## Introduction

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 *sensu*May (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.

Comparisons of richness, connectance and interaction strength among natural communities could test whether communities are constrained to exhibit only certain combinations of these parameters. Trade-offs should exist among richness, connectance and interaction strength, so that communities do not exhibit high values of all three. Martinez (1992) concluded that connectance is independent of species richness in well-resolved natural food webs (‘constant’ connectance). If eqn 1 is approximately correct, constant connectance implies that average interaction strength either declines with increasing species richness, or else is very low. Most authors argue that interspecific interactions generally are weak (Paine 1992; Goldwasser & Roughgarden 1993; Fagan & Hurd 1994; Wootton 1994, 1997; de Ruiter *et al*. 1995; Raffaelli & Hall 1996; Schmitz 1997; Berlow 1999; Müller *et al*. 1999; Freckleton *et al*. 2000; Roxburgh & Wilson 2000a; but see Sala & Graham 2002). However, existing data are difficult to interpret because ecologists often use different measures of interaction strength than May (1973) (Laska & Wootton 1998; Berlow *et al*. 1999). Wootton (2001) compared interaction strength *sensu*May (1973) in speciose coral reefs and depauperate mussel beds and found no significant difference.

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?