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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 460 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. 462 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- 464 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 466 The American Naturalist 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. 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