vol. 162, no. 2 the american naturalist august 2003 Spatial Dynamics in Model Plant Communities: What Do We Really Know? Benjamin M. Bolker, 1,* Stephen W. Pacala, 2 and Claudia Neuhauser 3 1. Zoology Department, University of Florida, Gainesville, Florida 32611-8525; 2. Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey 08544-1003; 3. Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, Minnesota 55108 Submitted March 14, 2002; Accepted December 20, 2002; Electronically published July 21, 2003 abstract: A variety of models have shown that spatial dynamics and small-scale endogenous heterogeneity (e.g., forest gaps or local resource depletion zones) can change the rate and outcome of competition in communities of plants or other sessile organisms. However, the theory appears complicated and hard to connect to real systems. We synthesize results from three different kinds of models: interacting particle systems, moment equations for spatial point processes, and metapopulation or patch models. Studies using all three frameworks agree that spatial dynamics need not enhance coexistence nor slow down dynamics; their effects depend on the underlying competitive interactions in the community. When similar species would coexist in a nonspatial habitat, endogenous spatial structure inhibits coexistence and slows dynamics. When a dominant species disperses poorly and the weaker species has higher fecundity or better dispersal, competition-colonization trade-offs enhance coexistence. Even when species have equal dispersal and per-generation fecundity, spatial successional niches where the weaker and faster-growing species can rapidly exploit ephemeral local resources can enhance coexistence. When interspecific competition is strong, spatial dynamics reduce founder control at large scales and short dispersal becomes advantageous. We describe a series of empirical tests to detect and distinguish among the suggested scenarios. Keywords: spatial, competition, competition-colonization, successional niche, phalanx, endogenous. Spatial heterogeneity at many different scales can change the rate and outcome of competitive interactions. In the past decade, attention has focused on the effects of short- * Corresponding author; e-mail: bolker@zoo.ufl.edu. Am. Nat. 2003. Vol. 162, pp. 135 148. 2003 by The University of Chicago. 0003-0147/2003/16202-020110$15.00. All rights reserved. distance dispersal and small-scale, endogenous resource heterogeneity such as light gaps in forests (Dalling et al. 1998). Empirical and theoretical studies have demonstrated the effects of small-scale endogenous heterogeneity on invasion, productivity, succession, and diversity in communities ranging from bacterial assemblages to forests (Chao and Levin 1981; Tilman 1993, 1994; Pacala and Deutschman 1995; Rees et al. 1996). Given that dispersal and competitive interactions are fundamentally restricted in space, often to scales only a few times the size of a single organism, we might expect short dispersal and small-scale heterogeneity to have nearly universal effects on community dynamics. The dynamical richness and technical challenge of spatial processes have encouraged the proliferation of mathematical models for endogenous spatial pattern formation (Levins and Culver 1971; Shigesada et al. 1979; Durrett and Levin 1998; Gandhi et al. 1998). However, most of the models have focused attention on a single class of phenomena (clustering and competition-colonization trade-offs) at the expense of other possibilities. In addition, studies that attempt any kind of analysis have made strong simplifying assumptions about the scales and nature of spatial competition and lead to controversy and confusion when different models have conflicting assumptions (Grace 1990). This article tackles three basic questions: How does endogenous spatial structure affect competition and coexistence in simple competition models? Do these effects differ depending on the way in which space is modeled? How can these theoretical predictions be tested in the field? We find, reassuringly, that a variety of different models all make qualitatively similar predictions, provided that they incorporate the basic building blocks of discrete individuals, local competition, and local dispersal (Durrett and Levin 1994). When interspecific competition is weak relative to intraspecific competition and species would coexist nonspatially (with global dispersal and competition) and when competing species have similar life histories, endogenous spatial structure may tip the community from coexistence to exclusion in the long term (Neuhauser and Pacala 1999;
136 The American Naturalist fig. 2). However, spatial structure can also change the rate of community dynamics and slow competitive exclusion (Neuhauser and Pacala 1999). Short dispersal can handicap a competitively dominant species and allow competitive subordinates to coexist with a short-dispersing dominant species; this phenomenon is a form of the classical competition-colonization tradeoff (CC; Tilman 1994; Holmes and Wilson 1998). Short dispersal can benefit a competitively subordinate species that reproduces faster than the dominant by allowing it to concentrate its reproductive effort in areas that are temporarily free from interspecific competition; this strategy has variously been called competition (Grime 1977), exploitation (Bolker and Pacala 1999), or a successional niche (Pacala and Rees 1998). Here, we call it the spatial successional niche (SSN), to avoid confusion with temporal successional niches that are maintained by disturbance. When interspecific competition is strong relative to intraspecific competition, short dispersal and small competitive neighborhoods benefit individuals by allowing local control of the environment. In spatial arenas, local founder control (dependence of competitive success within a small region on initial density) can still occur, but global founder control disappears; in a large enough habitat, one species will always win as long as both species start with positive densities everywhere in the habitat (Neuhauser and Pacala 1999). The combination of spatial localization and interspecific interactions strongly retards competitive dynamics (Gandhi et al. 1998). Next, we briefly review the different model frameworks that we and others have used to explore spatial community dynamics. We synthesize recent results from these different frameworks and show that all of them produce similar results that we can classify into the four basic phenomena previously described. We then outline a set of field experiments that test for and distinguish among various spatial phenomena that occur in natural communities in the absence of exogenous heterogeneity. (Although exogenous heterogeneity is obviously important in structuring natural communities, high-diversity communities do exist where no obvious exogenous heterogeneity is acting to maintain diversity.) Such experiments would fill a major gap because the rich theory of community spatial dynamics remains largely untested by manipulative experiments except in bacterial communities (Chao and Levin 1981; Korona et al. 1994; Rainey and Travisano 1998). Review and Synthesis of Spatial Phenomena Modeling Frameworks The wide range of simplifying assumptions about the structure of space and the nature of competition used by different models makes it hard to compare their predictions. We have found, however, that several different model frameworks that we and others have explored over the past few years converge independently to the same conclusions. Interacting particle systems (IPS) are