 1998).
Figure 1: A measure of the ease of coexistence of two otherwise equivalent
consumer species with mirror image consumption-rate parameters (C
and 1 ⫺ C) competing according to equations (5). The dotted line is for
the case C p 0.15, and the solid line describes a system where C p
0.35. The lines show the range of d1 permitting coexistence as a function
of d.
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The American Naturalist
Equations (5) assume a symmetric state where the two
competitors differ in the resource they capture most effectively and have mirror image consumption rates. If both
species capture resource i at a greater rate than resource
j, then a low enough value of d implies that resource j is
the only resource present at each of the two possible singleconsumer communities. In this case, d 1 max p d 1 min, and
coexistence is impossible. However, d 1 max ⫺ d 1 min increases
as soon as d is large enough that both resources are present
at one of the single-consumer equilibria. As soon as d is
large enough that both resources are present at both singleconsumer equilibria, d 1 max ⫺ d 1 min decreases, approaching
0 when d approaches 1. An example of this pattern is
shown in figure 2. The conclusion that coexistence occurs
over the broadest range of parameters at intermediate
mortality rates remains true regardless of the relative consumption rates of the two consumer species.
Three Resources. The two-resource model assumes that neither consumer species has a resource that it uses exclusively. Models of competition with exclusive resources have
been shown to have some special features in other contexts
(Schoener 1976; Abrams 1998). Adding a third resource
to the two-resource MacArthur model allows each species
to have an exclusive and a shared resource (e.g., consumer
1 uses resources 1 and 2, while consumer 2 uses resources
2 and 3). Here, I assume that the shared resource is captured at a rate 1 ⫺ C and the exclusive resources at a rate
C; complete overlap in resource capture rates occurs when
C p 0. When both di are relatively large, these low exploitation efficiencies mean that all resources are present
in the system and the equilibrium densities are described
by a Lotka-Volterra model with a p C 2/[C 2 ⫹ (1 ⫺ C)2]
(Abrams 1998). For this range of parameters, greater mortality reduces the range of resource requirements allowing
coexistence. However, when one of the di is small enough
that at least one of the resources is excluded, this LotkaVolterra approximation is no longer valid. The excluded
resource may be either the shared or exclusive resource,
and these have different consequences for the ease of coexistence. When C 1 0.5 (higher capture rate of the exclusive than of the shared resource), the exclusive resource
of the resident consumer (assume this is consumer 2 with
d 2 p d ) will go extinct due to apparent competition at
sufficiently low values of d. Whether the consumer 1 can
invade depends on whether it can increase when its own
exclusive resource is at carrying capacity and the shared
resource has a density of d/(1 ⫺ C). When C ! 0.5, the
shared resource is exploited more heavily, and it becomes
extinct at low d when only consumer 2 is present. In this
case, there is effectively no competition, and whether the
invading consumer species 1 can increase depends entirely
on whether it can increase when its own exclusive resource
Figure 2: The range of d1 that permits coexistence of two species characterized by consumption rate coefficients on resource 1 of C1 p 0.2 and
C2 p 0.4. Each consumer has a consumption rate of 1 ⫺ Ci on resource
2. Otherwise, the dynamics are given by equations (5). The figure shows
this range as a function of d p d2 , the resource requirement of the second
consumer. Coexistence is impossible for d ! 0.1333.
is at its carrying capacity. These phenomena are reflected
in the range of d1 values allowing coexistence. These two
cases (C 1 or ! 0.5) are illustrated in figure 3A and 3B.
Although the shapes of these curves differ from those in
figures 1 and 2, both panels of figure 3 support the generalization that increasing mortality initially either does
not affect or increases the range of parameters that allow
coexistence when consumer efficiency is high but decreases
the range when efficiency is low. In both two- and threeresource models, the differences between the behavior of
the MacArthur system and the Lotka-Volterra model when
d is low arise because of the exclusion of one or more
resources.
