Download Effects of altered resource consumption rates by one consumer

Ecology Letters, (2003) 6: 550–555
REPORT
Effects of altered resource consumption rates by one
consumer species on a competitor
Peter A. Abrams*
Department of Zoology,
University of Toronto, Toronto,
Canada
*Correspondence: E-mail:
[email protected]
Abstract
Non-mechanistic models of competition suggest that harming one of two competing
species will increase the population density of the other. These models also suggest that
any change in a fitness component of one competitor will make the densities of the two
competitors change in opposite directions. However, models of competition that
incorporate resource dynamics show that neither conclusion holds generally. Reducing
the consumption abilities of one competitor may decrease the population size of the
other by decreasing resource overexploitation by the first and thereby increasing its
density. It is also possible for decreased consumption abilities of one species to increase
the population densities of both species, when the increased density of the focal species
is offset by its decreased ability to consume the main resources of its competitor. Finally,
decreases in consumption may have the effects predicted by phenomenological models; a
decrease in the focal species and an increase in its competitor. Unstable systems may
exhibit more complicated patterns of changes in densities with changes in consumption
rates. These counterintuitive effects depend on the presence of overexploitation of biotic
resources, about which little is known. More generally, there have been few theoretical or
empirical studies examining the indirect effects of changes in consumption rates of a
focal species in a food web; these are termed Ôtrait-initiated indirect effectsÕ. A better
understanding of the potential consequences of altered consumption rates will be
important for understanding biotic shifts in communities undergoing environmental
change, and in using simple community modules to understand larger food webs.
Keywords
Attack rate, competition, consumer–resource system, indirect effects, overexploitation.
Ecology Letters (2003) 6: 550–555
INTRODUCTION
Much current understanding of competition is based on
models, such as the Lotka–Volterra, in which the resources
under competition are not represented explicitly. Such
models still constitute most of the coverage of competition
in modern textbooks (e.g. Case 1999). It has long been
known that models in which resource dynamics are
represented explicitly can have different outcomes and
dynamic behaviours (Abrams 1975, 1998, 2001; Schoener
1976; Hsu & Hubbell 1979; Vandermeer 1993). Nevertheless, the differences between these two types of models are
still not completely understood. This article will analyse the
responses of some simple consumer–resource models to
2003 Blackwell Publishing Ltd/CNRS
perturbations in the resource consumption rates of one
consumer species. Consumption rates are generally affected
by environmental variables (e.g. temperature), and are very
often reduced in response to predator addition (Abrams et al.
1996; Lima 1998; Peacor & Werner 2001). The response of
competing species to predation is an important and poorly
understood topic (Chase et al. 2002), and the changes in their
prey’s resource consumption rates are often likely to be an
important component of responses to predators. Although
general approaches to determining the population consequences of altered consumption rates (or other parameters)
were developed by Puccia & Levins (1985), these questions have seldom been examined for specific ecological
systems.
Consumption and competition 551
A generalized model of competition between homogeneous populations in which the model lacks an explicit
representation of resources has the following form:
dN1
¼ N1 f1 ðN1 ; N2 Þ;
dt
dN2
¼ N2 f2 ðN1 ; N2 Þ;
dt
ð1Þ
where the densities of the two competitors are N1 and N2
and their per capita population growth rates, fi, are
decreasing functions of the density of each species. The
growth rate functions, f, may depend on various parameters;
in the Lotka–Volterra model, these are intrinsic rates of
increase, carrying capacities, and competition coefficients.
However, because models having the form of eqn 1 generally lack parameters that explicitly reflect consumption
rates, the impact of these changes has received little attention. In the Lotka–Volterra model, for example, if species 1
suffers reduced resource consumption abilities, this
decreases both its per capita effect on its own growth rate,
measured by a11, and its per capita effect on species 2, a12.
However, determining the magnitudes of these decreases
requires a model of the resource consumption process.
Given the mutually negative (i.e. (), ))) effects produced by
changes in population size in eqn 1, it might be thought that
reductions in resource consumption by one consumer
would always increase its competitor’s population and
reduce its own population. However, that is not the case.
