Ecology, 80(7), 1999, pp. 2358±2372 ᭧ 1999 by the Ecological Society of America
COMPLEMENTARY FORAGING BEHAVIORS ALLOW COEXISTENCE OF TWO CONSUMERS
W. G. W ILSON,1,4 C. W. OSENBERG,2 R. J. SCHMITT,3 AND R. M. NISBET3 1Evolution, Ecology, and Organismal Biology Group, and Department of Zoology, Duke University, Durham, North Carolina 27708-0325 USA 2Department of Zoology, University of Florida, Gainesville, Florida 32611-8525 USA 3Department of Ecology, Evolution and Marine Biology, University of California±Santa Barbara, Santa Barbara, California 93106 USA
Abstract. We developed a mathematical model based on the microalgal±gastropod system studied by Schmitt, in which two coexisting consumers (Tegula eisini and T. au- reotincta) feed on a common resource. The two consumers differ in their foraging behavior and their ability to remove microalgae from rock surfaces. T. eisini is a digger, moving slowly and grazing the algae down to almost bare substrate, whereas T. aureotincta is a grazer, moving more quickly and leaving behind a larger fraction of the algal layer. These complementary foraging strategies result in a size-structured algal resource, with each size class differentially accessible to each of the consumers. Our model recognized three ac- cessibility states for an algal patch: a refuge (recently grazed by the digger and currently inaccessible to either consumer), a low level (exploitable only by the digger), and a high level (exploitable by both consumers). We assumed that all interactions between consumers and resources were linear and examined the relatively short time-scale dynamics of feeding, algal renewal, and individual consumer growth at ®xed densities of consumers. Thus, our model complemented related models that have focused on population dynamics rather than foraging behavior. The model revealed that coexistence of two consumers feeding on a single algal resource can be mediated by differences in the consumers' foraging modes and the resource structure that these behaviors create. We then estimated model parameters using data from Schmitt's experimental studies of Tegula. The ®ts to the experimental data were all very good, and the resulting parameter values placed the system very close to a narrow coexistence region, demonstrating that foraging complementarity in this system facilitates coexistence. The foraging trade-offs observed here are likely to be common in many consumer±resource systems. Indeed, mech- anisms similar to those we discuss have been suggested in many other systems in which similar consumers also coexist. This model not only demonstrates that such an argument is theoretically plausible, but also provides the ®rst application of the model, showing that the observed conditions for the Tegula system fall very close to the appropriate parameter space. Such quantitative tests are critical if we are to rigorously test the models developed to explain patterns of coexistence. Key words: exploitation competition; foraging behavior; foraging model; gastropods; herbivore coexistence; marine snails; microalgae; resource structure; Tegula spp.
INTRODUCTION stages (Briggs 1993, Briggs et al. 1993). In such cases, the single resource species can represent two (or more) Models of competition, and of the mechanisms that limiting resources and can permit coexistence of more allow coexistence of similar species, remain a corner- stone of community ecology. Considerable theoretical than one competitor species (Vance 1985). The for- work has focused on the general result that the number aging activities of consumers can also create hetero- of coexisting consumers cannot exceed the number of geneities or patchiness in space and/or time that would limiting resources (MacArthur and Levins 1964); many otherwise not exist. If consumers differ in their ability of these papers have led to reevaluation of the meaning to exploit the different patches, then under some con- of resources. For example, a single resource species ditions, this patchy distribution of a single resource in might be subdivided into different functional compo- a homogeneous environment can lead to coexistence nents such as roots and shoots of a plant (Haigh and of two (or more) consumers. Maynard Smith 1972), or different developmental Two general features of these models appear to be important in producing coexistence: (1) structuring of the resource into distinct subunits (development stages, Manuscript received 9 June 1997; revised 20 May 1998; accepted 22 September 1998. or patches of different quality distributed in space or 4 E-mail: [email protected] time), and (2) differential ability of the consumers to 2358 October 1999 DIGGER±GRAZER COEXISTENCE 2359 exploit these subunits (Haigh and Maynard Smith 1972, responses, rather than long-term population growth, Briggs 1993). Although the mechanisms suggested by this individual-level approach is particularly powerful these theoretical studies have been posited to operate and applicable to empirical studies. As a result, we are in a variety of ecological systems (e.g., McNaughton able to use the experimental data collected by Schmitt 1976, Brown 1986, 1989, Kohler 1992, Schmitt 1996, (1996) on the marine snails, Tegula aureotincta and T. Thrall et al. 1998) and thus promote coexistence, there eiseni, to parameterize the model and provide an em- have been no quantitative tests of these models to de- pirical test of the model's predictions. Analysis of data termine if the speci®c features of any real system are taken for each species in isolation ®xes all but one of appropriate to produce coexistence. the model's parameters and shows that the mixed sys- Tilman (1994) has recently extended the competitor± tem of T. aureotincta and T. eiseni sits close to a small refuge models developed in the 1970s (e.g., Levins and region of coexistence in the model's parameter space, Culver 1971, Horn and MacArthur 1972, Hastings in agreement with observed coexistence in the ®eld. 1980) to demonstrate that an in®nite number of plant Reexamination of several studies on coexisting inver- species (or other sessile consumers) can theoretically tebrate grazers provides additional con®rmation of the coexist on a single limiting resource within a physically model formulation and the assumed trade-off between homogenous environment. In Tilman's model, coex- extraction ability and patch-®nding ability. However, istence requires a trade-off between, for example, a these studies do not provide adequate information to plant's dispersal ability (i.e., its ability to colonize un- quantitatively test our model's predictions about the occupied patches, which have high nutrient availabil- relationship between foraging traits and coexistence. ity) and its competitive ability (i.e., its ability to deplete Quanti®cation of these parameters for additional pairs nutrients within a patch). Research in the Cedar Creek, of grazers that coexist, and more importantly, for pairs Minnesota prairie system provides evidence for this that do not coexist, will provide a critical next step in interspeci®c trade-off (Gleeson and Tilman 1990, Til- evaluating the utility of this model and the importance man 1990). Tilman (1994) suggested that similar in- of complementary foraging behaviors. terspeci®c trade-offs might also have relevance to the coexistence of mobile consumers, foragers that visit DIGGER±GRAZER MODEL many patches during a single foraging bout. An ``in- Our model addresses competitors utilizing a single ferior'' competitor could invade a system saturated resource, and was developed with reference to the ma- with a ``superior'' competitor species if it could ®nd rine snails Tegula eiseni and T. aureotincta, which ex- and exploit renewed patches (those of high resource hibit complementary foraging patterns. T. eiseni acts quality, not recently visited) before they are revisited as a digger, moving slowly and removing layers of (and hence depleted) by the superior competitor. microalgae down to almost bare rock, whereas T. au- In this paper, we present a simple mathematical mod- reotincta acts as a grazer, moving faster and removing el for an exploitation competition system consisting of a smaller fraction of the algal layer. These two species a single resource species and two mobile consumer compete exploitatively and coexist in subtidal rocky species with complementary foraging behaviors: ``dig- environments where predator densities are low gers,'' which can extract resources at low levels, and (Schmitt 1985, 1996). ``grazers,'' which are poorer within-patch exploiters, Most of the available data in this and other systems but can potentially ®nd good patches more quickly. The have arisen from short-term experiments measuring the model assumes a homogeneous environment within interrelations among consumer density, consumer which patchiness arises due to the foraging activities growth, and algal availability (Schmitt 1985, 1996). of the consumers. We assume that the consumers utilize This experimental focus on individual responses (prin- no other resource and have no direct interactions other cipally growth), and not long-term population dynam- than the exploitation competition mediated through the ics, means that existing theory addressing population resource. Our model includes parameters that describe dynamics cannot be used to interpret experimental re- the consumers' foraging behaviors and metabolic costs, sults. Instead, models focusing on short-term dynamics as well as the renewal characteristics of the algae. must be formulated and tied to the available experi- The model describes consumer growth and seeks mental data. Our consumer±resource model, formulat- conditions under which individuals of both consumer ed at behavioral time scales, is one such attempt and, species can exhibit non-negative growth. Thus, it iso- as a result, does not address long-term coexistence di- lates potential mechanisms of consumer coexistence rectly. Rather, as in the experiments, the model focuses operating on shorter time scales than classical models on short-term dynamics and uses short-term positive focusing on population dynamics. One feature of the individual growth as a proxy for long-term persistence system allowing this level of focus is that the time scale (i.e., long-term population growth). for resource reproduction is much shorter than the con- We assume that the resource located within a very sumers' generational time scales. Because most ex- small spatial region (a ``patch'') can be categorized into periments designed to investigate competitive inter- one of three levels (see Fig. 1, Table 1). The lowest actions and coexistence typically focus on individual level is unusable by either the digger or the grazer, and 2360 W. G. WILSON ET AL. Ecology, Vol. 80, No. 7
FIG. 1. Representation of the Tegula model. The low, middle, and high levels of density make up the size-structuring in the benthic algal mat imposed by the differential foraging strat- egies taken by the grazers and the diggers. Dot- ted lines indicate foraging: grazers consume the highest level, and diggers consume the middle and highest levels. Solid lines represent inter- action rates with variables and parameters de- ®ned in Table 1.