stochastic, continuous-time models with discrete individuals located in the cells of a square lattice; these models often offer qualitative insights into spatial dynamics rather than quantitative predictions (but see Silvertown et al. 1992 or Law et al. 1997 for counterexamples). Space is discrete rather than continuous in IPS models, but it is contiguous; competitive neighborhoods overlap and are connected in an explicitly spatial structure. Stochastic point processes also track discrete individuals that compete and disperse locally but assume that they occupy a single point in continuous (and, hence, also contiguous) space (Gandhi et al. 1998; Bolker and Pacala 1999; Dieckmann and Law 2000; see Diggle 1983 for the theory of static point processes). Stochastic point processes represent a low-density limit of IPS models where very few lattice cells are occupied but competition and dispersal also become long range. We have simulated stochastic point processes and also analyzed them with approximate moment equations that track the mean densities and spatial covariances within and between species (Bolker and Pacala 1999). Finally, in a limit where competition remains local but dispersal is global, we obtain noncontiguous patch models. These models differ from the metapopulation or patchoccupancy models used in many studies (Levins and Culver 1971; Tilman 1993) in that they allow multiple individuals and multiple species per patch. They allow some offspring to remain in the parental sites, which in turn allows spatial pattern to develop in the form of a multivariate distribution of the fractions of patches with particular combinations of species densities. Each of these approaches has its own strengths and weaknesses; using all three allows a choice of strengths and lends generality to the results. The IPS framework allows rigorous proof of the long-term coexistence or exclusion of species with different competitive and dispersal abilities in an infinite habitat (Durrett 1992; Durrett and Neuhauser 1994; Neuhauser and Pacala 1999) or approximation by means of pair approximations (Iwasa 2000). Spatial point processes provide the closest link to individual-based field studies and models, with all processes defined in terms of field-measurable life-history and spatial properties such as per capita fecundity and the density of seeds or offspring at a given distance from the parent. In addition, spatial moment equations allow analytical calculations of the shape and scale of equilibrium spatial patterns in a community. Finally, patch models provide a model framework that is much simpler to analyze or sim-
Spatial Dynamics in Model Plant Communities 137 ulate than either IPS or moment equations, and they give a point of comparison with popular and even simpler patch-occupancy or metapopulation models (Keeling 2000). We will discuss results from IPS, point processes, and patch models in order to demonstrate the strengths and weaknesses of each kind of model and to show how all three approaches converge on the same qualitative conclusions. Model Results: Classes of Phenomena Nonspatial Foundations. We start with the classical nonspatial Lotka-Volterra (LV) competition model: ( ) 1 2 ( n2 n1) n1 n2 ṅ1 p r1 1 b 12, K K ṅ2 p r2 1 b 21, (1) K K 2 1 where r i (for i p 1, 2) is the intrinsic exponential growth rate, K i is the carrying capacity when each species is grown in monoculture, and b ij gives the relative strength of within- and between-species competition. (Our definition of b ij gives the competitive effect of a density of species j relative to its own carrying capacity K j rather than relative to the target species density K i, which simplifies the notation; using the more common notation would replace b ij with aij p [K i/k j]bij.) The parameters in the model describe whole-population demography rather than individual life history and competitive abilities and as such may be misleading in an evolutionary context; for example, increasing density-independent fecundity or decreasing density-independent mortality would increase both r and K rather than just r. However, the model can also be easily derived from an individual-based competition model with parameters f i (fecundity), m i (density-independent mortality), and a ij (per capita effect of competition; Yodzis 1989; Royama 1992; Bolker and Pacala 1999). The population-level parameterization is simply more convenient for describing the criteria for species coexistence. Although the LV model is a caricature of real competition processes, it provides a useful framework for discussing the possible outcomes of competition (fig. 1). It covers scenarios ranging from simple single-resource competition (when b12b21 p 1) to strongly asymmetric com- petition, as is commonly incorporated in patch-occupancy models (when b12 r, b21 r 0). The invasion crite- rion the conditions on life-history, competitive, and spatial parameters that allow one species to invade an established monoculture of another species for species i to invade species j in this model is b! 1. With appropriate ij definitions, this invasion criterion also applies to a wider range of models, where per capita competition is a possibly nonlinear function of linear combinations of species densities (Chesson 2000b; B. M. Bolker, unpublished manuscript). The LV model allows any of the possible outcomes of two-species competition: dominance by either species, coexistence of both species, or founder control (exclusion of one species by another depending on initial conditions, usually exclusion of an initially rare species by an initially common one). If we derive the LV equations from an underlying model where individuals compete for a single limiting (but well-mixed) resource, we obtain the same equations, but the plane of possible competitive parameters shown in figure 1 shrinks to a hyperbola that allows only single-species dominance. Although many processes (e.g., allelopathy or responses to herbivory or pathogens) can move interactions off of the single-resource hyperbola and allow coexistence or founder control in a nonspatial model, we focus on how spatial competition allows coexistence when the underlying nonspatial rules would allow only one dominant species to persist. The LV parameter plane also helps organize the different possible effects of spatial pattern formation on competitive dynamics. The possible phenomena, which we discuss in the following sections, are spatial segregation and clustering when the parameters lie in the coexistence region of figure 1 (quadrat I), competition-colonization and spatial successional niche dynamics in the single-species dominance regions (quadrats II and IV), and phalanx growth in the founder control region (quadrat III). Before going any further, we should clarify the meaning of spatial coexistence. Spatial coexistence occurs when endogenous or exogenous spatial structure allows species to coexist indefinitely (in the absence of demographic stochasticity) where they could not if all individuals experienced the average conditions in the environment, including spatially averaged population densities. Real communities can be difficult to homogenize in this way. Conversely, determining the nonspatial outcome of competition is simple in a model where we can set the effective density of each species to its spatial mean density (although finding the nonspatial equivalents of patch-occupancy models can be surprisingly difficult). Heuristically, one can think about the nonspatial outcomes as the hypothetical result of competition if all individuals competed and dispersed globally. Spatial Segregation and Clustering. The first, simplest competitive scenario is where two species could coexist in the nonspatial, homogeneously mixed case (fig. 1, quadrat I). Although such coexistence is not possible under competition for a single essential resource, there are many sit-