Other Consumer-Resource Models
There are at least two aspects of the MacArthur system
that may influence the above analysis and for which relatively little empirical evidence exists: the assumption that
resource growth is logistic and the assumption that the
consumers have linear functional responses. The first includes the assumption that resources are self-reproducing
and noninteracting as well as having linear density dependence. The second implies zero handling time for resources, which is contradicted by the vast majority of empirical studies of functional responses (Hassell 1978;
Abrams et al. 1990; Gross et al. 1993; Messier 1994). This
section will examine the consequences of relaxing each of
these two assumptions. The other resource growth models
Mortality and Coexistence
463
ical investigations of the two- and three-resource versions
of the basic two-consumer model with v-logistic resource
growth did not reveal any cases where the form of the
relationship shown in figure 1 was changed greatly. Segments of the relationship were nonlinear, and the sizes of
the domains of the three segments changed with nonlinear
density dependence; however, the intermediate maximum
in conditions allowing coexistence did not. Because reduced
density dependence in resource growth increases the probability of resource exclusion, large values of v are characterized by resource exclusion over a broader range of d. This
implies that the range of d1 allowing coexistence increases
with d over a large fraction of the possible values of d when
v is large. In the limit, as v approaches infinity, resource
population dynamics approaches exponential growth with
a ceiling. In this case, one or the other of the resources is
excluded by apparent competition when there is only a
single consumer present and C ( 0.5. The range of d1 values allowing coexistence of two mirror image consumers is
equal to d(1 ⫺ 2C)/(C ⫺ C 2). In other words, the range of
d allowing coexistence increases in proportion to d over the
entire range of values that allows existence.
Figure 3: The range of parameters allowing coexistence in the threeresource version of MacArthur’s (1970, 1972) model. A, Range of d1
allowing coexistence as a function of d for two cases where the exclusive
resource is caught at a lower rate than the shared resource. B, Same
quantity for two cases where the exclusive resource is caught at a higher
rate than the shared resource.
considered are biotic resources with nonlinear density dependence, competing biotic resources, and abiotic resources. The alternative functional response that will be
considered is a Type II (decelerating) functional response.
The models retain MacArthur’s (1970, 1972) assumption
that resources are nutritionally substitutable. I do not examine models in which consumed resources interact nonlinearly to determine consumer fitness (e.g., Abrams and
Shen 1989; Grover 1997) or models in which the consumers exhibit adaptive foraging (Abrams 1982, 1999).
Nonlinear Density Dependence. The v-logistic model (Gilpin
and Ayala 1973) is dR/dt p rR[1–(R/K )v ], where v measures
the extent to which density dependence is concentrated near
the carrying capacity. When v ( 1, density dependence is
nonlinear, and when v k 1, density has little effect on per
capita growth rate until R approaches K. Extensive numer-
Competing Biotic Resources. The resources in the models
considered thus far have had independent dynamics in the
absence of the consumers. In many situations with biotic
resources, the resources compete with each other. Here, I
modify the two-resource MacArthur system considered
above by assuming that the two resources compete according to a Lotka-Volterra scheme with symmetric interactions,
characterized by a competition coefficient a ≤ 1. The effect
of competitive equivalence (a p 1) is clear; this converts
the two resources effectively into a single resource, which
makes coexistence impossible for any difference in utilization rates. However, for a ! 1, the competition can produce a mutualism between the consumers when they have
substantial differences in resource utilization abilities (Vandermeer 1980). Coexistence conditions can be determined
as in the previous models; the conditions are algebraically
rather complicated and will not be presented here. The
presence of competition between resources widens the range
of d values allowing coexistence (compared with a model
without resource competition) when C is relatively small
but has the opposite effect when C is close to one-half.
However, competition does not alter the result that sufficiently low values of d make coexistence difficult. As in the
basic MacArthur model, the range of d1 allowing coexistence
increases with increasing d when d is low and decreases with
increasing d when d is high.