Conversion efficiencies of resource j into individuals of
consumer i are given by bij and the per capita death rate of
consumer i is di. The parameter di may also be interpreted as
a scaled resource intake requirement for zero population
growth of species i. Coexistence at fixed densities requires
that there be some difference between the two consumer
species in the ratio of their two attack rates (Ci1/Ci2). For a
more detailed discussion of global stability and coexistence
in MacArthur-type models see Haygood (2002). A closely
related model was studied by Vandermeer (1993), who
showed that complex dynamics can occur when there is
relatively low overlap in the attack rate parameters between
the two consumer species. Here I examine the impact of a
change in one or both attack rates, Cij of species i on its
population size and that of its competitor. Even very simple
versions of eqn 1 with zero handling time lead to rather
complicated formulae for equilibrium densities. Therefore,
the following analysis will concentrate on cases with
symmetries in the initial parameter values that simplify
the analysis. These are: (r1 ¼ r2 ¼ r; K1 ¼ K2 ¼ K; d1 ¼
d2 ¼ d; b11 ¼ b21 ¼ b12 ¼ b22 ¼ 1). Numerical explorations of other parameters strongly suggest that the
qualitative results do not depend on these symmetries.
Analytical results for models with zero handling times are
supplemented with numerical results for systems with
significant handling times.
The equilibrium densities of the simplified four-species
system with all hij ¼ 0 are given below, assuming that all
these densities are positive:
A MODEL OF TWO PREDATORS (CONSUMERS)
WITH TWO SHARED PREY (RESOURCES)
2
2
2
2
C12 C21
C11 C22 C21 C12 C21 C22 þ C11 C22
þ d ðC11 C21 C21
þ C12 C22 C22
Þ
N^1 ¼
;
ðC11 C22 C12 C21 Þ2
The model considered here is an extension of MacArthur’s
(1970, 1972) consumer–resource model to include the
possibility of type 2 (Holling 1959) functional responses
(Abrams 1980):
2
2
2
2
C21 C12
C11 C22 C12 C12 C21 C11 þ C22 C11
þ d ðC22 C12 C12
þ C21 C11 C11
Þ
N^2 ¼
;
ðC11 C22 C12 C21 Þ2
X
dRj
Rj
Cij Rj Ni
P
¼ r j Rj 1 ;
dt
Kj
1þ
Cij hij Rj
i¼1;2
j¼1;2
0
1
X
bij Cij Rj
dNi
B
C
P
¼ Ni @
di A:
dt
1
þ
C
h
R
ij ij j
j¼1;2
ð2a,bÞ
j¼1;2
Most of the analysis considers the two-consumer–
two-resource version of eqn 2. The population densities
of the two resources are R1 and R2, and two consumer
populations are N1 and N2. The resources have logistic
growth with parameters ri and Ki. The consumersÕ functional
responses are defined by the parameters Cij denoting the
attack rate of consumer i on resource j, and hij denoting the
handling time for an item of resource j by consumer i.
(3a)
(3b)
R^1 ¼
C22 C12
d;
C11 C22 C12 C21
ð3cÞ
R^2 ¼
C11 C21
d:
C11 C22 C12 C21
ð3dÞ
I have assumed that Cii > Cij implying that
C11C22 > C12C21. A positive density of resource i also
requires that Cjj > Cij. A positive density of consumer
species i requires that its attack rates (Cii and Cij) be high
enough relative to those of species j, to make the numerator
of the expressions 3a and 3b positive.
Most of the possible effects of attack rates on densities
may be determined by examining a perfectly symmetric
system in which C11 ¼ C22 ¼ q, and C12 ¼ C21 ¼ 1 ) q,
with q > 0.5. The approach below simply calculates the
derivatives of equilibrium densities (eqn 3a, b) with respect
to attack rates when these symmetries are initially present.