138 The American Naturalist Figure 1: Invasibility and coexistence regions for the Lotka-Volterra model and the nonlinear (nonspatial) competition model, with regions of interest marked. uations such as competition for multiple resources or attack by host-specific pathogens that lead to coexistence even in a homogeneously mixed environment. In this case, we are interested in whether endogenous spatial structure can prevent (rather than enhance) coexistence and in how it changes the timescale of competitive dynamics. When interspecific competition is much weaker than intraspecific competition (both b 12 and b 21 are small; fig. 1, region d), space does not change qualitative outcomes of competition; species that coexist in the absence of space still coexist in the IPS. We can prove coexistence rigorously in the high-density limit, where fecundity is much greater than mortality for both species and there is no empty space left in the habitat (Neuhauser and Pacala 1999). In contrast, if interspecific competition is nearly as strong as intraspecific competition ( b! but 1) for one species or the other (fig. 1, regions a1 and a2), the combination of local competition and discrete individuals actually reduces coexistence in the high-density limit. In the shaded regions, where the stronger competitor would almost outcompete the weaker in the nonspatial model, the stronger competitor can actually exclude the weaker because of the discreteness of the competitive neighborhood. In this parameter regime, only a low density of the weaker species can survive in the nonspatial model. If the competition neighborhood is small enough, fewer than one individual could survive within a competitive neighborhood; because individuals are discrete in this model, the weaker species can never establish. Neuhauser and Pacala (1999) prove this phenomenon rigorously for the highdensity limit. Similar phenomena occur in patch models and in the spatial moment equations. In patch models, we restrict movement by reducing the probability of global interpatch movement rather than the size of overlapping neighborhoods, but the results are roughly comparable. To explore this effect, we used an approach that uses forward equations to track the entire probability distribution of patch occupancies in an infinite ensemble of patches (Renshaw 1991). Within a patch, population dynamics follow a stochastic process that would converge on the LV equations (1) for large patch sizes; deaths occur at a constant densityindependent rate, and fecundity decreases linearly according to the combination of con- and heterospecific population densities shown in equations (1) down to 0 when N K. If births are globally distributed among patches when they occur ( m p 1), the model converges to the LV model. If we restrict interpatch movement to 50% in a model with patch size (carrying capacity) K p 10, the co- existence region shrinks by approximately 10% ( 1/K) in the same regions where coexistence fails in an IPS (fig. 2). Reducing interpatch movement further (to 1%) more than doubles this contraction, and it reduces the size of the founder control region (see Phalanx Strategies ). In spatial point process models, intraspecific competition is always strengthened by a self-competition term (Bolker and Pacala 1999). An individual-based parameterization of the LV model starts with the statement that competition with an individual of species j at distance r decreases individual fecundity rate or increases individual mortality rate by an amount a ij U ij (r). (The connection with the population-level parameterization previously given is that K i p r i/aii and bij p [a ij/a ii][k j/k i].) In a randomly distributed or well-mixed conspecific population of density N i, the total strength of competition is a ii(ni U ii[0]), where the second term represents self- competition that does not occur in the nonspatial model. Self-competition changes competitive outcomes only if it changes b ij, which in turn changes only if the ratio of carrying capacities K i/k j p (r i aiiu ii(0))/(r j ajju jj(0)) is different from the original ratio K i /K j, which in turn requires U ii(0)/k i ( U jj(0)/k j. This criterion depends on the relative carrying capacities and the shape of the competitive neighborhood. In contrast, the more general neighborhood exclusion phenomenon proved for the IPS does not depend on details of spatial neighborhoods (Neuhauser and Pacala 1999) but may depend on the high-density limit. A final spatial phenomenon that does not alter the qualitative outcome of competition but does change the apparent strength of interspecific competition (changing the estimated location of a community in the parameter space shown in fig. 1) is the effect of spatial segregation among coexisting species. In nonspatial models with symmetric
Spatial Dynamics in Model Plant Communities 139 species, increasing interspecific competition (e.g., from increasing niche overlap) continuously increases the observed strength of competitive interactions between species. In the equivalent spatial model, however, interspecific spatial segregation also increases with increasing interspecific competition and reduces interspecific interaction. Removal experiments observing the increased performance of plants in an established community after the experimental removal of conspecific or heterospecific competitors are a standard assay of competitive strength. As we have shown (Pacala and Levin 1998), the endogenous spatial segregation generated by competition can lower the estimated strength of competition. For strongly competing, symmetric species (fig. 1, region b), spatial segregation is so strong that interspecific population dynamics are nearly neutral, as proved in the high-density limit by Neuhauser and Pacala (1999). This phenomenon might explain observed differences between quadrat-based and individualbased field estimates of interspecific competition based on removal experiments. Competition-Colonization Trade-Offs. A more familiar scenario is the case where one species is clearly competitively dominant in a nonspatial setting (e.g., in small-plot experiments pitting two heterospecific individuals against each other). Strong competitors may be better at preempting resources or may have some other advantage such as herbivore or pathogen resistance (Holt et al. 1994). In the LV model and other models that assume both spatial and temporal homogeneity (Tilman 1982), a superior competitor always wins, regardless of handicaps in fecundity or dispersal, so any coexistence must be the result of spatial structure. Our conclusions come from a variety of models that all converge to the LV in the appropriate nonspatial limit of large competition neighborhoods or patch sizes, so we can interpret nonspatial parameters such as relative strength of interspecific competition in terms of the LV parameters (fig. 1). In this case, we are looking at quadrats II and IV of figure 1, where one species dominates the other. One particular form of spatial coexistence is the competition-colonization trade-off (CC), where two or more species coexist in a spatial environment because the competitively dominant species have poorer colonizing ability: low fecundity, short dispersal, or both. The CC has been studied with a variety of models (IPS, spatial point