represents a resource refuge. A patch of resource at the the fractions of the area at each level: A0, A1, and A2, lowest level enters the middle level with transition rate for the lowest, middle, and highest levels, respectively,
␣0. This middle-resource level represents patches of with A0 ϩ A1 ϩ A2 ϭ 1. Resource consumption rates low algal cover that are exploitable by the digger but at each level depend linearly on the consumer popu- not the grazer. Middle-level patches enter the highest lation's biomass density (a Type I functional response) resource level (i.e., dense cover, or a thick mat) with and consumer-speci®c resource clearance rates. We as- transition rate ␣1, and become exploitable by both the sume that diggers have no complex foraging behaviors; digger and the grazer. Thus, the digger can extract nu- hence, their foraging rates are independent of resource trition from both the middle- and high-resource states habitat fractions. These assumptions lead to the model (and converts these patches into the low algal state), equations: whereas the grazer bene®ts only from the high-resource dA 0 ϭϪ␣ A ϩ B (A ϩ A ) (1a) state (converting it to the middle state). In the absence dt 00 DD 1 2 of consumers, all patches exist in the high level, or ungrazed state. Thus, by assumption, the environment dA 1 ϭ␣ A Ϫ␣A Ϫ BA ϩ BA (1b) is homogeneous, and spatial heterogeneity is generated dt 0 0 11 DD1 GG2 only by the local foraging activities of consumer, with dA2 the degree of local depletion of algae depending on the ϭ␣11A Ϫ DD2BA Ϫ GG2BA (1c) identity of the consumer (grazer or digger). dt
A unit area of resource habitat can be classi®ed into where BD ϭ NDMD and BG ϭ NGMG represent the digger
TABLE 1. Variables and parameters of the digger±grazer model.
Terms Description Units A) Model variables
A0 fraction of habitat at low level none A1 fraction of habitat at middle level none A2 fraction of habitat at high level none MD mass of individual digger consumer biomass/consumer MG mass of individual grazer consumer biomass/consumer ND number density of diggers consumers/area NG number density of grazer consumers/area BD digger biomass density (ND MD) consumer biomass/area BG grazer biomass density (NG MG) consumer biomass/area B) Model parameters
0 resource biomass density of low level resource biomass/area 1 resource biomass density of middle level resource biomass/area 2 resource biomass density of high level resource biomass/area Ϫ1 ␣0 transition rate from low to middle level time Ϫ1 ␣1 transition rate from middle to high level time D digger's resource clearance rate area/consumer biomass/time G grazer's resource clearance rate area/consumer biomass/time ␥D digger's resource conversion ef®ciency consumer biomass/resource biomass ␥G grazer's resource conversion ef®ciency consumer biomass/resource biomass Ϫ1 ␦D digger's metabolic rate time Ϫ1 ␦G grazer's metabolic rate time October 1999 DIGGER±GRAZER COEXISTENCE 2361
and grazer populations' biomass densities (biomass per extract the ®rst two resource biomass components, 21 area) resulting from a product of the respective number ϩ10 ϭ20. The last component, 0, is inaccessible densities ND or NG (number of consumers per area) and to either consumer. Note that multiplying Eqs. 3a and individual consumer mass MD or MG (biomass per con- b on both sides by their respective number densities sumer). Parameters ␣0 and ␣1 (per unit time) are the yields equivalent expressions for digger and grazer transition rates of the lowest and middle resource levels population biomasses. to the next highest level, and D and G (area cleared We reemphasize that this model is strictly a behav- per consumer biomass per time) denote the individual ioral time-scale model for consumer growth, not pop- digger and grazer clearance rates, i.e., the rates at which ulation dynamics. This distinction is made clear by diggers and grazers, respectively, clear substrate of the incorporating ND and NG as constants, rather than dy- algal resource. Note that only two of the three eqs. (1a± namical variables as in most competition models (e.g., c) are independent because the three algal fractions sum Tilman 1982, Briggs et al. 1993). As a result, equili- to one. bration arises as individuals adjust their size to the These expressions yield algal habitat fraction dy- point at which respiration balances consumption, rather namics on an areal basis (fraction of the habitat made than through changes in ND and NG (and the balancing up of each of the three algal types: low, medium, and of birth and death rates), as in a population dynamics high). These fractions are converted to resource bio- model. mass using the resource density parameters , , and 0 1 Invasion criteria 2 (resource biomass per area), giving the total biomass of algae, AT: We can recast our consumer±resource model into a model depending only on consumer biomasses by as- AT ϭ0 A0 ϩ1A1 ϩ2A2. (2) suming that algal dynamics occur on a fast time scale. Individual consumer dynamics depend on two pro- This assumption means that algal fractions are always cesses, consumption and respiration. Consumption rep- in equilibrium with the slowly changing consumer bio- resents a two-step process of converting resource bio- masses. The Appendix details this quasi-steady-state mass into consumer mass. Our model ®rst converts assumption and derives the digger±grazer competition patch-speci®c clearance rates into resource biomass equations involving only the consumer biomass den- consumption rates by multiplying by the density of sities: algae removed by the consumer. For example, a grazer 1 dMD␣ 100( GGB ϩ DDB ) ϩ 2001 ␣ ␣ of mass MG feeding on high-level resource patches with ϭ␥DD Ϫ␦ D MdtD(␣ϩ 1 GGB ϩ DDB )(␣ϩ 0 DDB ) a clearance rate GMGA2, reduces them to the medium level and changes the algal biomass density from 2 (4a) to 1. The consumed resource biomass is then (2 Ϫ 1 dMGGG2101␥ ␣␣ 1) GMGA2. Our model then converts resource biomass ϭϪ␦. Mdt (␣ϩB ϩ B )(␣ϩB ) G into consumer mass with ef®ciencies ␥D and ␥G (con- G 1 GG DD 0 DD sumer biomass per resource biomass). Net respiration (4b) is assumed to be directly proportional to consumer mass, with species-speci®c respiration rates ␦D and ␦G We de®ne coexistence to occur if the two equilibrium (per time). These two processes lead to expressions for single-consumer systems are invasible by the other individual digger and grazer masses, MD and MG, consumer (e.g., Holt et al. 1994). Evaluating this cri- terion is a two-step analysis. We ®rst examine how dMD individual growth varies as a function of conspeci®c ϭ␥D DM D( 10A 1ϩ 20A 2) Ϫ␦ DM D (3a) dt biomass density. Generally, individual growth will be positive at low biomass densities, declining to negative dMG ϭ␥GGM G212 A Ϫ␦ GM G. (3b) values as biomass increases. Single-consumer equilib- dt ria KD and KG de®ne the consumer biomasses at which