The results contradict the simple logic that favourable
2003 Blackwell Publishing Ltd/CNRS
552 P. A. Abrams
changes to species i increase its density and decrease that of
its competitor. Increasing the attack rate of consumer i on
its preferred resource (Cii) can have three qualitatively
different effects on the two consumer densities, depending
on the consumersÕ resource requirement, d. This is shown
by the following derivatives:
oN^i ð1 qÞð2q 1 d ð4q 1ÞÞ
¼
;
oCjj
ð2q 1Þ2
ð4aÞ
oN^j qð1 2qÞ þ d ð1 3q þ 4q2 Þ
¼
:
oCjj
ð2q 1Þ2
ð4bÞ
At very low resource requirements (d < (2q ) 1)/(4q ) 1)),
the species whose consumption increases suffers a decline in
population density, while its competitor increases. At
intermediate requirements, i.e. ((2q ) 1)/(4q ) 1) < d <
(2q ) 1)/(1 ) 3q + 4q2), both consumers decline in abundance as Cii increases. When requirements are high (i.e.
d > (2q ) 1)/(1 ) 3q + 4q2)), species i increases and species j decreases when Cii increases. (The opposite effects are
observed if Cii decreases.) Only the last of these three
outcomes causes the same directions of population change
as does increasing the population density of consumer
species i.
The underlying reason for these results is that a small d
implies overexploitation, making it possible for increased
consumption to decrease consumer density. Whether the
competitor population increases or decreases then depends
on the relative magnitudes of overlap in resource use and
overexploitation; i.e. on the relative magnitudes of the
indirect effects transmitted by each of the two resources.
It should be noted that a sufficient change in Cii may
cause exclusion of one of the two competitors. This is often
associated with exclusion of one of the two resources, and is
often accompanied by discontinuous changes in density of
both consumers. This occurs in the system illustrated in
Fig. 1A. This system is characterized by a low consumer
requirement, d, and the consumers are initially characterized
by equal degrees of specialization on opposite resources:
C11 ¼ C22 and C12 ¼ C21. Over most of the range of
possible reductions in C22 consumer 2 increases and
consumer 1 decreases. This is because consumer 2 gains
most of the benefit from the decreased overexploitation of
its major resource, and its increased population size results
in a larger impact on its own minor resource, which is
species 1Õs major resource. However, a sufficient reduction
in C22 results in exclusion of species 2; once C22 < 0.2,
species 2 is poorer at catching both resources and its density
drops from slightly less than its maximum density (over the
possible range of C22) to zero. This is immediately followed
by the exclusion of resource 1 as the result of apparent
competition with resource 2 via consumer species 1. The net
2003 Blackwell Publishing Ltd/CNRS
(A)
(B)
Figure 1 The population sizes of the two competing consumers as
the attack rate for consumer 2Õs major resource is reduced (from
right to left on the graph). The solid line is species 1 and the dashed
line is species 2. This is based on the symmetric version of eqn 2.
Panel A assumes zero handling time, while panel B assumes the
same parameters but with h ¼ 3 for all consumer–resource
combinations. Other parameters are d ¼ 0.1 and q ¼ 0.8. The
initial state is when C22 ¼ 0.8, which is identical to C11, and is
shown on the right hand side of the graph. The graph ends at
C22 ¼ 0.2 because at lower values, both N2 and R1 become
extinct; at that point N1 more than triples from approximately
0.08 to 0.25.
result is then a discontinuous jump in the density of
consumer 1 that more than triples its population. Adding a
handling time makes the dynamics either chaotic or cyclic
over a large part of the range of values of C22 where both
species coexist (Vandermeer 1993). Figure 1B shows how
handling times of hij ¼ 3 (which destabilize the system) alter
the results from the analogous stable model in Fig. 1A,
which lacks a handling time. Although there are quantitative
differences, the general pattern of population densities is
surprisingly similar in spite of the chaotic dynamics.
However, the abrupt extinction from near maximal densities
that befalls consumer 2 when C22 drops below 0.2 in Fig. 1A
does not occur when there is a significant handling time, as
in Fig. 1B.