processes, patch-occupancy models) and analytical tools (formal proofs [Durrett and Swindle 1991], pair approximation [Harada and Iwasa 1994], numerical studies [Holmes and Wilson 1998], and second-order moment equations [Bolker and Pacala 1999]). Competition-colonization trade-offs in patch-occupancy models allow a large number of species to coexist (Tilman 1994; Kinzig et al. 1999), but most fully spatial analyses have considered only twospecies interactions. The basic mechanism underlying all forms of CC is that colonization limitation of the dominant species leads to a lower overall (spatial mean) density of the dominant, which in turn allows the inferior species to invade. In contiguous-space models such as IPS and point processes, the dominant s density decreases because of spatial clustering, which enhances intraspecific competition and reduces mean density (Harada and Iwasa 1994; Holmes and Wilson 1998; Bolker and Pacala 1999). The CC is fundamentally spatial but is strongly enhanced if the dominant species also has low fecundity or high mortality so that its density is low (or it occupies only a small portion of the habitat) even in the absence of competition (Bolker and Pacala 1999). Formally similar trade-offs can also be constructed using nonspatial tradeoffs between seed size and number, for example (Adler and Mosquera 2000). In classical patch-occupancy models (Levins and Culver 1971), fecundity and dispersal ability are confounded in a single measure of nonlocal colonization (the number of offspring available to colonize outside the parent s site is determined by both the number of offspring produced Figure 2: Changes in coexistence regions with limited movement in the metapopulation Lotka-Volterra (LV) model. When all births are dispersed globally ( m p 1), the coexistence regions are equal to those of the stan- dard LV (fig. 1). When a fraction 1 m of births are retained locally, the coexistence and founder control regions (calculated by searching numerically for the b values where the invasion rate of each species is 0) shrink as shown. The triangles show the equivalent region of contraction for the interacting particle systems in the high-density limit. Parameters: intrinsic rate of increase r1 p r2 p 3.0, intrinsic reproductive number (competition-free expected offspring per generation) R 1 p R p 4.0, carrying capacity K p K p 10.0. 2 1 2
140 The American Naturalist and the fraction that disperse beyond the site boundaries). The distinction between low fecundity and short dispersal is important because the two forms of limitation have different implications; fecundity limitation is nonspatial while dispersal limitation is spatial. Fecundity and dispersal limitation combine to produce colonization, or recruitment limitation, which can be tested by adding recruits (seeds or young juveniles) to a site and observing whether population density increases. While the influence of recruitment limitation has been extensively tested in both marine and terrestrial communities (Clark and Ji 1995; Clark et al. 1999; Hubbell et al. 1999; Warner et al. 2000), the effects of dispersal limitation have begun to be studied more recently (Dalling et al. 1998; Ehrlen and Eriksson 2000; Jacquemyn et al. 2001; Verheyen and Hermy 2001; Webb and Peart 2001). There are biological as well as technical reasons to expect difficulty in separating fecundity from dispersal limitation. Late-successional species that invest heavily in structures for resource competition may, as a result, have fewer resources to invest in dispersal structures for their offspring. They are also under less pressure to colonize because their survival does not depend on escaping competition, although competition both with kin and with other conspecific neighbors will always give some reasons to disperse. They may also choose to produce a small number of large, short-dispersing seeds that have higher germination probabilities and early growth rates (Ezoe 1998; Levin and Muller-Landau 2000). Finally, these species are able to maintain long-term control of their local resource environment and may be playing a phalanx strategy against other strong competitors. Although CCs have been intensively studied by theoreticians, their influence is actually limited in continuousspace models. In simple models of homogeneous environments, CCs are limited to a small region of parameter space where strong competitors have extremely low fecundity and short dispersal (Bolker and Pacala 1999). When fecundity is realistically high (intrinsic reproductive number or lifetime reproduction in the absence of competition greater than two), intraspecific spatial clustering is weak or even negative and leads to evenly spaced populations and rules out a CC (Bolker and Pacala 1997). Competition-colonization trade-offs are more important in models such as patch-occupancy models that assume some spatial structure or heterogeneity, but even these models require the dominant species to be such a poor colonizer that its patch occupancy never gets high enough to exclude the inferior competitor. We suspect that exogenous heterogeneity and disturbance, which both increase the variance in population densities above the level that can be generated by endogenous processes, will enhance the effects of CC; for example, the apparent CC identified by Tilman (1988) in Minnesota sand plain communities was driven by disturbance at the scale of agricultural fields. Whether it is safe to extend the same framework we have used for endogenous dynamics to include disturbance and exogenous variability is an open question, although Chesson s (2000a) framework for spatial coexistence, which we discuss later, does lump endogenous and exogenous processes. Spatial Successional Niches. Competition-colonization trade-offs are not the only form of spatial coexistence under asymmetric competition; competitively inferior species can also gain from short dispersal. In previous work, we have termed this scenario a successional niche (Pacala and Rees 1998) or an exploitative strategy (Bolker and Pacala 1999); here, we call it a spatial successional niche (SSN). Even when the dominant species does not cluster (which it must do to make a CC work), the inferior species can persist by dispersing its offspring nearby to exploit local, endogenous heterogeneity in the resource environment. The SSN strategists gain the greatest advantage from short dispersal by reproducing quickly to exploit gaps before competitively superior species arrive and take over. We have found this result in both moment equations for point process models, where it appears as part of the spatial advantage arising from spatial segregation between species (Bolker and Pacala 1999), and IPS. In IPS, the SSN requires that the inferior species reproduce on a fast timescale and that the empty spaces unfilled by the dominant species percolate form unbroken corridors across the environment. When these conditions hold, the inferior species can always find empty space to occupy; the corridors do open and close because of stochastic birth and death in the dominant species, but because of its fast timescale, the inferior species can never be trapped and become extinct. (While it would seem that long-distance dispersal benefits the inferior species in this case, it is true only if dispersal is somehow restricted to corridors and avoids the majority of the habitat that is occupied by the dominant species.) As with the CC, the SSN encompasses both spatial mechanisms (local dispersal) and nonspatial mechanisms (fast growth; Pacala and Rees 1998). Grime s (1977) competitive species exploit a nonspatial successional niche by growing rapidly. Local dispersal and rapid growth reflect different geometries of local growth. New growth of a modular organism can be distributed horizontally, as in clonal swards and grasses, or vertically, as in woody trees. The asymmetric and time-dependent nature of heightstructured competition makes vertical growth qualitatively different from horizontal growth and harder to model; there are no analytical models that fully incorporate the trade-offs among vertical growth, horizontal growth, and