Parameters ij ϭi Ϫj represent the reduction in individual growth is zero (Fig. 2A). We next examine algal biomass density caused by consumers; for ex- individual growth as a function of the biomass density ample, in the previous example, grazers reduce high- of the other species. If each species has positive in- level resource patches by 21 ϭ2 Ϫ1. Using these dividual growth at the other species' equilibrium (see
ij parameters, we can clarify the differential utilization Fig. 2B), then each single-species equilibrium is in- of the resource by the consumers. For example, the net vasible and we conclude that the species can coexist. resource biomass contained in a unit area of high-level Two criteria result from the test for coexistence (see resource patches, 2, can be represented as three com- Appendix). The ®rst criterion speci®es when grazers ponents, 2 ϭ21 ϩ10 ϩ0, or upon expanding, 2 can invade a system of diggers. Following the previous
ϭ (2 Ϫ1) ϩ (1 Ϫ0) ϩ0 ϭ2. The ®rst com- two-step analysis, we ®rst solve the digger equilibrium ponent, 21, represents the resource biomass that graz- values from Eqs. 1b, 1c, and 3a, using the relation A0 ers extract from high-level patches, whereas the diggers ϭ 1 Ϫ A1 Ϫ A2 and BG ϭ 0. The resulting condition 2362 W. G. WILSON ET AL. Ecology, Vol. 80, No. 7
The second invasibility criterion speci®es when dig- gers are able to invade a grazer-only system. The Ap- pendix shows that diggers will invade if
␦␦DG Ͻ10 ϩ . (6) ␥DD ␥ GG
If we de®ne the digger's RD* ϭ␦D /␥DD, then we can relate Inequality (6) to the more general R* rules: dig-
gers will invade if theirRD* is less than the resource accessible to them in the grazer-only system. This ac- cessible resource is composed of two parts, the middle- level resource unavailable to the grazer, and the equi- librium high-level resource at which the grazer persists,
equivalent to the grazer'sRG* . As we will examine in detail, coexistence occurs when the grazer, a superior competitor on the best re-
source (i.e., A2G*(0, K ) Ͻ A 2D*(K , 0)), leaves behind the middle-level resource that the digger exploits in com- pensation for its inferior competitiveness on the high- level resource. Thus, coexistence arises because of a trade-off at the individual scale between area-intensive FIG. 2. Schematic relationships between consumer and area-extensive foraging. The complementary for- growth rates as functions of (A) conspeci®c and (B) hetero- speci®c consumer biomass densities necessary for digger± aging behaviors produce a structured resource with an grazer coexistence. Equilibrium biomass densities KD and KG overlap in consumer utilization, and thereby permit co- are the conspeci®c biomass densities (in the absence of het- existence. erospeci®cs) at which consumer growth rates are zero. If, as pictured, both consumer species have positive growth rates APPLICATION TO TEGULA at these heterospeci®c biomass densities, then the single-con- sumer systems are mutually invasible. Parameter estimation using laboratory data In this section, we obtain parameter estimates for the Tegula system from the experiments reported by for positive individual grazer growth, dMG/dt Ͼ 0, Schmitt (1996) and test whether the predictions of our yields model are consistent with the observations. In those experiments, snails of uniform size were collected from ␦G Ͻ21A*( 2K D, 0) (5) the ®eld and were used to create two replicates of 10 ␥ GG different densities (0±42 snails/0.1 m2) in ¯ow-through whereA2D*(K ,0) is the equilibrium high-level resource laboratory tanks lined with terra-cota tiles that had been fraction of the digger-only system (see Eq. A.17). Mul- previously colonized by algae. A subset of the collected tiplication ofA2D*(K ,0) by 21 yields the total amount snails was dried for 48 h at 60ЊC and weighed to obtain of resource accessible to the grazer. initial dry tissue masses (excluding shell and opercu- We can relate Eq. 5 to R* concepts (Tilman 1982). lum). After 73 d, the experiment was terminated, snails The R* rule, usually applied to competition for a com- were removed from their tanks, dried, and weighed, mon resource pool, is that the competitor with the and algal density was estimated by scraping tiles and smallest R* outcompetes the others. In our model, the determining the amount of chlorophyll a (see Schmitt R* rule must be slightly modi®ed because the resource (1996) for details). Changes in snail mass across the is structured into components that are differentially ac- treatments ranged from mass gains of up to 60% in the cessible by the two species. In this case, a species' R* low-density snail treatments to losses of up to 25% in only includes the resource accessible to it, excluding the high-density snail treatments. Variation between resource components that it cannot utilize. Hence, from replicates was up to 20% for both snail mass gains and
Eq. 5, we de®ne the ratio RG* ϭ␦G /␥GG to represent ®nal algal densities. the amount of accessible resource biomass needed for Results of the experiment provide the ®nal algal den- an individual grazer to balance intake and metabolism; sities at the end of the experiment and the change in if a grazer's available resource falls below this level, snail mass during the experiment as a function of the it experiences negative growth. Hence, if diggers de- number of snails per tank. To relate the data to our plete the amount of resource biomass accessible to the model, we assume that the ®nal algal densities repre- grazers belowRG* , then the grazers cannot invade. Con- sent algal densities that have equilibrated to the grazing versely, grazers can invade if their accessible resource activities imposed by the snail treatment; the transient biomass in the digger-only system is greater thanRG* . times to reach equilibrium were short so that we could October 1999 DIGGER±GRAZER COEXISTENCE 2363 use the observed change in snail mass to approximate the instantaneous rates of change in snail mass. All of the experimental treatments used to parameterize the model dealt with grazers and diggers isolated from het- erospeci®cs.
Algal abundance We estimated the ®rst set of parameters by ®tting the model's predicted dependence of algal biomass on snail biomass to data for algal biomass. Our model provides algal fractions as a function of ®xed consumer densities by setting the time derivatives in Eqs. 1b, c to zero and using the condition A0 ϩ A1 ϩ A2 ϭ 1. These equilibria depend on snail biomass; thus, we assume that the algal equilibrium densities respond quickly to the slowly changing snail biomass. Schmitt's experimental studies of Tegula did not provide direct estimates of the three algal fractions. Instead, his studies estimated the total resource density, which we relate to the model by mul- tiplying each algal fraction by the appropriate resource biomass density parameter:
AT(BD, BG) ϭ0A0(BD) ϩ1A1(BD, BG)
ϩ2A2(BD, BG) (7) (see Eq. A.4 for explicit dependencies).