Consumption and competition 553
In the symmetrical model under consideration, increasing
both of a given speciesÕ attack rates proportionally by the
factor z can increase or decrease its own density, as shown
by
oN^j ð2q 1Þ2 þ 2d ð1 3q þ 3q 2 Þ
¼
;
ozj
ð2q 1Þ2
(A)
ð5aÞ
which can be positive or negative, depending on the
magnitude of d. However, such increases in both attack rates
always decreases the density of its competitor, as shown by
oN^j 2dqð1 qÞ
¼
:
ozi
ð2q 1Þ2
ð5bÞ
(B)
Figure 2A illustrates a case where proportional increases
in both of consumer 1Õs attack rates decrease both its
density and the density of its competitor over the range of
parameters where both species exist (species 2 is excluded
when species 1Õs capture rates have increased threefold).
Adding a handling time (Fig. 2B) again greatly complicates
the dynamics. Here, the system is initially stable, and
increases in both of species 1Õs C values by a factor z
increase its population and decrease the competitor’s
population. However, the system becomes unstable at a
proportional increase, z, slightly >1.1, and both species then
decrease with greater consumption abilities of the focal
consumer. Other dynamic transitions occur at still larger
values of z, and there are small ranges of z where increasing
species 1Õs consumption increases the mean density of
species 2. When the increase in species 1Õs attack rates
exceeds a factor of 3, species 2 abruptly becomes extinct,
and the density of species 1 jumps to a much larger value.
Another possibility is a change in the attack rate on the
resource that is initially caught at a lower rate (Cij where
i „ j). The effects of a small change in Cij on the two
consumer densities are given by
oN^j 1 þ 3q 2q2 þ d ð2 5q þ 4q2 Þ
¼
;
oCji
ð2q 1Þ2
ð6aÞ
oN^j qð1 2qÞ þ dqð4q 3Þ
¼
:
oCij
ð2q 1Þ2
ð6bÞ
An increase in a given consumer’s own consumption of its
ÔminorÕ resource always increases its own density, as may be
verified from eqn 6a, noting that q > 0.5. An increase in the
other consumer’s attack rate of its minor resource always
decreases the focal species (this follows from eqn 6b
because d < 1 and 0.5 < q < 1). Again, non-equilibrium
dynamics can modify this pattern for some parameters.
Given the daunting range of dynamics possible in this
system when the equilibrium is unstable (Vandermeer 1993),
no attempt will be made to measure interspecific interactions over a broad range of parameter space where cycles or
Figure 2 The mean population sizes of two competitors as a
function of proportional increases in both of the attack rates of
competitor 1. The solid line is species 1 and the dashed line is
species 2. The model is given by eqn 2. The parameter values
common to all graphs are: ri ¼ Ki ¼ Bi ¼ 1; C22 ¼ 0.375;
C21 ¼ 0.125; di ¼ 0.09. Before any increases C11 ¼ 0.375 and
C12 ¼ 0.125. In panel A hij ¼ 0, so all systems are stable. In panel
B, hij ¼ 5; here the initial system is stable, but there are several
different types of dynamics that occur over the range of increased
attack rate values shown in the graph.
chaos occur. However, most cases examined had relatively
modest differences compared with similar stable systems
(Figs 1 and 2).
DISCUSSION
It is tempting to think that what is good for one of two
competing resource consumers will increase its density and
decrease the density of the other consumer. However, the
models analysed here show that this is not always the case.
When handling times are near zero, proportional increases
in both attack rates of one consumer always decrease the
density of the other consumer. However, even in this case,
the overall response of densities is often counterintuitive,
because the focal species itself often decreases in density
with increases in its consumption rates. Moreover, the
2003 Blackwell Publishing Ltd/CNRS
554 P. A. Abrams
prediction that the competitor always decreases in this
example is sensitive to the assumption that the two
consumers lack any exclusively utilized resources. In a
similar model with three resources, in which each consumer
uses one shared and one exclusive resource, increases in
both consumption rate parameters of consumer i often
produce an increase in consumer species j. While the models
analysed here include only two consumers and two
resources, the same pathways of indirect effects exist in
any multiple consumer–multiple resource model of competition; overexploitation of a major subset of resources allows
decreases in the consumption rates of a given species to
increase its own density, and potentially decrease the density
of its competitor.
