Jeremy Fox, NERC Centre for Population Biology, Imperial College, Silwood Park, Ascot, Berkshire SL5 7PY, UK. Tel. + 44 (0) 20 75942471. Fax: + 44 (0) 1344 873173. E-mail: firstname.lastname@example.org
1Theory predicts that communities comprising many species connected randomly by strong interactions are unlikely to exhibit stable, feasible equilibria (i.e. positive equilibria to which all species will return following perturbation), making the observed complexity of natural communities very surprising. This prediction has never been tested experimentally in speciose communities.
2We tested this prediction using long-term data from experimental communities of bacteria, protists and small metazoans in laboratory microcosms. Initial communities varied in species richness ( S ), composition and connectance ( C ; the fraction of possible interspecific interactions that are actually realized).
3Many initial communities lost species to extinction (unstable and/or infeasible communities), but population densities and species compositions subsequently stabilized.
4We used logistic regression to describe the probability that an initial community would be stable and feasible as a function of complexity [( SC ) 1/2 ]. Probability of a stable, feasible equilibrium declined with complexity, as predicted by theory.
5We also found that communities converged in connectance over time, so that at the end of the experiment connectance was independent of species richness. Monte Carlo simulations indicated that this result was highly improbable if extinctions occurred at random with respect to connectedness. Connectance is also independent of species richness in well-resolved natural food webs, a pattern usually ascribed to phylogenetic and/or physiological constraints on diet breadth. However, such constraints cannot explain our results. Trophic interactions causing non-random extinctions may provide a novel explanation for patterns in the connectance and species richness of whole food webs.
The connections between the complexity and stability of ecological communities remain a major focus for ecological research (Tilman 1996; McGrady-Steed, Harris & Morin 1997; McCann, Hastings & Huxel 1998). That the nature of the connections will depend on the precise definitions of ‘stability’ and ‘complexity’ has long been recognized (May 1973). The relationships between some aspects of complexity and stability are increasingly well understood (e.g. species richness and temporal stability of ecosystem function, Cottingham, Brown & Lennon 2001). However, the relationship between stability and complexity as originally defined in the seminal work of May (1973) remains obscure.
May (1973 ) defined stable communities as those where each population, if perturbed a small distance from its equilibrial density, would return eventually to that density. He concluded that stability was probable if
i ( SC ) 1/2 < 1 (eqn 1)
and improbable otherwise. Here, S is species richness and C is connectance. Interaction strength is assumed to follow a normal distribution with mean 0 and standard deviation i. Increasing complexity [defined by May (1973) as increases in S, C and/or i] decreases the probability of stability.
Equation 1 derives from probably the simplest set of assumptions that can be made about food web structure. Changing the assumptions changes the predicted relationship between complexity and stability. For example, May (1973 ) assumed randomly connected interaction webs, while real food webs are highly non-random ( Williams & Martinez 2000 ). The relationships between species richness, connectance, interaction strength and stability observed in real, non-random webs may not reflect the average relationships expected for randomly connected webs ( DeAngelis 1975 ; McMurtrie 1975 ; Yodzis 1981 ; Hogg et al. 1989 ; Haydon 1994 ; McCann et al. 1998 ). May (1973 ) assumed that all species are equally self-damped, although simulation studies suggest eqn 1 is robust to random variation in self-damping terms ( McMurtrie 1975 ). Variation in the degree of self-damping can enhance stability, particulary if weakly self-damped species interact with strongly self-damped ones ( Haydon 1994, 2000 ). May (1973 ) assumed implicitly that all equilibria were feasible (i.e. all species have positive equilibrial densities), but feasibility is at least as crucial for community persistence as is stability ( Roberts 1974 ; Law & Blackford 1992 ). Gilpin (1975 ) found that the probability of feasibility also declines with increasing complexity in random webs.