In the absence of both diggers and grazers (BD ϭ 0,
BG ϭ 0), A0 ϭ 0, A1 ϭ 0, and A2 ϭ 1, thus, from Eq.
7, the total resource density is AT(0, 0) ϭ2. Schmitt's 2 experiments yielded 2 ϭ 1.81 g Chl a/cm for the algal density in the absence of snails at the end of the experiment. To estimate additional parameters from algal density data, we used results obtained when each species was established at a gradient of densities. These results FIG. 3. (A) Algal standing stock density plotted as a func- yielded relationships between total algal biomass and tion of the inverse consumer biomass density for both the grazer (T. aureotincta, ⅷ) and the digger (T. eiseni, Ⅺ). Model consumer biomass. We simpli®ed the model predic- analysis predicts linear relationships for high consumer den- tions by using a Taylor series expansion of Eq. 7 giving, sities, and yields the model parameters 0, 1, ␣1/G, and ␣0/ to ®rst-order approximation, linear functions relating D. (B) Algal density plotted as a function of the untrans- total algal biomass density to digger (or grazer) bio- formed snail densities. The solid lines depict Eqs. A.7 and mass: A.10, using the parameters determined from the linear ®ts to the data of Fig. 1. Three trial values for the transition rate
ratio, ␣1/␣0 ϭ 0.04 (lower curve), 0.06 (middle curve), and ␣0 1 ATD(B ,0)ഡ ϩ 0(Ϫ 1 0) (8a) 0.08 (upper curve), generate three curves for the digger. DDB
␣1 1 2 ATG(0, B ) ഡ ϩ 1(Ϫ 21) (8b) gave an estimate of ϭϪ0.068 (R ϭ 0.97). This B 0 GG value is biologically implausible; hence, we re®tted the
(see derivations of Eqs. A.11 and A.8). Parameters curve constraining the intercept to be zero (0 ϭ 0) for were estimated by linear regressions of ®nal algal den- all subsequent analyses. Resulting linear ®ts to the al- sity on the inverse of ®nal snail biomass density (Fig. gal±snail biomass data were very good (R2 ϭ 0.89, 3A). The intercepts of Eqs. 8a and b provide estimates 0.95). Comparing the ®tted functions with Eqs. 8a and for 0 and 1, respectively. Using the i's, the slopes 8b yields the parameter values: 1 ϭ 0.0888 g Chl 2 Ϫ3 2 provide estimates for ␣0 /D and ␣1/G. a/cm ; ␣0 /D ϭ 6.13 ϫ 10 g dry mass/cm ; and Ϫ4 2 We excluded results from the lowest snail densities ␣1/G ϭ 1.89 ϫ 10 g dry mass/cm (Table 2).
(NG ϭ 3 and ND ϭ 3), due to clear deviations from the These parameter estimates, however, are based on linear approximation (which works best when N is linear approximations to the exact expressions for large). A further complication arose using Eq. 8a to ®t AT(BD, 0) and AT(0, BG). We can test our linearization the digger data, because the intercept was negative and scheme by substituting the parameter estimates into the 2364 W. G. WILSON ET AL. Ecology, Vol. 80, No. 7
TABLE 2. Experimentally determined values for the param- Ϫ1 Ϫ5 0.00907 d ; ␥D␣0 ϭ 4.12 ϫ 10 (g dry mass´(g eters appearing in Eqs. 1±3. Ϫ1 Ϫ1 Ϫ6 Chl a) ´d ); and ␥G␣1 ϭ 8.37 ϫ 10 (g dry mass´(g Ϫ1 Ϫ1 Model Chl a) ´d ). These parameters, estimated from linear parameters Tegula values approximations, were then placed back into the full theoretical expressions and compared with the untrans- 0 0.0 g Chl a/cm2 formed experimental data (Fig. 4B). There was a good ®t between the ®tted theoretical curves and the exper- 1 0.0888 g Chl a/cm2 imental data for the grazer. The digger relationship
2 1.81 again depends on the transition rate ratio, ␣1/␣0, and a g Chl a/cm2 good ®t between the theoretical curves and the exper- Ϫ3 ␣0 /D 6.13 ϫ 10 imental data was observed for ␣1/␣0 ϭ 0.06. g dry mass/cm2
Ϫ4 Plausibility of parameter estimates ␣1/G 1.89 ϫ 10 g dry mass/cm2 The parameter estimates (Table 2) obtained using Ϫ5 ␥D ␣0 4.12 ϫ 10 Schmitt's (1996) experimental data can be checked for Ϫ1 Ϫ1 g dry mass´(g Chl a) ´d plausibility by comparisons with independent data. In Ϫ6 ␥G ␣1 8.37 ϫ 10 g dry mass´(g Chl a)Ϫ1´dϪ1
␣1/␣0 0.06 Ϫ1 ␦D 0.00177 d Ϫ1 ␦G 0.00907 d Notes: The data of Schmitt (1996) leave only one uncon- strained parameter, an absolute measure of the algal transition rate (␣0 or ␣1), which has no effect on the coexistence con- ditions. full forms of these expressions (see Eq. A.4). However, the full form for AT(BD, 0) depends on the transition rate ratio ␣1/␣0. Thus, we next compared the observed relationships with theoretical curves for several values,
␣1/␣0 ϭ 0.04, 0.06, and 0.08 (Fig. 3B). The value ␣1/␣0 ϭ 0.06 provided a good visual match to the untrans- formed digger data, but we presume the uncertainty in the transition rate ratio, ␣1/␣0, is relatively large. Consumer growth We next sought estimates for the remaining param- eters of the model by examining the relationship be- tween snail growth and snail density. Thus, as detailed in the Appendix, we obtained the following linear ap- proximations for individual consumer growth depen- dent on inverse consumer biomass:
1 dMD 1 ϭ␥D100 ␣ Ϫ␦ D (9a) MdtDD B
1 dMG 1 ϭ␥G211 ␣ Ϫ␦ G. (9b) MdtGG B Intercepts of Eqs. 9a and 9b provide estimates for the metabolic losses ␦D and ␦G, respectively. Using the i's determined from the algal ®ts and the slopes from Eqs. FIG. 4. (A) Consumer growth curves for both T. aureo- tincta (ⅷ) and T. eiseni (Ⅺ) plotted against the inverse con- 9a and b, we can estimate the aggregate parameters sumer biomass density. Model analysis suggests linear rela- ␣0␥D and ␣1␥G, respectively. tionships, and the regression coef®cients determine model Snail growth was linearly related to the inverse of parameters ␦D, ␦G, ␥D␣0, and ␥G␣1. (B) Consumer growth data snail biomass density (Fig. 4; as in Fig. 3A, data from plotted against the consumer biomass densities. The solid lines plot the predicted growth rates using model parameters the lowest snail densities were excluded). Linear re- obtained through the ®ts in Figs. 1 and 3. As in the algal plot, gression (R2 ϭ 0.92 and 0.97) yielded estimates for the three trial values for the transition rate ratio, ␣1/␣0 ϭ 0.04, Ϫ1 remaining model parameters: ␦D ϭ 0.00177 d ; ␦G ϭ 0.06, and 0.08, are considered for the digger curve. October 1999 DIGGER±GRAZER COEXISTENCE 2365 this section, we examine the plausibility of our esti- mates of clearance rate, patch transition rate ratio, met- abolic rates, and conversion ef®ciencies to determine if they are consistent with other known biological as- pects of the Tegula system. Clearance rates.ÐUsing the previously discussed values, we ®nd the ratio of the grazer-to-digger algal clearance rates, G/D ϭ 1.95, showing that the grazer encounters high-density algae patches at almost twice the rate of the digger. Schmitt (1996) performed ex- periments that measured the area cleared per unit time for each grazer. The ratio of those values was ϳ2.1, in rough agreement with our estimate. Our predicted value depends on our determination of the transition rate ra- tio, ␣1/␣0, which has an uncertainty of Ն10%; likewise, Schmitt's data have uncertainties of ϳ10%. Algal transition rate ratio.