If death rates (or more generally, resource requirements
for replacement) are sufficiently high, the responses of
densities to consumption rates in the system analysed here
match the intuition based on the (),)) form of the
competitive interaction. If the resources do not reproduce
themselves (e.g. resources enter the system at a rate
independent of current density and eventually leave the
system if not consumed) overexploitation is impossible,
and again, responses match intuition from phenomenological models. Previous work has shown that overexploitation
of resources can lead to counterintuitive relationships
between the amount of overlap in resource use and the
strength of competition between competing consumer
species (Abrams 1998). Overexploitation is also critical in
determining how mortality will affect the coexistence of
competitors (Abrams 2001). Thus, the key question is, how
often do consumers in natural communities overexploit
their biotic resources? In spite of considerable interest in
this topic in the 1970s (e.g. Slobodkin 1974), we still are
unable to provide many examples where overexploitation
has been clearly demonstrated in natural predator–prey
systems. This is not because it has been shown not to
occur; it is simply that it has not been a popular topic for
experimental ecology. The large magnitude of the decrease
in prey densities caused by predators in many laboratory
microcosms strongly suggests that overexploitation can be
common, as does the large effect of predators on prey
densities that has been observed in field manipulations (Sih
et al. 1985). In a notable recent study, Peacor (2002)
demonstrated that bullfrog tadpoles overexploit algal
populations in mesocosms. This work showed that the
reduction in algal consumption rates caused by the
presence of caged predators increased tadpole growth rates.
The interactions considered here represent an example of
a much larger class of interactions that can be termed Ôtraitinitiated indirect effectsÕ. There is a considerable literature
dealing with Ôtrait-mediated indirect effectsÕ, in which a
change in the trait of an intermediate species that interacts
with both species i and j transmits an effect between them.
2003 Blackwell Publishing Ltd/CNRS
With Ôtrait-initiatedÕ effects, the chain of effects is started by
a change in one or more traits (here, consumption rates),
rather than a population density. The mediating factors
associated with intermediary species may be densities or
traits or both. As a change in an ecologically significant trait
in a focal species almost always changes its own population
density, there are often two different types of pathways
involved in the indirect effect of one species on another. For
example, if C11 declines, there is a pathway to consumer 2
via resource 1, which is affected by both the change in C11
and the subsequent change in density, N1. There is also a
second pathway via resource 2 that is only affected by the
change in N1. The presence of these different pathways is
one of the reasons such trait-initiated effects are more
complicated than those that are initiated by change in a
population density. As predators can cause greater changes
in foraging traits than mortality rates of prey (Peacor &
Werner 2001), theory similar to that developed here is
likely to be required to understand the impact of predators
on the densities of competing consumers. More generally,
theory describing potential trait-initiated effects is likely to
be necessary to extend our understanding of simple sets of
interacting species to larger food webs (Schoener 1993; Holt
1997).
I know of no empirical evidence in support of the
prediction that decreased resource consumption by one
consumer will increase the density of the other, but there are
few experimental manipulations of consumption rates in
competitive systems. Although there are examples of positive
correlations in the densities of competitors, this is also
expected to occur as a simple consequence of common
responses to an environmental variable. However, consumption rates should be easy to manipulate in experimental
microcosms when species have different temperature optima
for activity, and have been manipulated in laboratory
predator–prey systems (Luckinbill 1973). The main reason
why the effects described here might be rare is that
overexploitation of biotic resources might be rare. As noted
above, we have little evidence for judging this question.
ACKNOWLEDGMENTS
This work was supported by a grant from the Natural
Sciences and Engineering Research Council of Canada and
by a J.S. Guggenheim Fellowship to the author.
REFERENCES
Abrams, P.A. (1975). Limiting similarity and the form of the
competition coefficient. Theor. Pop. Biol., 8, 356–375.