Spatial Dynamics in Model Plant Communities 141 dispersal (Pacala et al. 1993; Pacala and Tilman 1994). In contrast to CCs, where nonspatial traits (high fecundity) and spatial traits (good dispersal) co-vary in existing species, nonspatial aspects of successional niches may go along with spatial CCs. For example, early-successional canopy trees often have fast growth, possibly corresponding to a nonspatial successional niche; long dispersal, corresponding to a spatial CC; and high fecundity, which benefits both strategies. There are too many other trade-offs in plant strategies (seed mass and number, germination probability, dispersal ability, etc.) to do a complete accounting or a complete comparative analysis (Rees 1996), but the partitioning of spatial strategies into categories that are robust to differences in model structure is an important first step. All three of our modeling approaches (IPS, pointprocess, and patch models) suggest that SSN strategists must have a short generation time relative to their competitors. We suspect, therefore, that SSNs will be more important for strongly asymmetric competition between life forms such as grasses and trees rather than for withinguild competition or competition between more similar guilds such as early- and late-successional trees. Although competitive interactions between grasses and trees are often assumed to reflect a trade-off between utilization of water and light, grasses could also persist through a smallscale SSN strategy (and through an interaction with fire frequency in many ecosystems). As in the case of CC, disturbance may also increase the strength of SSN (preliminary explorations of patch models suggest that both CC and SSN increase with mild amounts of correlated within-patch disturbance (B. M. Bolker, unpublished manuscript), but adding either disturbance or other forms of exogenous heterogeneity blurs the distinction between SSN and other forms of spatial coexistence such as temporal successional niches. It may seem hard to distinguish between CC and SSN strategies; among other things, loose semantics allow one to say that any spatial persistence strategy represents a colonization advantage. However, there is a simple operational test that distinguishes CC from SSN. If a focal species benefits from CC, increasing its dispersal distance will give it minor benefits (because it has good colonization ability already), while increasing a competitor s dispersal distance will hurt the focal species by eliminating the colonization advantage. In contrast, increasing the dispersal distance of a focal species that benefits from SSN will hurt it by removing its ability to retain seeds in good areas; in this case, increasing the competitor s dispersal will have little effect. Phalanx Strategies. Finally, we come to the scenario where interspecific competition is stronger than intraspecific competition (either above the single-resource hyperbola in fig. 1 or more strictly in the upper right quadrat, regions c1 and c2). Here, spatial dynamics affect both the timescales of community dynamics and the outcome of competition. Strong interspecific competition leads to strong spatial segregation in this parameter regime, which in turn slows down competitive dynamics. Gandhi et al. (1998) have shown that in a spatial point process starting from random initial conditions, monospecific patches form in the first phase of competitive dynamics as the locally denser species in each neighborhood excludes the sparser species. Thereafter, the stronger competitor encroaches on patches of the weaker competitor but at a drastically reduced pace that would be equivalent to decades or millennia in most ecological systems. The qualitative change in competitive dynamics is that unlike in the nonspatial model, founder control never occurs; in infinite space and time, the stronger competitor (species 1 in region c1 or species 2 in region c2 of fig. 1) always wins eventually. Neuhauser and Pacala (1999) rigorously proved that the founder control region is reduced in the high-density limit model; in a long-range limit, one can show that the founder control region disappears. From an ecological point of view, however, one may observe either local founder control dominance of a local region by a weaker competitor that started at higher density or, at a slightly larger scale, apparent coexistence stability of large monospecific patches over ecological timescales. Some authors have concluded on the basis of simulation studies that different species can coexist in this regime (Solé and Bascompte 1997; Molofsky et al. 2001), even though competitive exclusion can be proved rigorously for long enough timescales; moment equations and pair approximations also give the wrong result in this regime (strong interspecific competition) because they underestimate the effects of large-scale spatial structure (Iwasa et al. 1998). While indefinite coexistence is impossible in this regime, apparent coexistence over many generations is a real possibility (Frelich et al. 1993). These timescale effects apply more generally and weaken interspecific competition relative to intraspecific competition whenever strong spatial segregation occurs. Spatial segregation among similar species could amplify the effects of recruitment limitation that have been shown to slow down competitive exclusion in some cases (Hurtt and Pacala 1995). In particular, recent theories of the dynamics of neutral communities (Hubbell 2001) could be strengthened and reconciled with the apparent importance of competition on an individual scale, through the effects of small-scale interspecific and interguild spatial segregation. If species within a guild are symmetric and similar in their ability to compete for resources, endogenous spatial seg-
142 The American Naturalist regation could slow dynamics to effective neutrality on ecological timescales (although Chesson and Huntly [1997] argue broadly that mechanisms that simply slow exclusion are not enough to account for diversity). This phenomenon could be tested by analyzing the neighborhood densities within a community to see whether interspecific association (within guilds) is much rarer than expected from a random spatial distribution, although the confounding effects of environmental heterogeneity would have to be taken into account. We call the scenario of strong spatial segregation in the founder-control region phalanx growth, by analogy with the phalanx/guerrilla dichotomy described by observers of growth forms in clonal plants (Lovett-Doust 1981). In nutrient-rich habitats, clonal plants tend to form tightly aggregated patches, whereas in nutrient-poor habitats, they extend their stolons and disperse new ramets farther from existing ones. Spatial moment equations show that this strategy works best when the intrinsic reproductive number (offspring per generation in a noncompetitive environment) is large (Bolker and Pacala 1999), which corresponds well with a nutrient-rich environment. This empirical observation suggests that a spatial competition strategy may be an