Saturating Functional Response. The final modification of
the biotic resource model considered here consists of the
most common alternative form for the functional re-
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The American Naturalist
sponses, the Holling disk equation. This complicates the
analysis because low values of d frequently produce limit
cycles when only one consumer is present, and chaotic
dynamics frequently occur with two consumers (Vandermeer 1993). The basic model we have been considering
thus far becomes
dN1
CR1 ⫹ (1 ⫺ C)R 2
p N1
⫺ d1 ,
dt
1 ⫹ hCR1 ⫹ h(1 ⫺ C)R 2
(6a)
dN2
(1 ⫺ C)R1 ⫹ CR 2
p N2
⫺ d2 ,
dt
1 ⫹ h(1 ⫺ C)R1 ⫹ hCR 2
(6b)
[
[
]
]
dR1
CR1N1
p R1(1 ⫺ R1v ) ⫺
dt
1 ⫹ hCR1 ⫹ h(1 ⫺ C)R 2
(1 ⫺ C)R1N2
⫺
,
1 ⫹ h(1 ⫺ C)R1 ⫹ hCR 2
(6c)
dR 2
CR 2 N2
p R 2(1 ⫺ R v2 ) ⫺
dt
1 ⫹ h(1 ⫺ C)R1 ⫹ hCR 2
⫺
(1 ⫺ C)R 2 N1
.
1 ⫹ hCR1 ⫹ h(1 ⫺ C)R 2
resource is present and it undergoes cycles. Higher values
of d eventually result in successive shifts to stable dynamics, presence of the second resource with cycles, and presence of both resources with a stable equilibrium point.
Figure 4 presents a representative relationship between d
and the maximum range of d1 permitting coexistence for
the case of C p 0.3, h p 5, and v p 1. The curve is approximately parabolic, but there is a significant irregularity
near the d value where there is a transition from a stable
system with only one resource (at lower d ) to an unstable
system with both resources present (at higher d ). A range
of h values between 0 and 10 and a range of C between
0 and .5 have been examined for equations (6), and the
result that a maximum range of d1 allows coexistence at
intermediate mortalities appears to be general.
Abiotic Resource Growth. The most commonly used model
of abiotic resource growth (MacArthur 1972) is the chemostat model:
dR
p I ⫺ ER ⫺ CRN,
dt
(6d)
The only new parameter in these equations is the handling
time, h, required to consume a resource item; h is assumed
to be identical for both resources. The assumption of equal
handling times is consistent with the implicit assumption
of equal values (caloric or nutritional) of the two types.
These assumptions avoid the potential complications of
the need to incorporate optimal foraging into the model.
When the equilibrium with a single consumer is unstable,
the invasion conditions must be determined numerically.
In addition, results relating mutual invasion to coexistence
are not available. However, numerous simulations have
not revealed any cases where mutual invasion fails to produce coexistence. In any event, the results presented for
this model are more limited in scope than those for the
previous models. However, the basic patterns remain
similar.
If the values of d are initially low, higher mortality, d,
in equations (6) first increases and then decreases the range
of d1 that allows coexistence. The relationships between
the initial d and the permissible range of d1 are no longer
piecewise linear, and the shape of the relationships changes
when the system becomes stable or unstable. In many
cases, there are several transitions between stability and
instability as the value of d in a single-consumer system
is increased from values close to 0 to the maximum value
that will allow coexistence in the absence of competition.
Very low values of d imply that only the less consumed
(7)
where I is the rate at which resources enter the system.
Resources leave at a rate that is proportional to a (positive)
rate constant, E, multiplied by resource density. In the above
equation, the consumer species (with density N ) has a linear
functional response, but Type II responses do not make
significant differences in the following results. This form of
resource growth has been used extensively to represent
plants competing for nutrients (e.g., Tilman 1982; Grover
Figure 4: Ease of coexistence as a function of mortality for a model with
logistic resources and Type II consumer functional responses (eqq. [6]).
The curve describes the range of d1 that will allow coexistence for an
example in which C p 0.3 and h p 5.
Mortality and Coexistence
1997). It has also been used to model resources that are
living organisms when those organisms have a refuge in
space, time, or life-history stage (e.g., Abrams 1977).
I analyzed the conditions for consumer coexistence for
models comparable to those discussed above, except that
equation (7) described resource growth. The two resource
dynamics parameters, I and E, do not have a qualitative
effect on the relationship between mortality, d, and conditions for coexistence. The basic results for two-resource
models are shown in figure 5. Either high or low mortality
reduces the range of differences in d1 that permit coexistence, as shown in figure 5. Appendix B gives the formulas for maximum and minimum values of d1; these
results show that the greatest range of d1 values allowing
coexistence occurs when d p I/(2E), that is, when d is
exactly halfway between its minimum value of 0 and its
maximum of I/E. The maximum range of d1 must approach 0 when d approaches either 0 or I/E. It is likely
that most species have resource requirements significantly
below the maximum that would allow them to exist when
resources are at their maximum densities; in other words,
d K I/E. If so, the results argue that mortality is more
likely to increase than to decrease species diversity in communities competing for a set of shared abiotic resources.
