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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. 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