All these alternative models of stability and complexity are motivated by the fundamental assumption that unstable web configurations are unlikely to be observed in nature. This assumption is largely unproven; stability sensuMay (1973) may be irrelevant to explaining the structure of natural communities. Natural communities might be unstable but nevertheless persistent or ‘permanent’ (Law & Blackford 1992). However, speciose, highly connected communities are likely to be impermanent as well as unstable (Morton & Law 1997; Chen & Cohen 2001). Natural communities might not even be closed dynamic systems at all, but rather open systems with structures reflecting the influence of the surrounding biogeographical region (Ricklefs & Schluter 1993).
All these models can be viewed as alternative approximations to the more complex reality of natural communities (Wimsatt 1987). The empirical question is, which approximations are adequate? Unfortunately, relevant data are scarce, although two studies suggest that non-random patterns of weak and strong interactions stabilize natural communities (de Ruiter, Neutel & Moore 1995; Roxburgh & Wilson 2000a, 2000b). Empirical data are difficult to collect in part because stability is difficult to assess; an indirect approach would be to examine an index of stability. Simulations of simpe models by Taylor (1992) suggest that temporal variability of population dynamics is a good index of instability. Tilman (1996) found that variability of population dynamics increased with species richness in experimental plant communities, suggesting that more speciose communities were less stable, but the relationship was extremely weak. Lawler (1993a) and McGrady-Steed & Morin (2000) found little or no relationship between population dynamic variability and species richness in aquatic microcosm communities.
Another approach, complementary to comparing richness, connectance and interaction strength among natural communities, would be to assemble controlled experimental communities with known initial species richness and food web structure (Hall & Raffaelli 1996). The stability of the subsequent community dynamics could be compared to theoretical predictions. We have taken this approach by examining data from experimental microcosm communities of bacteria, protists and small metazoans (McGrady-Steed et al. 1997). These organisms have short (< 48 h) generation times, facilitating collection of long-term population dynamic data. Their diets are well known, allowing accurate description of food web structure. By maintaining closed communities under controlled conditions, we can exclude the effects of environmental variation in order to focus on the consequences of species interactions. Previous experiments used this approach to examine the stability of species-poor communities (Luckinbill 1979; Lawler & Morin 1993; Lawler 1993a,1993b; Weatherby, Warren & Law 1998). Here we use communities with a greater range of species richness and food web structures to ask the following questions: how does complexity [defined here as the square root of the product of species richness and connectance, (SC)1/2; see below] change over time? Are more highly connected species at greater risk of extinction (as might be expected if high connectance is destabilizing)? How does the probability that a web will be feasible and stable vary with complexity?
Materials and methods
MICROCOSM ASSEMBLY, MAINTENANCE AND SAMPLING
McGrady-Steed et al. (1997 ) assembled replicate aquatic communities of varying richness and composition in laboratory microcosms. Microcosm assembly, maintenance and sampling methods are described in detail elsewhere ( McGrady-Steed et al. 1997 ; Fox, McGrady-Steed & Petchey 2000 ; McGrady-Steed & Morin 2000 ), so only essential details are repeated here. Bacteria, protists, rotifers and gastrotrichs were added in different species combinations to 250 mL screw-capped Ehrlenmeyer flasks containing 100 mL of nutrient medium and four wheat seeds. Incubators maintained constant environmental conditions (23 °C, 14 : 10 light:dark cycle). Weekly replacement of 7% of the medium and bi-weekly replacement of one wheat seed provided supplemental nutrients.
The experiment involved a total of 39 eukaryotic species, used to create initial communities of varying richness (3–32 eukaryotic species) and composition (McGrady-Steed et al. 1997). All 39 eukaryotic species were isolated from the Rutgers Display Garden Pond (New Brunswick, NJ, USA). This comprised all the protists and protist-sized metazoans that could be grown in the laboratories and differentiated (based on appearance and behaviour) under a dissecting microscope. All communities shared the same bacterial species pool. Bacterial species could not be identified confidently, so we treated all bacteria as a single taxon.
