## Introduction

The relationships between species interaction strength, species diversity and ecosystem stability are of great importance to conservation biology in the face of the increased anthropogenic pressure on natural ecosystems through species extinctions and additions (McCann 2000). Early, mainly verbal, theories on the diversity–stability relationship supported the view that complex and more diverse communities are more stable than simple and less diverse ones (MacArthur 1955; Elton 1958). These ideas were later challenged by May (1972, 1974), who used random Jacobian matrices to investigate the question of stability of ecological communities. He used an elementary result from theoretical physics to show that the stability of ecosystems tends to decrease with the number of species, thus refuting that ‘complexity begets stability’. This result, however, apparently contrasts with the high species richness found in nature.

The question of how diversity and stability are related in natural systems remains unanswered, mainly because empirical studies are still uncommon. Recent empirical studies (Paine 1992; Fagan & Hurd 1994; Raffaelli & Hall 1992; Berlow 1999) have shown that weak interactions are prevalent in natural communities. Hardly any of these studies measure interaction strength (*sensu*May 1974), which results in a quite difficult interpretation of the empirical data. Theoretical investigations have pointed to mechanisms that can explain the stabilization effect of weak interactions on community dynamics (Ives & Jansen 1998; McCann, Hastings & Huxel 1998; McCann 2000). Notably, McCann *et al*. (1998), working with simple food web models, found that weak interactions tend to stabilize a community by suppressing the destabilizing effect of strong interactions. This takes place through the limitation of energy flow in a possibly strong predator–prey interaction. To our knowledge nobody has ever demonstrated experimentally community stabilization via any of the mechanisms proposed by McCann *et al*. (1998).

In an intelligent and industrious attempt to examine the relationship between diversity and stability in natural and agricultural soils, de Ruiter, Neutel & Moore (1995) found that the strengths of the interactions among species were patterned in a way that promoted stability. Roxburgh & Wilson (2000) arrived at similar conclusions studying competition in a lawn community. Both of these studies examined interaction strength, *sensu*May (1974), the elements of the Jacobian matrix.

Studying a single trophic level and attempting to explain the high diversity of natural communities, the importance of competition strength on species diversity may broadly be categorized as follows (modified from Tokeshi 1999).

- 1Competition does not occur at all, and thus it is unimportant.
- 2Competition occurs, but is intrinsically not sufficiently strong to influence the state of co-existence, or previously strong competition has left communities characterized by niche partitioning and therefore weak interspecific interactions: ‘the ghost of competition past’.
- 3Competition occurs and can potentially influence the state of co-existence. However, there may be factors or patterns of interaction strength that prevent it from exerting a significant influence upon co-existence. That is, competition may be intense at the level of two individuals but may be unimportant on net population growth (
*sensu*Welden & Slauson 1986).

In this work we set out to re-examine and assess the effect of interaction strength and species richness on the stability probability of randomly constructed interaction matrices for competitive systems.

We did this by creating random interaction matrices (sometimes referred to as community matrices) for communities and studied the feasibility of a positive and stable equilibrium. This methodology is similar to the one used by May (1972, 1974) to show that the stability of an ecological community depends on the variance of the interaction strength. There are, however, some subtle but important differences between May’s approach and ours. We constructed interaction matrices, used these to assess equilibrium feasibility and hence realize species richness, and then proceeded to examine the Jacobian matrix for stability. May (1972, 1974) studied random Jacobian matrices (confusingly, these are also referred to as community matrices; Case 2000) from which the stability of the equilibrium could be inferred.

The use of a Jacobian matrix assumes that a positive equilibrium exists. Therefore May’s model provides an important result about the stability of ecological interactions, but gives no information about the relation between interaction strength and feasibility of the equilibrium. This limits the use of the model to study species richness. The use of random Jacobian matrices is convenient from a mathematical point of view, as results are available that link the distribution of the matrix entries to the probability of the equilibrium being stable. It can be justified by assuming there exists some model that could be parameterized to give the right Jacobian matrix. However, the disadvantage of this approach is that without knowing the appropriate model, it not possible to compare the model to ecological data.

Another limitation of May’s work is the assumption that the elements of the Jacobian have zero mean. This condition might be appropriate for food web models, but not for competitive systems, for instance. It has been shown that relaxation of this assumption may have important consequences for the stability of ecosystems (Hogg, Huberman & McGlade 1989; Haydon 1994).

We constructed communities by creating random interaction matrices instead of Jacobian matrices. In ecology information is often presented in the form of interaction coefficients and these coefficients can be measured (Bender, Case & Gilpin 1984). The advantage of this approach is that our results can be related much more directly to measurable quantities. We also relaxed the assumption for the mean of the interactions to be zero. We asked whether ecologically plausible structures of the interaction matrix have significant effects on the stability probability of the resulting competitive system.