Abrams, P.A. (1980). Consumer functional response and competition in consumer–resource systems. Theor. Pop. Biol., 17,
80–102.
Consumption and competition 555
Abrams, P.A. (1998). High competition with low similarity and low
competition with high similarity: The interaction of exploitative
and apparent competition in consumer–resource systems.
Am. Nat., 152, 114–128.
Abrams, P.A. (2001). The effect of density independent mortality
on the coexistence of exploitative competitors for renewing
resources. Am. Nat., 158, 459–470.
Abrams, P.A., Menge, B.A., Mittelbach, G.G., Spiller, D. & Yodzis, P.
(1996). The role of indirect effects in food webs. Food Webs:
Integration of patterns and dynamics. Chapman and Hall, New York.
pp. 371–395.
Case, T.J. (1999). An Illustrated Guide to Theoretical Ecology. Oxford
University Press, Oxford, UK.
Chase, J., Abrams, P.A., Grover, J.P., Diehl, S., Chesson, P., Holt,
R.D. et al. (2002). The effects of predators on competition
between their prey. Ecol. Lett., 5, 302–313.
Haygood, R. (2002). Coexistence in MacArthur-style consumer–
resource models. Theor. Pop. Biol., 61, 215–224.
Holling, C.S. (1959). The components of predation as revealed by a
study of small mammal predation of the European pine sawfly.
Can. Ent., 91, 293–320.
Holt, R.D. (1997). Community modules. In: Multitrophic interactions
in terrestrial systems. (eds Gange, A.C., Brown, V.K.) British
Ecological Society Symposium. Blackwell, Oxford, UK.
pp. 333–350.
Hsu, S.B. & Hubbell, S.P. (1979). Two predators competing for
two prey species: An analysis of MacArthur’s model. Math.
Biosci., 47, 143–171.
Lima, S.L. (1998). Stress and decision making under the risk of
predation: Recent developments from behavioural, reproductive,
and ecological perspectives. Adv. Stud. Behav., 27, 215–290.
Luckinbill, L.S. (1973). Coexistence in laboratory populations of
Paramecium aurelia and its predator, Didinium nasutum. Ecology, 54,
1320–1327.
MacArthur, R.H. (1970). Species packing and competitive equilibria
for many species. Theor. Pop. Biol., 1, 1–11.
MacArthur, R.H. (1972). Geographical Ecology. Princeton University
Press, Princeton, NJ.
Peacor, S.D. (2002). Positive effects of predators on prey growth
rate through induced modifications of prey behaviour. Ecol.
Lett., 5, 77–85.
Peacor, S.D. & Werner, E.E. (2001). The contribution of traitmediated indirect effects to the net effects of a predator. Proc.
Nat. Acad. Sci. (USA), 98, 3904–3908.
Puccia, C.J. & Levins, R. (1985) Qualitative Analysis of Complex Systems. Harvard University Press, Cambridge, MA.
Schoener, T.W. (1976). Alternatives to Lotka–Volterra competition: Models of intermediate complexity. Theor. Pop. Biol., 10,
308–333.
Schoener, T.W. (1993). On the relative importance of direct versus
indirect effects in ecological communities. In: Mutualism and
Community Organization: Behavioural, Theoretical, and Food Web
Approaches (eds Kawanabe, H., Cohen, J.E., Iwasaki, K.). Oxford
University Press, Oxford, UK. pp. 365–411.
Sih, A., Crowley, P., McPeek, M., Petranka, J. & Strohmeier, K.
(1985). Predation, competition, and prey communities: A review
of field experiments. Ann. Rev. Ecol. Syst., 16, 269–311.
Slobodkin, L.B. (1974). Prudent predation does not require group
selection. Am. Nat., 108, 665–678.
Vandermeer, J.H. (1993). Loose coupling of predator–prey cycles:
Entrainment, chaos, and intermittency in the classic MacArthur
consumer–resource equations. Am. Nat., 141, 687–716.
Manuscript received 20 January 2003
First decision made 26 February 2003
Manuscript accepted 10 March 2003
2003 Blackwell Publishing Ltd/CNRS