ÐOur estimate of the patch transition rate ratio, ␣ /␣ ϭ 0.06, is roughly FIG. 5. The coexistence region for the Tegula system, 1 0 with algal parameters ®xed to the values determined through equal to our ratio of the resource density parameters, the ®ts in Figs. 1 and 4. The condition separating the coex- 10/21 ϭ 0.052. If the two ratios were identical, then istence region and digger-only region depends on the tran- ␣010 ϭ␣121; i.e., the algal biomass growth rates (the sition rate ratio; boundaries are shown for four values of the product of patch transition rate and algal biomass dif- ratio. The heavy line separates the coexistence region from ference) would be the same for the two patch types. the grazer-only region. When the transition rate ratio ap- proaches zero, the coexistence region is nonexistent. Esti- Hence, our estimated patch transition rate ratio, ␣1/␣0, mates for the ratios ␦G /(10␥GG) and ␦D /(10␥DD) from the re¯ects the longer time required to produce the greater Tegula data, denoted by the circle, place the system just out- abundance of algae in the high level. The close match side the coexistence region, but within a 5% change in the between the two estimated ratios suggests consistency estimated ratios. within our set of parameter estimates. Conversion ef®ciencies.ÐThese data allow us to pre- dict the grazer-to-digger conversion ef®ciency ratio, that arises because transporting a mass m over some
␥G/␥D. The ®ts yield 3.4, meaning that for every unit distance with speed requires a power output of ⑀m, of algae consumed by each consumer species, grazers where ⑀ is a proportionality constant. If we assume that ®x 3.4 times more of the algal biomass into consumer the greatest portion of Tegula's metabolic rate arises biomass than do diggers. We are unaware of the phys- from these transportation costs, then we would expect iological basis for these underlying conversion ef®- equality between the metabolic rate ratio and the power ciency differences. output ratio, ␦G/␦D ϭGmGG/DmDD. Schmitt (1996)
Metabolic rates.ÐThe estimated ratio of grazer and reports a measured speed ratio of G/D ϭ 7.0, close to digger metabolic rates is ␦G/␦D ϭ 5.1. Metabolic rates that of the estimated metabolic rates. If we assume that can be compared to relative movement speeds because the coef®cients G and D are identical (due to a lack metabolic rates increase with increasing speed for or- of information), then we expect the ratio of the snail ganisms ranging from terrestrial animals (Schmidt- masses (including shells) to be mG/mD ϭ 0.73. For the Nielsen 1995, Calder 1996) to marine gastropods snails used in Schmitt's experiments, individual bio- (Newell and Roy 1973, Denny 1980, Houlihan and In- masses were 2.75 g for T. aureotincta and 3.52 g for nes 1982). Respiration rates for the winkle Littorina T. eiseni, giving the ratio of total wet masses as ϳ0.78, littorea demonstrated a ®ve- to sevenfold increase be- in close agreement with our estimate. tween the quiescent and active states (Newell and Roy 1973), and the trochids Monodonta articulata, M. tur- COEXISTENCE binata, Gibbula richardi, and G. rarilineata demon- There are four possible competitive outcomes: (1) strate twofold increases when active in air (Houlihan extinction of both species; (2) persistence of only the and Innes 1982). There are at least two potential grazer; (3) persistence of only the digger; and (4) co- sources for metabolic costs in the gastropod Tegula. existence of the digger and grazer (Fig. 5). The out- First, Denny (1980) vividly demonstrates the important comes depend on the resource parameters and the ag- contribution of mucus production to overall locomotion gregate digger and grazer parameters, ␦D /␥DD and costs in the terrestrial gastropod, Ariolimax columbi- ␦G/␥GG. Persistence for each species requires a simple anus, and this per unit distance cost scales directly with balance between consumption and metabolism, i.e., speed. Alternatively, Denny (1980) demonstrated for ␦D /␥DD ϭ RDG* Յ 20 and ␦G /␥GG ϭ R* Յ 21. Within A. columbianus that the internal power generated in- the region where each species can persist in the absence creases linearly with the slug's speed, a relationship of competition, we have three competitive scenarios: 2366 W. G. WILSON ET AL. Ecology, Vol. 80, No. 7 the digger wins, the grazer wins, or both species co- and to deplete food within a patch. Coexistence is pos- exist. sible because consumption produces patchiness in the Coexistence occurs in the narrow region indicated shared resource, such that local resource densities fa- by the overlap of the two invasibility conditions (Eqs. voring each species exist in the environment. As such, 5 and 6; Fig. 5). The region's size depends on the ratio our system can be viewed as a spatial analogue to the of the patch transition rates (␣1/␣0). In the limit ␣1/␣0 temporally heterogeneous one modeled by Armstrong → 0, the boundaries for the two invasibility conditions and McGehee (1980). are equivalent and there is no overlap: one species or Our theoretical approach was motivated, in part, by the other always wins. However, as ␣1/␣0 increases, the the large empirical literature on exploitation compe- coexistence region expands because the grazer's in- tition between species of mobile herbivores. Numerous vasion boundary pushes into the digger's persistence examples of competition for microalgae between her- region. This coexistence region attains its maximal area bivores in benthic marine and freshwater environments → as ␣1/␣0 ϱ. have been reported (Branch 1984, Carpenter 1990, Substituting the parameter values listed in Table 2 Robertson 1991, Underwood 1992, McClanahan et al. into the invasibility criteria yields the conditions that 1996, Schmitt 1996). For example, competitors from must be satis®ed for coexistence. Numerically, the re- one of the more comprehensively studied systems in- sultant conditions are 2.43 Ͻ 2.33 (grazer invasion cri- volving two species of marine snails in the genus Teg- terion; Eq. 5) and 2.97 Ͻ 3.31 (digger invasion crite- ula (Schmitt 1985, 1996) have persisted together with rion; Eq. 6). Hence, the estimated parameters predict relatively small ¯uctuations in abundances (Ͻ20% of grazer exclusion, but a change of only 5% in the grazer the long-term mean) for at least four population turn- invasion condition's numerical values, well within the overs on a reef with intense competition (Schmitt 1985, error associated with the parameter estimates, would 1996). yield coexistence. A near-universal feature of empirical studies of com- petition for microalgae is the focus on individual per- DISCUSSION formance, e.g., body growth, fecundity, and mortality A sizable body of theoretical work has focused on of the herbivores. The studies of Tegula by Schmitt whether a single resource species can be ``partitioned'' (1985, 1996) were no exception. A speci®c goal of our into different limiting components and thereby allow study