important part of the community dynamics of competing clonal swards. As we will discuss later in more detail, we could test this prediction by relocating ramets randomly to see whether they gain a competitive advantage by forming patches or whether the patches have more to do with the spatial scale and variability of the resource environment. Empirical Tests No matter how clearly it seems to explain the organization of competitive communities, any categorization of spatial strategies stands or falls on its ability to be tested in the field. While recruitment limitation (which could arise from any combination of fecundity, microsite, or dispersal limitation) has been both empirically tested and noted in the field (Clark and Ji 1995; Clark et al. 1999; Hubbell et al. 1999), tests of explicitly spatial phenomena are rarer. These tests usually consist of carefully calibrating a spatial model and then using it to contrast the theoretical estimate of the nonspatial behavior of the community with the dynamics (productivity, species coexistence, etc.) actually observed (Pacala and Silander 1987, 1990; Pacala and Deutschman 1995; Rees et al. 1996; Clark et al. 1998). A related approach uses more generic models of competition to extract competition coefficients from observational data on an appropriate scale (Law et al. 1997; Freckleton and Watkinson 2001); these estimates could, in principle, be compared with nonspatial estimates averaging competition over the entire habitat to observe the effects of endogenous spatial structure on the effective strength of intra- and interspecific competition. We know of only one published example of a manipulative experiment designed to test endogenous spatial mechanisms in plant communities (Stoll and Prati 2001), although there are several such tests of bacterial communities, as previously cited (Chao and Levin 1981; Korona et al. 1994; Rainey and Travisano 1998). We propose empirical tests to discriminate between different mechanisms of spatial coexistence. These range from purely observational tests of field systems to calibration of data-driven models to manipulative experiments. Clearly, there is no magic bullet and no substitute for the natural history and experimental work required to establish the basic processes that operate in a community. We hope, however, that this discussion will clarify what observations are necessary to establish the existence of spatial competition mechanisms and clarify their nature in a particular plant community. One obvious shortcoming of the tests we propose is that they distinguish only among different endogenous spatial phenomena; they attribute any spatial structure observed to endogenous processes. In experimental systems, one can eliminate exogenous heterogeneity by the usual means of tilling, shade cloth, uniform watering, or fertilization, and so forth, but one can never be absolutely sure that the system has really been homogenized. We feel that exploring endogenous mechanisms is a first step toward understanding the full complexity of plant communities, including both endogenous and exogenous heterogeneity and their interactions. How can we discriminate among the different spatial scenarios: symmetric species, competition-colonization, SSN, and phalanx growth? We might already know something about the relative strengths of inter- and intraspecific competition and the relative competitive abilities of the two species; then, we could tentatively locate the system on the plot shown in figure 1, which would narrow the possibilities. However, figure 1 is qualitative rather than quantitative, and it would be difficult in practice to reduce the complexity of a real community to a point in the parameter plane. In addition, finding competitive asymmetry between two species would not resolve whether CCs, SSNs, or both were operating. Instead, we suggest a series of qualitative experiments that will discriminate among the different alternatives. The most straightforward way to test the effects of local dispersal and spatial structure is to relocate individuals so that their local competitors become a random sample of the population. (Note that these experiments are very different from classical recruitment limitation experiments, which supply additional seeds or recruits to an area rather than randomizing the spatial patterns of individuals.) All
Spatial Dynamics in Model Plant Communities 143 Table 1: Experiments for discriminating different spatial scenarios Randomize species 1 Randomize species 2 Conclusion 2 f 20/F Species 2 maintained by CC 10/F 1 f Species 1 maintained by CC 1 f No change Species 1 maintained by SSN No change 2 f Species 2 maintained by SSN 1 f 2 f Phalanx/spatial segregation Note: Up and down arrows denote increases and decreases, respectively, in fecundity, growth, or survivorship relative to control treatments; 0 denotes no change for one species. CC p competition-colonization trade-off; SN p spatial successional niche. of the experiments we suggest are, unfortunately, practically limited to communities of small- or medium-size plants, both by the logistics of transplanting trees and by the timescales required to see an effect, which are on the order of the generation time; forest ecologists will have to continue to calibrate models to determine spatial effects. For logistical reasons, for the ability to distinguish ecologically distinct individuals (those unconnected by a network of underground structure), and in order to avoid transplant shock, it will be easiest to do these experiments with plants that reproduce primarily from seed. Annual plants would be best of all; everything that we describe in terms of transplants could be done by much easier seed manipulation. Randomizing the positions of all individuals is a test of spatial coexistence itself and reduces the system to its nonspatial equivalent by forcing all individuals to compete equally (or at least randomly) with all other individuals. The control treatment is to swap the positions of conspecifics randomly, which will mimic the effects of the randomization treatment but preserve the hetero- and conspecific neighborhood densities of all individuals. If randomization leads to a significant change in densities or population persistence, we would then want to see which spatial mechanism is operating. We modify the previous experiment by randomizing the position of only one species at a time (always controlling for transplant effects by digging up and replanting the sedentary species in the same positions or randomly interchanging the positions of sedentary individuals; table 1). If the focal species benefits from randomization and increases in relative abundance or drives the other species to extinction, we conclude that it is naturally colonization limited. If the focal species is hurt by randomization and decreases in relative abundance or becomes extinct, we conclude that it exploits endogenous spatial pattern in the environment to survive. If one species clearly dominates in the control treatment and the other exploits spatial pattern, then we conclude that it occupies a SSN. If neither species dominates and both are hurt by randomization, we can infer a phalanx scenario. If neither species benefits consistently from the experimental treatment but the timescale of competitive dynamics accelerates significantly, we infer that the system is symmetric and experiences either spatial segregation of coexisting species or phalanx competition. In the spatial segregation case, randomization will not change the coexistence or diversity of the community, but it will increase resilience