Eukaryotic species were assigned randomly to initial communities, subject to three constraints. First, as far as possible, each community initially contained species from all major trophic groups (primary producers, herbivores, bacterivores and predators). Secondly, certain voracious predators (e.g. Dileptus anser L., Didinium nasutum Müller) that would have eliminated many prey species rapidly (Luckinbill 1979; Petchey 2000; personal observation) were excluded from the experiment. Thirdly, each herbivore and predator in an initial community could consume at least one other eukaryotic species in that community. The third constraint ensured that herbivores and predators did not starve immediately, but did not guarantee their persistence (McGrady-Steed et al. 1997; Fox et al. 2000). Other than the second and third constraints, no attempt was made to ensure that the species in a community could coexist. The third constraint makes initial species richness somewhat confounded with species composition and trophic structure (e.g. some initially depauperate communities lacked predators). To minimize confounding, McGrady-Steed et al. (1997) used several different species combinations to create alternative communities of initially identical richness within the lower portion of the richness gradient. Each initial community was replicated 1–5 times; replication varied due to minor contamination early in the experiment (see Fox et al. 2000). We treated contaminants as members of initial communities, following Fox et al. (2000). The experiment involved 24 initial communities of unique (but overlapping) compositions (described in Fox et al. 2000). Initial communities were randomly assigned to bottles.
Eukaryotic species were sampled approximately once every 4 days by counting all individuals in a ∼0·5 mL sample. This sample volume allows reliable detection of populations as sparse as 1–2 individuals/mL (most populations were much denser (McGrady-Steed & Morin 2000)). Bacteria were monitored with weekly plate counts. The experiment lasted 6 weeks, representing dozens to hundreds of eukaryote generations.
DEFINING FOOD WEB STRUCTURE
We assumed that all direct interspecific interactions were trophic (consumer–resource) interactions. Many studies of protist microcosms support this assumption (e.g. Kaunzinger & Morin 1998; Fox 2002). Given this assumption, the species richness and connectance of a community can be derived from knowledge of the species present and their diets. Diet information came from feeding trials and our own experience culturing these organisms. The only species for which we lack diet information are the primary producers and bacteria. Density compensation among primary producers suggests interspecific competition (McGrady-Steed & Morin 2000), but the resource(s) for which competition occurs are unknown. We defined our food web structures assuming that all primary producers consume a shared resource (‘light’), to reflect the possibility of competition among primary producers. ‘Light’ is a basal species in our food webs (i.e. it does not consume other species). We treated all bacteria as a single species which consumes ‘detritus.’‘Detritus’ is the other basal species in our food webs. Our assumptions about the diets of primary producers and bacteria are the most realistic assumptions that can be made on the basis of the available information. These assumptions should not affect our conclusions about variation in structure among food webs, since all webs included primary producers and bacteria. Some species-poor communities actually comprised two separate food webs (e.g. a web of algae and ‘light’, and a web of detritus, bacteria, and bacterivores). For simplicity, we treated separate webs as if they were unified webs.
We calculated the species richness (S), connectance (C), and complexity of each initial community, and the final values after six weeks of development (many webs lost species to extinction during the experiment; see below). Each feeding link (species A eats species B) adds two effects to the community matrix (an effect of A on B, and an effect of B on A) (May 1973; note that mutual predation – species A eats B and vice-versa – and cannibalism are absent from our communities). Connectance sensu May (1973) is given by C= 2 L/S(S − 1), where L is the number of links in the web. Although interaction strength affects stability and feasibility (May 1973; Gilpin 1975), we lacked data on interaction strength, and so measured complexity as (SC)1/2.