was to test our model using data from the Tegula coexistence of more than one species of consumer. Co- system. Our desire to use available data to test model existence has been shown to be possible, theoretically, predictions dictated that we construct a ``behavioral'' if the limiting resource ¯uctuates on an appropriate or ``within-generation'' time-scale model. time scale between states that alternatively favor dif- Our criterion for ``coexistence'' and, hence, the ab- ferent consumer species. Such ¯uctuations could arise sence of competitive exclusion, rests on the ability of from external forces (Levins 1979) or from the activ- a single individual of each consumer to have positive ities of the consumers themselves (Koch 1974a, b, body growth when the population biomass of the other Armstrong and McGehee 1980). Coexistence also is species is at equilibrium with the resource. Parameter possible when different components of a limiting re- values calculated from the equilibrium equations of our source simultaneously exist in a single environment. model, using growth data for each Tegula species in For example, Briggs et al. (1993) showed that more isolation (Schmitt 1996), indicate that this system of than one parasitoid species could coexist on a single competitors lies on the edge of parameter space that host species by attacking different developmental allows coexistence. This result suggests that the mech- stages of the host. Tilman (1994) found that coexistence anism that we modeled may provide a suf®cient, al- of sessile organisms (plants) competing for a single though perhaps not complete, explanation for their co- limiting resource could occur in a physically homo- existence. As we will discuss, features of the Tegula geneous environment in which spatial heterogeneity in species that promote coexistence may be prevalent the resource arises solely from resource extraction ac- among other pairs of benthic herbivores that compete tivities and death of the consumers. The essential fea- exploitatively for microalgae. Long-term processes ture of Tilman's model that allows coexistence is a might well alter coexistence prospects. For example, trade-off in the abilities of species to colonize unoc- reproduction requires a separate allocation of resources cupied patches and deplete resources within a patch. that we have not considered. Likewise, ¯uctuations in We have extended this perspective to mobile con- resource productivity and consumer mortality and dis- sumers capable of visiting many patches during a feed- turbances may play a role in long-term coexistence. ing episode where patchiness in the limiting resource These potentially important processes must be consid- arises from the foraging activities of the consumers. ered for any conclusive determination, either empirical We found that mobile consumers with complementary or theoretical, of long-term coexistence. foraging attributes can coexist in an otherwise homo- The connection between our criterion for coexis- geneous environment. A key to coexistence is a trade- tence, based on individual performance, and the more off in the consumers' ability to locate abundant food traditional view is relatively straightforward for eco- October 1999 DIGGER±GRAZER COEXISTENCE 2367
FIG. 6. Microalgal abundance vs. inverse consumer density from a variety of marine and freshwater systems. (A) Data from Underwood (1984) on the Nerita atramentosa (snail, grazer)±Cellana tramoseria (limpet, digger) system from the marine intertidal shore in eastern Australia. Data are from his Experiment 1, mid- and lower intertidal zones. (B) Data from Steinman et al. (1987) on ash-free dry mass of algae in a freshwater stream system in North America: Juga silicula (snail, digger) and Dicosmoecus gilvipes (caddis¯y larva, grazer). (C) Data from R. J. Schmitt and S. S. Anderson (unpublished manuscript)on the marine intertidal shore along the western United States using the limpets Lottia digitalis (digger) and L. strigatella (grazer). (D) Data from the freshwater system of Kohler (1992), using ash-free dry mass of algae for Glossosoma nigrior (caddis¯y larva, digger) and Baetis tricaudatus (may¯y larva, grazer). logically closed populations in which population dy- cies with ecologically open populations than are the namics are determined solely by the demographic rates traditional criteria that focus on population dynamics. of local individuals. However, populations of many ma- An exception occurs when food is not the limiting re- rine and freshwater species that consume microalgae source and the overall abundance of a consumer is lim- are open by virtue of bipartite life cycles. For example, ited by the supply of offspring (which is independent most benthic marine organisms have early develop- of local fecundity). In this case, the ability of colonists mental stages that disperse widely as plankton, and to reach sexual maturity is not a reasonable coexistence many freshwater herbivores are the larvae of highly criterion. However, food was limiting in the Tegula mobile, terrestrial-dwelling adults. The traditional con- system that we modeled; individual growth of both cepts of competitive exclusion and coexistence are consumers on a natural reef tripled when the competing problematic for species in which colonization of local species was removed (Schmitt 1985). Furthermore, ob- populations is from external sources; regardless of the servations that the individual growth of Tegula under outcome of local competition, continued invasion of ambient conditions was near the ``knife-edge'' between an inferior competitor can preclude total exclusion positive and negative growth provide further evidence (Schmitt 1996). that microalgae were limiting (Schmitt 1985, 1996) and Indeed, our coexistence criterion, focusing on indi- that competition was important in determining coex- vidual growth, is more appropriate for competing spe- istence in this system. 2368 W. G. WILSON ET AL. Ecology, Vol. 80, No. 7
FIG. 7. Consumer growth vs. inverse consumer density from a variety of marine and freshwater systems. (A) Data from Creese and Underwood (1982) on the Siphonaria denticulata (snail, grazer)±Cellana tramoseria (limpet, digger) system from the marine intertidal shore in eastern Australia. (B) Data from Underwood (1984) on the Nerita atramentosa (snail, grazer)± Cellana tramoseria (limpet, digger) system from the marine intertidal shore in eastern Australia. (C) Data from R. J. Schmitt and S. S. Anderson (unpublished manuscript) on the marine intertidal shore along the western United States using the limpets Lottia digitalis (digger) and L. strigatella (grazer). (D) Data from the freshwater system of Kohler (1992) for Glossosoma nigrior (caddis¯y larva, digger) and Baetis tricaudatus (may¯y larva, grazer). The data for Baetis near 3 cm2 and 6 cm2 were confounded by low survivorships (20% and 40%, respectively) through the course of the experiment.
