The conventional wisdom, however, has by no means always received support, and has been undermined in particular by the analysis of mathematical models. A watershed study was that by May (1972). He constructed model food webs comprising a number of species, and examined the way in which the population size of each species changed in the neighborhood of its equilibrium abundance (i.e. the local stability of individual populations).

resilience and resistance local and global stability dynamic fragility and robustness

Figure 20.7 Various aspects of stability, used in this chapter to describe communities, illustrated here in a figurative way. In the resilience diagrams, X marks the spot from which the community has been displaced.

Low resilience

High resilience

Low resistance

High resistance

Low resistance

High resistance

Low local stability Low global stability

Low local stability High global stability

High local stability Low global stability

High local stability High global stability

Dynamically fragile

Stable combinations

Stable combinations

Environmental parameter 2

Dynamically robust

Environmental parameter 2

Each species was influenced by its interaction with all other species, and the term ß^ was used to measure the effect of species j's density on species i's rate of increase. The food webs were 'randomly assembled', with all self-regulatory terms (ßii, ßj, etc.) set at -1, but all other ß values distributed at random, including a certain number of zeros. The webs could then be described by three parameters: S, the number of species; C, the 'connectance' of the web (the fraction of all possible pairs of species that interacted directly, i.e. with P^ non-zero); and P, the average 'interaction strength' (i.e. the average of the non-zero P values, disregarding sign). May found that these food webs were only likely to be stable (i.e. the populations would return to equilibrium after a small disturbance) if:

the conflicting results amongst the models at least suggest that no single relationship will be appropriate in all communities. It would be wrong to replace one sweeping generalization with another.

Otherwise, they tended to be unstable.

In other words, increases in the number of species, in connect-ance and in interaction strength all tend to increase instability (because they increase the left-hand side of the inequality above). Yet each of these represents an increase in complexity. Thus, this model (along with others) suggests that complexity leads to instability, and it certainly indicates that there is no necessary, unavoidable connection linking stability to complexity.

Other studies, however, have sug-many models defy the gested that this connection between conventional wisdom complexity and instability may be an artefact arising out of the particular characteristics of the model communities or the way they have been analyzed. In the first place, randomly assembled food webs often contain biologically unreasonable elements (e.g. loops of the type: A eats B eats C eats A). Analyses of food webs that are constrained to be reasonable (Lawlor, 1978; Pimm, 1979) show that whilst stability still declines with complexity, there is no sharp transition from stability to instability (compared with the inequality in Equation 20.1). Second, if systems are 'donor controlled' (i.e. Py > 0, Pjj = 0), stability is unaffected by or actually increases with complexity (DeAngelis, 1975). And the relationship between complexity and stability in models becomes more complicated if attention is focused on the resilience of those communities that are stable. While the proportion of stable communities may decrease with increased complexity, resilience within this subset (a crucial aspect of stability) may increase (Pimm, 1979).

Finally, though, the relationship between species richness and the variability of populations appears to be affected in a very general way by the relationship between the mean (m) and variance (s2) of abundance of individual populations over time (Tilman, 1999). This relationship can be denoted as:

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