We determined both stability and feasibility because theory suggests both are likely to constrain community structure (Gilpin 1975). In the long run, we do not expect to observe communities that lack either feasibility or stability. We determined stability and feasibility by inspecting community dynamics. Initial communities can be thought of as communities that have been perturbed away from their equilibrium densities. Most initial communities subsequently lost species to extinction. We considered initial communities that lost species to be unstable (sensu May 1973) and/or infeasible (sensuGilpin 1975), since initial communities that lost species either lacked a feasible equilibrium, or else had a feasible equilibrium but failed to return to it (i.e. were unstable). In treating initial communities that lost species as unstable and/or infeasible, we implicitly assumed that alternative states were absent, so that initial stability and feasibility did not depend on initial species abundances. Published evidence and our own experience suggests that alternative states are rare or absent in protist microcosms (Vandermeer 1969; Grover 1997; Weatherby et al. 1998).
Most extinctions occurred early in the experiment, so that species richness, composition, and population densities changed very little over the second half of the experiment (McGrady-Steed et al. 1997; McGrady-Steed & Morin 2000). Stable compositions and population densities were maintained over the latter part of the experiment despite periodic small perturbations (weekly removal of medium; see above). We therefore assumed that all six-week-old-final communities had reached stable, feasible equilibrium points. Although our assumptions about stability and feasibility may not be strictly correct, the interesting question is whether the basic trade-offs between complexity and stability/feasibility identified by May (1973), Gilpin (1975), and others nevertheless provide a robust explanation for the relationship between stability and complexity in these data.
How does complexity change over time? We compared the complexity ([ SC ] 1/2 ) of initial and final webs using a sign test. We expected complexity to decline over time, since communities could only lose species. While species loss might increase connectance, increased connectance is unlikely to completely offset species loss because connectance is constrained by species’ diet breadths.
Are more highly connected species at greater risk of extinction? Species vary in their contributions to food web complexity ( Williams & Martinez 2000 ). Since eqn 1 predicts that high connectance is destabilizing, we asked whether initially unstable/infeasible webs achieve stability and feasibility by losing highly connected species more frequently than would be expected by chance. We tested this hypothesis with a randomization procedure. From each initially unstable community we constructed 100 randomized final communities. We created randomized final communities by randomly deleting species from the initial community until we had deleted as many species as were actually lost from that initial community during the experiment. We did not allow extinctions of detritus, light, or bacteria. Detritus and light are externally supplied and so cannot go extinct, and bacteria never go extinct under our experimental conditions. The average final connectance of the randomized communities gives the final connectance expected under the null hypothesis that extinctions occur independently of how many links a species has. Randomizations were conducted using MathCad + 8·0 Professional Edition for Windows (MathSoft, MA, USA). We regressed observed on expected final connectance to test for deviations from a 1 : 1 relationship.
How does the probability that a web will be feasible and stable vary with complexity? We averaged data across replicates of the same initial composition to calculate the probability that each of our 24 initial community types would exhibit a stable, feasible equilibrium. We then used logistic regression to describe the probability of that an initial web would be stable and feasible as a function of complexity. We included only initial data in the analysis, to avoid problems of nonindependence between initial and final communities.
How does complexity change over time? Complexity declined over time; initial webs were significantly more complex than final webs (sign test, P < 0·001). None of the 83 webs increased in complexity over the course of the experiment. Most (67) declined in complexity; the others (16) did not change.
Are more highly connected species at greater risk of extinction? Observed final connectances deviated significantly from the values expected if extinctions occurred independent of food web position ( Fig. 1a ). Webs expected by chance alone to have low connectance generally had higher connectance than expected, while webs expected by chance alone to have high connectance generally had lower connectance than expected. The differences between observed and expected final connectance increased significantly with increasing final species richness ( Fig. 1b ). Speciose communities had significantly higher final connectance than expected, while depauperate communities had lower final connectance than expected.
Initial connectance declined with initial richness (linear regression, P < 0·01) (Fig. 2a), but final connectance was independent of final richness (linear regression, 0·15 > P > 0·20) (Fig. 2b). Lack of a significant relationship between observed final connectance and richness contrasts with the negative relationship expected if extinctions occurred independent of food web position (linear regression of expected final C on final S, y= 0·359–0·007x, R2 = 0·62, P < 0·001).