In the case of three resources, the abiotic resource model
again supports conclusions similar to those derived from
the biotic model. Increased mortality increases the range
of d1 values allowing mutual invasion when d is low and
decreases this range when d is large. Even when d becomes
very small, the allowable range of d1 does not approach 0
because sufficiently low d will always allow coexistence
based on use of the exclusive resource alone. If the tworesource abiotic model is modified so consumer species
have Type II functional responses, the formulas for maximum and minimum d1 values are complicated (P. A.
Abrams, unpublished data), but the qualitative results
change very little. The range of d1 allowing mutual invasion
again approaches 0 as d approaches 0 or as d approaches
the maximum value allowing existence in the absence of
the competitor. The potential range of d1 increases with d
over slightly greater than one-half of the range of possible
values of d. Thus, mortality often makes coexistence more
probable. (It should be noted that, in the cases of threeresources or Type II functional responses, there do not
exist proofs that mutual invasion implies coexistence, although no counterexamples were found in this study.)
More Consumer Species. Models with three or more consumer species require three or more resources to allow coexistence at stable densities, and a larger number of invasion
conditions must be examined to determine which species
are able to coexist. A variety of three-consumer–threeresource models have been analyzed (P. A. Abrams, un-
465
Figure 5: Ease of coexistence as a function of mortality for a model with
abiotic resources (eq. [7]) and linear consumer functional responses. The
curves are the range of d1 that will allow coexistence for two values of
similarity in resource use. The dashed line is low similarity (C p 0.15),
and the solid line is high similarity (C p 0.35).
published data), and for all cases examined, a maximum
range of efficiencies allows coexistence at intermediate
mortalities.
Related Theoretical Conclusions
Can the Preceding Results Be Extended to the Effects
of Predators on Competing Prey?
If predators impose density-independent mortality on prey,
the results reviewed above are unchanged. However, precise
density independence is unlikely when mortality is due to
predation; it requires that the predators have linear functional responses to all prey species and that equilibrium
predator density be independent of their prey consumption.
It is unlikely that these assumptions will often be satisfied
exactly. However, approximate density independence could
arise if the predator has a saturating functional response
and experiences direct density dependence. In this case, the
increase in predator population size caused by the presence
of an additional prey species is limited by the direct density
dependence and may be approximately offset by the increase
in the level of predator satiation. When a generalist predator’s population density is determined primarily by its prey
consumption, the predator effectively represents another
shared limiting factor of the prey species. This may increase
or decrease the mean overlap of two prey species over all
limiting factors (Abrams 1983; J. Chase, P. A. Abrams, J. P.
Grover, et al., unpublished manuscript). Special types of
predation (e.g., switching) produce a frequency-dependent
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advantage for rare prey, which greatly expands conditions
for coexistence (Roughgarden and Feldman 1975; Hambäck
1998).
What Does the Theory Predict Regarding ProductivityDiversity Relationships?
The parameter d in the biotic resource models analyzed
here is inversely related to the carrying capacities of the
two resources. Thus, the results presented here also predict
maximal numbers of coexisting species at intermediate
productivities. In particular, if the environment is sufficiently productive, then decreasing productivity should increase species diversity. There are many examples illustrating this pattern (e.g., Rosenzweig and Abramsky 1993;
Rosenzweig 1995) but also a large number of cases that
display monotonic relationships (Abrams 1995; Waide et
al. 1999). Recently proposed explanations for unimodal
relationships (Jansen and Mulder 1999; Kassen et al. 2000)
have not been based on traditional resource partitioning.
There are now a large number of possible explanations for
unimodal relationships, and the challenge is to determine
which are responsible for particular observed relationships.