or the speed at which the community recovers from a change in relative population densities. In the phalanx case, randomization will increase the speed of extinction and possibly induce founder control (as in Chao and Levin s [1981] experiments), although the timescales of plant communities may be too slow to see this effect. In this case, if seeds or seedlings (preferably grown in a common garden) are available, we can assemble communities from scratch by planting or transplanting individuals of different species in predetermined random or structured configurations. These assembled communities distinguish between clustering in the coexistence region and clustering in the founder control or phalanx region of figure 1. These cases are hard to separate by nonmanipulative methods because spatial segregation slows the rate of population dynamics severely. In the coexistence region, single individuals or small clusters of an invading species planted within a monoculture of a resident species should increase their local density and/or expand over the course of a few seasons; in the phalanx region, they should disappear as they are overwhelmed by the greater local density of the resident. If interspecific dynamics are too close to neutrality, it could still be impossible to tell coexistence from exclusion, but this protocol at least offers a way to separate individual-level neutral dynamics from the reduced interspecific interactions caused by spatial segregation. Stoll and Prati (2001) sowed herbaceous plants in random and intraspecifically aggregated mixtures and measured biomass and reproductive output; they found that stronger competitors performed worse and weaker competitors performed better in the aggregated treatment. In our terminology, this experiment shows the results of both CC (reduced performance of stronger competitors) and SSN (increased performance of weaker competitors; if the
144 The American Naturalist overall neighborhood density is maintained, intraspecific aggregation implies interspecific segregation). A completely different way to test and measure the strength of different spatial scenarios is to calibrate a phenomenological or mechanistic individual-based model of the plant community by measuring local dispersal and neighborhood competition strengths. Ideally, such a model should also be tested, either against independent observational data or (even better) by using it to predict the outcome of controlled experiments (see van den Bosch et al. 1988 for an example from plant epidemiology). Once the model is calibrated and tested, one can run any experiment in the model, including making competition or dispersal global for one or both species, and can measure the effect of endogenous spatial structure on the persistence or invasibility of different species. Such a model can test and discriminate among any of the scenarios listed. Completely field-calibrated models have strong advantages but require a great deal of effort to parameterize and test; only a few models of this sort have actually been constructed and used to test spatial dynamics (Cain et al. 1991; Pacala et al. 1993; Rees et al. 1996). These experiments will require considerable investment and, as we previously discussed, are practically limited to small plants that reproduce from seed and mature and reproduce quickly. These specifications obviously rule out many important natural communities (forests, old-field communities dominated by long-lived perennials such as Solidago, etc.), but they are a starting point. We can always fall back on carefully calibrated models for those communities that resist experimental manipulation, but we should start with systems where we can do qualitative, manipulative experiments that are independent of specific models, and, if we do use models, we can check them against independent experimental (not just observational) evidence. Given the potential importance of spatial dynamics and given the near complete absence of experimental tests, experiments such as the ones we have suggested should be well worth the effort. Discussion We believe that our work and the work of many other mathematicians and mathematical ecologists over the past few years have finally made it possible to understand the basic spatial phenomena that underlie both simple strategic models and more complicated models used to understand applied ecological issues such as the generation and conservation of biodiversity. The results presented here suggest that we can understand many of the basic phenomena that underlie spatial competition equally well in patch, lattice, or point-process models. This is good news; in particular, it strengthens the case that the conclusions of the voluminous literature on patch dynamics could carry over, qualitatively, to landscapes that are not inherently divided into discrete patches. Models with explicit neighborhood structure such as lattices or point processes will still be needed to make quantitative or explicitly spatial predictions (such as exploring spatial scales and patterns of dispersal) and measure the relative strengths of different phenomena, but we can hope that patch models will suffice for elucidating basic spatial processes. While we have intentionally titled this article Spatial Dynamics in Model Plant Communities, it is reasonable to ask how important we think the various mechanisms outlined and endogenous spatial structure in general will be in real communities. We argued in the introduction to this article that short scales of dispersal and competition are ubiquitous; conversely, so are exogenous disturbance and heterogeneity, which might be expected to swamp endogenous effects. Some important natural and experimental systems such as the Cedar Creek Long Term Ecological Research site or the Rothamstead Park Grass experiment apparently support high diversity with minimal exogenous heterogeneity. However, these represent only a few (perhaps special) systems, and one can never be sure that exogenous heterogeneity is really absent; color polymorphisms in Linanthus parryae, long cited as an example of endogenous pattern in population genetics but now believed to be maintained by subtle exogenous heterogeneity, are but one cautionary example (Schemske and Bierzychudek 2001). More evidence for the importance of some kind of spatiotemporal mechanism, either endogenous or exogenous, comes from reviews of plant competition experiments and observations (Goldberg and Barton 1992; Gurevitch et al. 1992; Freckleton and Watkinson 2001). These reviews suggest that, in general, intraspecific competition is not much weaker than interspecific competition (as would be expected from models of resource partitioning), which rules out region Id in figure 1 but does not necessarily rule out regions Ia and Ib. (Then again, Rees et al. [1996] did find that interspecific competition was relatively weak.) These reviews did not generally try to separate systems on the basis of competitive symmetry (quadrats II and IV vs. quadrats I and III in fig. 1); however, asymmetry is at least anecdotally common in natural communities (two examples are the communities studied by Law et al. [1997] and Tilman [1988]). We do not know, in general, the likelihood of competitive near equivalence (b 12 or b21 1), which would put species pairs near the boundary of quadrat I, where strong segregation can occur and where small spatial effects could have important qualitative differences. (Hubbell [2001] argues for equivalence or near equivalence and bases his argument on macroscopic patterns of diversity rather than on competition experiments.)