How does the probability that a web will be feasible and stable vary with complexity? Logistic regression found a significant negative relationship between complexity and probability of stability in the initial data ( G -test, P < 0·001; Fig. 3 ).
Stability and feasibility vs. complexity
Theoretical work on stability and complexity has motivated a great deal of empirical research, but most of this research is at best indirectly related to the theory. To our knowledge, our results represent the first experimental demonstration that the probability that a community will exhibit a stable, feasible equilibrium declines with complexity over a broad range of community complexity. This qualitative trend is consistent with the classic predictions of May (1973) and Gilpin (1975).
Many authors argue that local stability sensu May (1973) may not be essential for community persistence. Non-equilibrium dynamics can maintain a high diversity of coexisting species under certain circumstances (e.g. Armstrong & McGhee 1980). In general, communities might persist so long as all species tend to remain at some positive finite density (possibly varying in time) when their initial densities are positive and finite. Morton & Law (1997) define such communities as ‘permanent.’ However, theory indicates that the probability of permanence is also a declining function of community complexity (Morton & Law 1997; Chen & Cohen 2001). Our experiment confirms the broad theoretical consensus that increasing the species richness and connectance of a community typically increases the number of constraints that must be satisfied for all species to persist in the long run.
Our results are also consistent with a simple null model that assumes that there is a constant per-species extinction probability, so that more speciose communities have a higher probability of losing at least one species (= unstable/infeasible). However, previously published analyses and experiments indicate that extinction probabilities are not independent of community richness and composition in this and similar systems, so this simple null model can be rejected (Weatherby et al. 1998; Fox et al. 2000; Fox 2002).
Interestingly, the initial data imply that communities with (SC)1/2 > ∼1·8 have essentially zero probability of stability, but many final communities stabilized at (SC)1/2 > 1·8. Why did the final communities stop losing species at such high levels of complexity? One possibility is that strongly interacting species are especially prone to extinction, so that mean interaction strength in each community declined over time (Kokkoris et al. 1999). This hypothesis predicts that initial communities with the same composition should tend to lose the same species (the strong interactors), thereby maintaining similar final compositions. This prediction can be tested by comparing the compositions of final communities with a similarity index. Jaccard index scores for pairs of final communities with the same initial composition ranged from 0·6 to 0·9, indicating more similar final compositions than expected by chance (J. W. Fox unpublished). However, repeated loss of the same species from a given initial community also is consistent with numerous alternative hypotheses unrelated to interaction strength. Existing models of extinction risk and interaction strength also assume that all species occupy a single trophic level (Kokkoris et al. 1999). In a food web, interaction strength sensu May (1973) will change following species loss, due to changes in the densities of the remaining species and redistribution of feeding effort by the remaining predators. Interaction strength sensu May (1973) may also be inversely correlated with connectedness, since predators that feed on more prey species (i.e. more highly connected predators) will necessarily feed at lower per-capita rates on each prey species (McCann et al. 1998; note that McCann et al. (1998) define interaction strength differently than May (1973)). To our knowledge, no existing theory directly addresses the question of how interaction strength sensu May (1973) should be expected to change during the assembly of communities with several trophic levels.
Our results should be treated with caution for at least two reasons. First, the true relationship between complexity and the probability of a stable, feasible equilibrium may be poorly estimated by the initial data. This would not be surprising given the small sample size. The initial food web configurations used by McGrady-Steed et al. (1997) are only a small fraction of all possible configurations, or even of all biologically reasonable configurations (i.e. configurations that do not include predators with no prey). Secondly, we assumed that alternative states were absent. Repeating the experiment with different initial densities might yield a different relationship between complexity and stability if alternative states are common. However, published evidence for alternative states in protist microcosms is weak (Vandermeer 1969; Grover 1997; Weatherby et al. 1998; but see Long & Karel 2002).