Discussion
Mortality applied to consumer species can increase their
probability of coexistence by increasing the diversity of
their resources. In the case of biotic resources, there is a
qualitative reduction in diversity due to resource exclusion
when consumers are efficient. In the case of abiotic resources, the reduction is quantitative, with some resources
being depressed much more than others. For mortality to
favor coexistence, it is necessary that consumers be efficient enough to significantly depress resource diversity in
the absence of the added mortality. Efficiency implies that
both intrinsic mortality rates and metabolic requirements
are low enough that the resources are reduced significantly
below the densities they would reach in the absence of
consumption. (The necessary magnitude of the reduction
depends on the details of the model.) When consumer
mortality is initially high, still greater mortality generally
has minor effects on resource diversity and will always
make it more difficult for species to exist, with or without
competitors. The precise shape of the relationship between
mortality and ease of coexistence depends on the presence/
absence of exclusively used resources, the form of the resource growth functions, and the consumer functional responses. When consumers initially have low resource
requirements, there is always a maximum range of efficiencies allowing coexistence at intermediate levels of additional mortality. The “intermediate disturbance hypothesis” (Connell 1975) is a recurring idea in the literature
about competition, and the current results seem to extend
the realm of validity of that hypothesis, at least if “disturbance” is either correlated with, or equivalent to, the
mean level of density-independent mortality. To the extent
that the diversity of species on the top trophic level of a
community is determined by interspecific competition for
renewable resources, one would expect the greatest diversity on that level to occur in environments with intermediate mortalities.
To make predictions about the effects of environmental
variables on species diversity, these variables must be translated into effects on both mortalities and metabolic rates
in order to estimate their effect on d. If species that are
naturally more efficient also suffer greater increases in
mortality because of the negative environmental factor,
then it is possible that environmental harshness increases
the similarity of the di values of different species as well
as reducing the range of values that permit coexistence.
This could reverse predictions about effects on species
diversity. However, these considerations do not apply when
harshness represents uniform mortality rates imposed by
the experimenter.
Is there empirical evidence for the phenomena discussed
here? Yodzis (1986, 1989) compiled what is apparently the
most recent analysis of studies of the effects of mortality
on species diversity of competitors suffering that mortality.
Unfortunately, most of the studies have been done on
organisms that compete for space or for a resource that
is spatially localized and monopolizable (Yodzis 1986). In
this case, mortality often increases the effective number
of resources when space occupied by one species can be
taken over by other species. Two studies reviewed by Yodzis
(Addicott 1974; Risch and Carroll 1982) had nonselective
mortality and exploitative competition for renewable resources. Both of these studies found that species diversity
always decreased with increased mortality. This is not necessarily inconsistent with the results presented here, provided the lowest mortality levels in those studies were
already greater than those producing the maximum coexistence bandwidth. Unfortunately, this cannot be determined from the information reported in those articles.
Empirical studies that examine a range of different imposed mortality rates would be the best way to examine
the hypotheses generated by the models presented here.
The current results still do not provide a justification
for the widespread view that any factors that increase resource densities make it easier for the competing consumers of those resources to coexist. For example, in a recent
review of experimental studies of the interaction of predation and competition, Gurevitch et al. (2000, p. 445)
interpret their findings as showing that “by weakening
competition, predation may prevent competitive exclusion, as is suggested by the theory of predator-mediated
Mortality and Coexistence
coexistence.” Because it represents higher mortality, predation will often weaken competition in the sense of the
experiments reviewed by Gurevitch et al. (2000); that is,
in the presence of predators, competitor removal will have
a smaller absolute effect on the resources available to other
consumer species. As a consequence, effects of competitors
on quantities such as size or growth rate are likely to be
smaller in the presence of generalist predators than in their
absence, which is what was found in most of the studies
reviewed by Gurevitch et al. (2000). However, this does
not translate into a greater probability of coexistence because the absolute amounts of resources are not important
determinants of coexistence. Higher resource densities are
required to offset the increased mortality in the presence
of the predator, so they do not automatically make it more
likely that a species will persist (Chesson and Huntly 1997;
J. Chase, P. A. Abrams, J. P. Glover, et al., unpublished
manuscript). What is important for coexistence is the effect of the predator on the diversity of resources in the
system when one or more competing species are rare. The
results presented here suggest that this effect is generally
positive provided that the consumer has sufficiently low
total resource requirements for zero population growth.