A primary assumption of our model is that the ver- effects of grazing by co-occurring may¯y nymphs (ge- tically structured nature of the microalgal resource re- nus Ameletus) and caddis¯y larvae (Neophylax)on sults from the foraging activity of the herbivores. Vari- freshwater periphyton. May¯y larvae, which use brush- ation in the effect of one herbivore on the performance like labial palps to feed, removed microalgae from the of another frequently has been attributed to differences upper layer of the periphyton, but had no discernible in the degree to which microalgae were cropped close effect on the lower adnate layer. By contrast, the cad- to the substratum (Underwood 1978, 1984, Branch and dis¯y larvae, which have hard, scraper-edged mandi- Branch 1980, Underwood and Jernakoff 1981, Creese bles, were able to depress periphyton in both the upper and Underwood 1982, McAuliffe 1984, Fletcher and and lower layers (Hill and Knight 1988; also see Mc- Creese 1985, Hill and Knight 1987, 1988, Steinman et Auliffe 1984). al. 1987, Kohler 1992, Schmitt 1996). Such differences Available empirical evidence also suggests that the often can be traced to morphological variation in feed- degree to which a consumer depletes microalgae in a ing structures (Underwood 1978, Branch and Branch patch varies inversely with the area that it grazes per 1980, Underwood and Jernakoff 1981, Creese and Un- unit time (e.g., Underwood 1977, 1978, Steinman et derwood 1982, Steneck and Watling 1982, Black et al. al. 1987, Kohler 1992, Schmitt 1996). The apparent 1988, Hill and Knight 1988, Steneck et al. 1991). For prevalence of this covariation among mobile herbivores example, Hill and Knight (1987, 1988) compared the prompted Schmitt (1996) to suggest that benthic con- October 1999 DIGGER±GRAZER COEXISTENCE 2369 sumers may be constrained in their abilities to both those components. Here, we have shown that the for- maximally extract food in each patch harvested and aging activities of mobile consumers can structure the maximize the area that they graze per unit time (also shared resource and lead to ®ne-scale spatial hetero- see Underwood 1978). These differences may represent geneity in an otherwise homogeneous environment. a general trade-off for harvesting food under different The heterogeneity results from differences in the ability resource levels (Schmitt 1996); area-intensive grazing of consumers to extract food within a patch, and in (i.e., depressing microalgae to a low level in a patch) movement rates while foraging. Coexistence is possible should confer a relatively greater intake rate of food when the ability to ®nd patches of abundant food varies when or where food is scarce (a thin layer of microal- inversely with the ability to deplete resources in a gae) in the environment, whereas area-extensive for- patch. To the extent that is known, such a trade-off aging (i.e., feeding over a larger area per unit time) between resource extraction and patch-®nding abilities should result in a greater intake rate when food is abun- appears to be common for co- occurring consumers that dant. (at least potentially) compete for microalgae. It remains If such a trade-off is common, then other pairs of to be determined whether the mechanism modeled here co-occurring benthic consumers should be expected to is a general feature that contributes to coexistence of affect microalgal densities in a pattern similar to that such species in nature. observed for Tegula. This contention is supported by four examples taken from the literature (representing ACKNOWLEDGMENTS two freshwater and two marine systems), in which in- We would like to thank Cherie Briggs, Ed McCauley, Bill dependent information was presented that allowed us Murdoch, Shane Richards, and Knut Schmidt-Nielsen. The work was funded, in part, by grants from the U.S. Of®ce of to classify each species in a pair of consumers as an Naval Research (N00014-93-10952) and the National Science area-intensive forager (digger) or area-extensive for- Foundation (OCE 95-03305, DEB-95-28445). ager (grazer). The pattern of effects of intraspeci®c variation in consumer pressure on microalgal density LITERATURE CITED was qualitatively similar both among the four pairs of Armstrong, R. A., and R. McGehee. 1980. Competitive ex- herbivores (Fig. 6) and with that seen for Tegula (Fig. clusion. American Naturalist 115:151±170. 3). At low herbivore densities (i.e., high area per con- Black, R., A. Limbery, and A. Hill. 1988. Form and function: size of radular teeth and inorganic content of faeces in a sumer), the area-extensive species reduced microalgae guild of grazing molluscs at Rottnest Island, Western Aus- to a greater extent than did area-intensive foragers. tralia. Journal of Experimental Marine Biology and Ecol- However, increases in herbivore densities produced ogy 121:23±35. comparatively smaller declines in microalgae for area- Branch, G. M. 1984. Competition between marine organ- isms: ecological and evolutionary implications. Oceanog- extensive than for area-intensive foragers (Fig. 6). As raphy and Marine Biology Annual Reviews 22:429±593. a consequence, area-intensive species depressed food Branch, G. M., and M. L. Branch. 1980. Competition be- to a lower level at some high consumer density. tween Cellana tramoserica (Sowerby) (Gastropoda) and A remaining issue is whether the area-intensive± Patirella exigua (Lamarck) (Astroidea), and their in¯uence area-extensive foraging dichotomy results in predict- on algal standing stocks. Journal of Experimental Marine Biology and Ecology 48:35±49. able and consistent patterns of density-dependent Briggs, C. J. 1993. Competition among parasitoid species growth responses of co-occurring herbivores. Evidence on a stage-structured host and its effect on host suppression. from the literature on other pairs of consumers suggests American Naturalist 144:372±397. that the qualitative patterns of density-dependent Briggs, C. J., R. M. Nisbet, and W. W. Murdoch. 1993. Co- growth exhibited by Tegula (Fig. 4) may be a general existence of competing parasitoid species on a host with a variable life cycle. 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APPENDIX DIGGER±GRAZER MODEL ANALYSIS
Ϫ1 Resource equations ration rates ␦D and ␦G (time ). These assumptions yield Our model, formulated at behavioral time scales of con- dM sumers, assumes three discrete resource levels with habitat D ϭ␥ M ( A ϩ A ) Ϫ␦ M (A.5a) dt D D D 10 1 20 2 D D fractions A0, A1, and A2 (A0 ϩ A1 ϩ A2 ϭ 1). Resource con- sumption is directly proportional to resource density and con- dMG sumer biomass density. These assumptions lead to resource ϭ␥GGM G212 A Ϫ␦ GM G (A.5b) dt equations