COMPARISONS WITH OTHER STUDIES
A few previous studies of stability and complexity in protist microcosms have related initial community complexity to the probability that at least one species will go extinct. These studies conclude either that the probability of an extinction declines with initial complexity (Lawler 1993a), or is independent of initial complexity (Lawler 1993b; Weatherby et al. 1998). We used a broader range of initial complexity than previous studies, and so have more confidence in our conclusions.
The observed decline in the probability of stability and feasibility with increased complexity contrasts with the results of McGrady-Steed & Morin (2000). They also examined the data of McGrady-Steed et al. (1997), and concluded that temporal variability in population dynamics (an index of instability, Taylor 1992) was independent of species richness. The contrast may be more apparent than real. Most extinctions occurred early in the experiment, and so most of the population dynamic data were taken from communities in which composition had stabilized and densities were near their equilibrial values. Population dynamic variability would not be expected to vary strongly with richness at equilibrium.
CHANGES IN CONNECTANCE OVER TIME
We have no explanation for the intriguing result that final connectance tended to exceed expectation in speciose communities, but fall short of expectation in depauperate communities. This result implies that less-connected species are at high risk of extinction in speciose communities, while highly connected species are at high risk of extinction in depauperate communities. The partial confounding of species richness and composition in the experiment (initially depauperate communities lacked predators) only strengthens the result. The predators had broad diets (= highly connected), and tended to go extinct more frequently than species in other trophic groups (J. W. Fox, unpublished). Predators might be at high risk of extinction due to small population sizes (McGrady-Steed & Morin 2000), but extinctions of highly connected predators from initially speciose communities should have caused final connectance in those communities to fall short of random expectations, rather than exceed expectations. Further, the positive relationship between deviation from expected final connectance and final richness (Fig. 1b) remains significant for communities with < 13 or ≥ 13 species (results not shown), and so is not due entirely to differences in initial composition between speciose and depauperate communities.
Observed changes in connectance meant that, while initial connectance was a declining function of initial richness (Fig. 2a), final connectance tended to a mean value of ∼0·28, independent of final richness (Fig. 2b). This is a slightly different measure of connectance than the directed connectance reported in recent food web studies (Martinez 1991, 1992). Our mean connectance of ∼0·28 is equivalent to directed connectance of ∼0·14 (Martinez 1991). Well-resolved natural food webs also have mean directed connectance of ∼0·14, independent of richness (constant connectance, Martinez 1992). The precise correspondence between mean final connectance in our experiments and mean directed connectance in natural food webs is surely coincidental, but the constant connectance found in both data sets is intriguing. Constant final connectance would not be expected if extinctions occurred independently of food web position. We are unaware of a published theoretical explanation for constant connectance. Most existing food web models either assume constant connectance (e.g. Williams & Martinez 2000), or explain why food webs should be restricted to certain combination of connectance, species richness, and interaction strength (e.g. May 1973). Constant connectance in natural food webs might reflect phylogenetic and/or physiological constraints on diet breadth (N. Martinez, personal communication), but this does not explain why our webs converged on constant connectance over time. Future experiments should test the robustness of constant connectance with experiments on webs exhibiting a larger range of initial connectance and richness than used here, and by conducting experiments that involve invasions of novel species as well as extinctions of resident species.
Our results provide one of the first experimental demonstrations that the probability that a community will exhibit a stable, feasible equilibrium declines with complexity, and the first demonstration that communities converge on constant connectance over time. Future progress will depend on measuring interaction strength, and on new theory directly addressing the question of how the relationships between stability, feasibility, species richness and connectance are likely to change as communities assemble.
We received financial support from NSF grant DEB 9806427 to Peter J. Morin and Tim Casey. Comments from Owen Petchey, Dave Raffaelli, Peter de Ruiter, and an anonymous referee improved the manuscript.