The analysis here has considered coexistence that is
only generated by resource partitioning within a spatially
and temporally homogeneous habitat. It is known that
the presence of endogenously generated cycles in consumer-resource systems can allow the coexistence of two
species on a single biotic resource (Koch 1974; Armstrong
and McGehee 1976, 1980). Cycles can also allow coexistence of a number of species that use abiotic resources
(Armstrong and McGehee 1980; Huisman and Weissing
2001). When coexistence is dependent on cycling, the
effect of mortality on stability must be considered in
analyzing its effects on coexistence. Stabilization or destabilization could alter the effects based on resource diversity that are discussed here. Similarly, predicting the
consequences of mortality in metapopulation models or
spatially explicit models would require analysis of appropriate models. However, because the mechanisms reviewed here could operate within each subpopulation of
a metapopulation, similar mortality-coexistence relationships should be possible.
The results presented here represent an additional argument against using the Lotka-Volterra model to draw
inferences about resource competition. The assumption of
constant competition coefficients is unlikely to be satisfied
in most systems involving exploitative competition for renewable resources. For the question examined here, the
Lotka-Volterra model fails to reflect the reduction in resource diversity produced by efficient consumers. It is this
reduction in resource diversity that hampers coexistence
at low levels of consumer mortality.
467
The conclusions reached here differ from those reached
in several previous theoretical studies. Slobodkin (1961),
Abrams (1977), and Holt (1985) analyzed the two-species
Lotka-Volterra model. Slobodkin (1961) and Holt (1985)
both showed that mortality that was proportional to the
intrinsic rate of increase, r, in the standard parameterization of the Lotka-Volterra model did not affect conditions for coexistence. However, there is no reason to expect
this proportionality to hold, and it certainly would not
hold with experimentally imposed uniform mortality.
Abrams (1977) found some cases with unequal competition coefficients, where maximum overlap could increase
as mortality increased, but this occurred over only a limited range of (low) mortality rates. These previous studies
have all used a parameterization of the Lotka-Volterra
model that, unlike equations (1), did not allow the densityindependent component of fitness to be varied independently of the density-dependent component. Wootton
(1998) analyzed a consumer-resource model with two
competing resources; however, his graphical treatment did
not allow quantification of parameter ranges allowing coexistence. Chesson and Huntly (1997) present a very general analysis, which demonstrates that mortality may have
a variety of effects on coexistence in different models. Although they stress that harshness per se need not favor
coexistence, they point out that it can do so when there
are nonlinear competitive effects. Most consumer-resource
models of the sort considered here do produce nonlinear
effects.
The mechanism described here—lower consumer densities generating resource diversity—is likely to be an important scenario by which density-independent mortality
could promote the coexistence of consumer species competing for nutritionally substitutable renewable resources.
However, there are other important mechanisms whereby
mortality can promote coexistence. One example is when
the mortality actually expands the number of resources
(often true under competition for space); another is when
the mortality is caused by predators acting in a density
dependent manner (J. Chase, P. A. Abrams, J. P. Glover,
et al., unpublished manuscript; see above). Nevertheless,
the mechanism described here, mortality increasing resource diversity, seems sufficiently general that it should
be considered as a possible hypothesis in most cases where
mortality increases the number of coexisting competitors.
Acknowledgments
This work was conducted as part of the Competition Theory Working Group, supported by the National Center for
Ecological Analysis and Synthesis, a center funded by the
National Science Foundation (grant DEB-94-21535); the
468
The American Naturalist
University of California, Santa Barbara; and the state of
California. I am also grateful to the University of Toronto
and the Natural Sciences and Engineering Research Coun-
cil of Canada for financial support and to T. Day, D.
DeAngelis, W. Wilson, and two anonymous reviewers for
their comments on earlier drafts of the manuscript.
