The loss of a species from an ecological community can set up a cascade of secondary extinctions that in the worst case could lead to the collapse of the community. Both deterministic and stochastic mechanisms may be involved in such secondary extinctions. To investigate the extent of secondary extinctions in ecological communities following the loss of a species, we here develop a community viability analysis. We introduce a measure called the "quasi-collapse risk" that is defined as the probability that the number of species in a community falls below some defined value within a fixed period of time following the loss of a species. We develop deterministic and stochastic methods for finding post-extinction communities. We use these methods to investigate the relationship between diversity (species richness) and quasi-collapse risks in model communities. It is shown that, in a deterministic context, communities with more species within trophic levels have a larger fraction of species remaining in post-extinction communities. This benefit of species richness is to a large extent lost in the presence of demographic stochasticity. The reason for this is a negative relationship between population density and species diversity. We also show that communities become increasingly triangular in shape as secondary extinctions take place, due to greater extinction risk of species at higher trophic levels. We argue that this new approach holds some promise for identifying fragile ecosystems and keystone species.
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