where ij ϭi Ϫj represent the resource biomass differences dA0 ϭϪ␣00A ϩ DDB (A 1ϩ A 2) (A.1a) between levels i and j. Substituting the quasi-steady-state dt resource fractions from Eq. A.3 into Eq. A.5, we obtain
dA1 1 dMD␣ 100( GGB ϩ DDB ) ϩ 2001 ␣␣ ϭ␣00A Ϫ␣ 11A Ϫ DD1BAϩ GG2BA (A.1b) dt ϭ␥DD Ϫ␦ D MdtD(␣ϩ 1 GGB ϩ DDB )(␣ϩ 0 DDB )
dA2 ϭ␣A Ϫ BAϪ BA (A.1c) ϭ f (BDG, B ) (A.6a) dt 11 DD2 GG2
1 dMG01␣␣ where BD ϭ NDMD and BG ϭ NGMG represent the digger and ϭ␥GG21 Ϫ␦ G grazer biomass densities (biomass per area) resulting from a MdtG(␣ϩ 1 GGB ϩ DDB )(␣ϩ 0 DDB ) product of the respective number densities ND or NG (number ϭ g(BDG, B ). (A.6b) of consumers per area) and individual consumer mass MD or Ϫ1 MG (biomass per consumer). Parameters ␣0 and ␣1 (time ) Single-consumer systems are the transition rates of the lowest and middle-resource We are interested in two experimental treatments, grazers levels to the next highest level, and D and G (area cleared per consumer biomass per time) denote the individual digger only (BD ϭ 0) and diggers only (BG ϭ 0), considered in turn. and grazer clearance rates. In both treatments, we linearize the resource density and con- Quasi-steady-state resource.ÐWe assume, throughout our sumer growth functions to simplify data analysis. analysis, that changes in resource abundance occur quickly Grazers-only system.Ð1. Algal response.ÐWhen only relative to the longer time scales associated with the consum- grazers inhabit the system (BD ϭ 0), the total resource density ers. This assumption enables equilibrium analysis of the re- (Eq. A.4) simpli®es to source equations, providing equilibrium resource fractions as ␣12 a function of consumer biomass. Thus, setting the time de- ϩ1GB rivatives in Eqs. A.1b and c to zero gives G ATG(0, B ) ϭ . (A.7) ␣1 0 ϭ␣01211DD1GG2(1 Ϫ A Ϫ A ) Ϫ␣A Ϫ BAϩ BA (A.2a) ϩ BG G 0 ϭ␣A Ϫ ( B ϩ B )A (A.2b) 11 DD GG 2 Linearization proceeds by assuming that the consumer density and leads to is a large quantity, allowing a series expansion,
B ␣12 1 DD ␣12 A (B ) ϭ (A.3a) ϩ1GB 0D ϩ1 ␣ϩ0DDB GGB GGB ATG(0, B ) ϭϭ 1 ␣1 ␣0GG( B ϩ DDB ) ␣1 1 ϩ A1D(B , B G) ϭ (A.3b) ϩ BG B (␣ϩ1GGDD0DDB ϩ B )(␣ϩB ) GGB GG
␣␣01 ␣12 ␣ 1 A2D(B , B G) ϭ (A.3c) ഡ ϩ1 1 Ϫ (␣ϩ1GGDD0DDB ϩ B )(␣ϩB ) GGB GGB for the three resource fractions, given consumer biomass den- ␣1 1 sities B and B . These equations provide the total resource ഡ ϩ121(Ϫ) , (A.8) D G B density after multiplying each fraction by the appropriate re- GG source biomass density parameters, 0, 1, and 2 (resource keeping terms up to ®rst-order in 1/BG. In the expansion, we biomass per area), representing the amount of resource bio- make use of the standard Taylor series expansion, 1/(1 ϩ x) 2 3 mass contained in a unit area if the habitat fraction Ai ϭ 1. ഡ 1 Ϫ x ϩ x Ϫ x ϩ ...,forͦxͦ Ͻ 1. Total resource density is then 2. Grazer growth.ÐWhen in isolation from diggers (BD ϭ 0), the grazer's biomass dynamics (Eq. A.6b) can be expressed as AT(B D, B G) ϭ 0A 0(B D) ϩ 11A (B D, B G) ϩ 22A (B D, B G)
1 dMG211GG␣␥ ϭ [0DDB (␣ϩ 1 GGB ϩ DDB ) ϭϪ␦G MdtG1GG␣ϩB
ϩ␣10( GGB ϩ DDB ) ϩ␣␣ 201] ␣␥21 1 G G ഡ Ϫ␦G Ϭ [(␣ϩ1GGDD0DDB ϩ B )(␣ϩB )]. GGB (A.4) 1 ഡ ␣␥21 1 G Ϫ␦ G (A.9) Consumer equations BG
Consumed resource biomass is converted into individual where the ®nal expression is valid for ␣1 K GBG, providing consumer mass with ef®ciencies ␥D and ␥G (consumer biomass a linear dependence of growth on 1/BG. per resource biomass), and diggers and grazers have respi- Diggers-only system.Ð1. Algal response.ÐWhen only 2372 W. G. WILSON ET AL. Ecology, Vol. 80, No. 7
diggers inhabit the system, BG ϭ 0 and the total resource 1 dMD100GG2001␣( K ) ϩ ␣␣ density is given by ϭ␥DD Ϫ␦ D Ͼ0 MdtD01GG␣ (␣ϩK ) ␣ ␣ 2 12 D 10 D 2 ␣␥21 1 G G ϩϩ1D0DB ϩ B Ϫ␣ϩ␣␦10 1 20 1 G ␣␣00 ␣ 0 ␣ 0 ␦G ATD(B ,0)ϭ . ␥DD Ͼ␦ D ␣D1D ␣␥21 1 G G 1 ϩ BDDϩ B ␣␣␣000 ␦␦GD (A.10) ϩ10 Ͼ ␥GG ␥ DD Series expansion for high digger density follows the calcu- ␦D lation performed for the grazers-only case, ϩ10 21A 2(0, K G) Ͼ . ␥DD A (B ,0) TD (A.14) ␣ ␣ 22 ␣ The ratios of metabolic rate to resource utilization represent 12 D 10 D2 0 ϩϩ1D0DB ϩ B each snails' resource availability required for persistence. ␣␣00 ␣ 0 ␣ 0 DDB ϭ Digger invasion of the grazer-only system occurs if inequality 2 (A.14) is satis®ed, and requires that the digger's persistence ␣␣D1D 0 1 ϩ BDDϩ B level be less than the equilibrium resource available to it. In ␣␣␣000DDB this case, resource available to the digger is composed of two 2 parts: the resource at the middle level, which is unused by ␣10 ␣ 0 ␣ 12 ␣ 0 ϩ01 ϩ ϩ the grazer, and the equilibrium resource density at the high- ␣0DD0B ␣ DDB resource level, shared with the grazer. ϭ Grazers invading diggers.ÐGrazers will be able to invade ␣␣01 the equilibrium digger-only system if 1 ϩ 1 ϩ DDB DDB 1 dMG ϭ␥GG212A (K D,0)Ϫ␦ G Ͼ0 MdtG ␣10 ␣ 0 ␣ 0 ␣ 1 ഡ ϩ01 ϩ 1 Ϫ 1 Ϫ ␣0DDDDDDB B B ␦G Ͻ21A 2(K D, 0) (A.15) ␥GG ␣0 1 ഡ ϩ010(Ϫ) . (A.11) where K is the steady-state digger biomass and A (K ,0)is B D 2 D DD the equilibrium high-level resource fraction of the digger- only system. Thus, this invasion condition requires that the 2. Digger growth.ÐWhen in isolation from grazers, the equilibrium high-level resource fraction left behind by the digger's biomass changes according to diggers be greater than the persistence level of the grazers. This grazer invasion condition requires solving the full set 1 dMD100DD2001␣B ϩ ␣␣ of equations determining the digger equilibrium, ϭ␥DD Ϫ␦ D MdtD0DD1DD(␣ϩB )(␣ϩB ) dA 1 ϭ 0 ⇒ ␣ (1 Ϫ A (K ,0)Ϫ A (K , 0)) dt 01D2D (␣10 0 DB D) ഡ ␥DD Ϫ␦ D (DDB )( DDB ) ϭ (␣ϩ1DD1DK )A (K , 0) (A.16a) 1 dA2 ⇒ ഡ ␥D100 ␣ Ϫ␦ D (A.12) ϭ 0 ␣11A (K D,0)ϭ DD2KA(K D, 0) (A.16b) BD dt 1 where the ®nal expression is valid for high digger densities. dMD ⇒ ϭ 0 10A 1(K D,0)ϩ 20A 2(K D,0) MdtD Invading single-consumer systems ␦ ϭ D (A.16c) Our analysis assumes a very limited and strict de®nition ␥DD for coexistence: if in each steady-state, single-consumer sys- which yields tem, the other consumer would have a positive growth rate, 1 ␦D then the two species coexist. A2D(K ,0)ϭϪ 101DA (K , 0) (A.17) Diggers invading grazers.ÐWhen the grazers exist alone ␥20 D D and are at equilibrium, the grazer biomass achieves a steady- where state value K determined by g(0, K ) ϭ 0, or, 1 y G G A (K ,0)ϭ ͙y2 Ϫ 4xz Ϫ 1D 2x 2x 1 dM ␦␣ Gϭ 0 ⇒ Gϭ 21 1 ␣21 1 20 MdtGGG1GG␥ ␣ ϩK x ϭϪ1 ␣10 0 10 ␣ (␥Ϫ␦) K ϭ 121GG G. (A.13) G ␦D 20 ␣ 1 202␦ D ␦GG y ϭ 1 ϩϩϪ ␥10 D D 10 ␣ 0 10 ␥ 10 D D Under these conditions, diggers face a growth rate given by ␦␦DD f (0, KG) (Eq. A.6a), which must be positive for successful z ϭϪ. 2 20 invasion, ␥10DD ␥ DD