APPENDIX A
Invasion and Coexistence in the Two-Consumer MacArthur Models (Eqq. [5])
Coexistence is determined by the ability of each consumer species to invade when the other is at equilibrium with
the resources. These invasion conditions are identical to the conditions for a stable interior equilibrium, and results
in Case and Casten (1979) show that such an equilibrium will be globally stable. Each invasion condition depends on
whether one or two resources are present at the single-consumer equilibrium. The formulas for resource densities in
the two-resource model when both are present with consumer 1 (which consumes resource 1 at a rate C) are given
by
R1 p
1 ⫺ 3C ⫹ Cd ⫹ 2C 2
,
1 ⫺ 2C ⫹ 2C 2
(A1a)
R2 p
2C 2 ⫺ C ⫹ d(1 ⫺ C)
.
1 ⫺ 2C ⫹ 2C 2
(A1b)
We can arbitrarily assume that C ! 0.5 , which implies that R2 is the only resource that can be driven to extinction by
consumer 1; extinction occurs when d ! (C ⫺ 2C 2)/(1 ⫺ C). If extinction of resource 2 occurs, R1 p d/C. The equilibrium resource densities with consumer species 2 present are given by the above formulas with the subscripts reversed.
The requirement that each species be able to increase when it is rare and the other is at equilibrium leads to the
following expressions for the maximum value of d1 that will allow coexistence when initially d 1 p d 2 p d:
2C(1 ⫺ C)d ⫹ 1 ⫺ 4C ⫹ 4C 2 (1 ⫺ C)d
,
.
1 ⫺ 2C ⫹ 2C 2
C
[
d 1 max p Min
]
(A2)
Similarly, the minimum value of d1 that allows invasion by species 2 when both species are initially characterized by
the same d is
4C ⫺ 1 ⫺ 4C 2 ⫹ d(1 ⫺ 2C ⫹ 2C 2) Cd
,
.
2C(1 ⫺ C)
1⫺C
[
d 1 min p Max
]
(A3)
When d is sufficiently low, the second argument of both of the above functions applies, and d max ⫺ d min p d(1 ⫺
2C)/(C ⫺ C 2). In other words, the range of d1 permitting coexistence of both species increases linearly with d when d
is sufficiently low and approaches 0 as d approaches 0.
In the model with three resources, the expressions for the equilibrium resource densities are identical to those given
above, but there is no requirement that C ! 0.5. The shared resource is excluded by one consumer if C ! 0.5 and
d ! (C ⫺ 2C 2)/(1 ⫺ C). When C 1 0.5, the exclusive resource goes extinct in a single-consumer system when d !
(3C ⫺ 1 ⫺ 2C 2)/C. If d is sufficiently low, both d 1 max and d 1 min will be determined by systems in which only one
resource is present, which simplifies the invasion conditions. If d is extremely low, it is possible for competitive exclusion
of a consumer to be impossible because it can maintain a viable population on its exclusive resource alone. This leads
to a more complicated set of contingent invasion conditions, which are most easily understood by the graphs in figure
3.
Mortality and Coexistence
469
APPENDIX B
Conditions for Coexistence with Two Abiotic Resources
These conditions are again derived by determining resource densities when one consumer is present at equilibrium
and determining whether the other consumer can invade. These conditions can be shown to be identical to those for
the existence of a positive interior equilibrium. The results of Ballyk and Wolkowicz (1993) show that such an
equilibrium point is globally asymptotically stable given linear consumer functional responses. The conditions for
coexistence produce the following expressions for the maximum and minimum values of d1 when consumer 2 is
characterized by a resource requirement of d and when the two consumers have mirror image consumption rates of
C and 1 ⫺ C on the two resources:
d 1 max p
d 1 min p
d[⫺(1 ⫺ 2C)2dE ⫹ (1 ⫺ 2C ⫹ 2C 2){2CI ⫺ 2C 2I ⫹ 冑[dE(1 ⫺ 2C)]2 ⫹ 4[IC(1 ⫺ C)]2}
2C(1 ⫺ C){2CI ⫺ 2C 2I ⫹ 冑[dE(1 ⫺ 2C)]2 ⫹ 4[IC(1 ⫺ C)]2}
(dE ⫺ I)(1 ⫺ 2C ⫹ 2C 2) ⫹ 冑4dEC 2(1 ⫺ C)2(2I ⫺ dE) ⫹ (dE ⫺ I)2(1 ⫺ 2C ⫹ 2C 2)
2CE(1 ⫺ C)
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