Spatial Dynamics in Model Plant Communities 145 As for founder control (quadrat III in fig. 1), we know of no direct evidence for this scenario in plant communities (even in clonal plant communities where one might expect little to no intraspecific competition between ramets), although it has been found in bacterial communities (Chao and Levin 1981). Despite all the effort that has been put into field studies of competition, we still cannot make strong a priori predictions about what kinds of endogenous effects we expect. Caveats The models previously discussed are simple strategic or toy models (Nisbet and Gurney 1982). As such, they neglect many important complexities such as exogenous spatial and temporal heterogeneity (including disturbance and habitat fragmentation) and nonlinear per capita effects of competition (Chesson 2000a, 2000b), which are certain to shape natural competitive communities. How can we be sure that these phenomena do not somehow reverse our conclusions or that we have not missed some important strategic spatial axis? We have explored nonlinear patch models (like the patch models previously discussed but with nonlinear per capita competitive effects), with and without correlated disturbance that drives an entire patch population extinct simultaneously. In the absence of disturbance, SSN strategies still dominate CCs over much of parameter space. Alternatively, disturbance, which makes the model more similar to patch-occupancy models by emptying entire patches simultaneously, strengthens the effect of CCs (B. M. Bolker, unpublished manuscript). The only novel phenomenon seen in these nonlinear disturbance models is the effect of different curvatures in the effect of competition, which allows one species to do better in rare gaps while the other does better in common, medium-density patches (relative nonlinearity; Chesson 2000b). Peter Chesson (1984, 1985, 1990, 1994, 2000a, 2000b) has extensively explored and categorized the (largely exogenous) effects of spatial and temporal heterogeneity, and their interactions, in patch models. While Chesson s results may not carry over to contiguous-space models unchanged, the framework is sufficiently general that we can expect that many of the phenomena will be the same. Chesson s scheme divides spatiotemporal mechanisms into three general categories: relative nonlinearity, which allows some species to capitalize on rare, high-quality sites while others exploit average sites; temporal or spatial storage effects, where carryover from good sites and time periods allows an inferior competitor to persist; and densitygrowth correlation, where a species manages to inhabit good sites and time periods preferentially, either actively (through directed dispersal) or passively (by retaining propagules in good habitats). In terms of this framework, the SSN (and interspecific spatial segregation generally) represents a positive density-growth covariance for the inferior species; by reproducing on a fast timescale and retaining its propagules locally, a plant species induces a positive association between population density and good (enemy-free) habitat. In contrast, the CC represents negative density-growth covariance; in intraspecific competition, more individuals are necessarily found in bad (highdensity) neighborhoods. We have found that in the presence of exogenous heterogeneity, short dispersal adds an additional component to the density-growth correlation (B. M. Bolker, unpublished manuscript). Chesson s results suggest that nonlinearity, disturbance, and heterogeneity will add new spatiotemporal mechanisms but not qualitatively change the ones discussed here. In contrast to disturbances and heterogeneity, which can be understood at the level of a patch, the effects of fragmentation a particular form of exogenous heterogeneity that partially or completely cuts patches off from colonization are explicitly spatial and, thus, harder to understand. They are captured in some ways by our models when movement rates drop very low in patch models for example but a full exploration of fragmentation probably requires explicitly spatial models (Bascompte and Solé 1997; Fahrig 1998). Conclusions We are strongly encouraged by the convergence of three structurally different types of models IPS, spatial point processes, and patch models on the same qualitative conclusions. This convergence strengthens our conclusions, and it suggests, as one would hope, that the fine details of mathematical models do not affect qualitative predictions. However, there are two strong caveats to this statement. First, quantitative predictions of competitive outcomes in natural communities definitely will depend on choosing an appropriate structure and spatial scale with which to model the system. Second, there are important details as well as unimportant ones ( Everything should be made as simple as possible but not simpler ; Albert Einstein); as we stated at the outset, the properties of individual discreteness and local competition and dispersal are vital. In addition, models that describe only site occupancy or assume that sites are always filled at some stage in the annual cycle may miss important aspects of spatial dynamics, although increasing the number of different types of patches to allow temporary within-patch coexistence (Pacala and Rees 1998) or multiple resource levels (Wilson et al. 1999) is one way around this problem. We have shown that despite the potential richness of spatial dynamics, it is possible to generalize about the ef-
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