How can species diversity affect ecosystem stability?
Generally speaking, greater species diversity (alpha diversity) leads to greater ecosystem stability. This is termed the "diversity–stability hypothesis."
Warning: the simple answer is greater species diversity leads to increased stability, but this is not the complete answer.
The diversity-stability hypothesis states that ecosystems with greater species diversity are more stable. Increased alpha diversity (the number of species present) generally leads to greater stability, meaning an ecosystem that has a greater number of species is more likely to withstand a disturbance than an ecosystem of the same size with a lower number of species.
This hypothesis was controversial for a time as it had not actually been tested. It is generally accepted today, although much remains to be understood about the nuances and complexities of diversity, which can be measured in multiple ways, and stability, which can also be measured in more than one way.
A 2007 review of over 50 empirical studies on diversity-stability concluded that we cannot expect to reach a general conclusion about diversity and stability (Ives and Carpenter, 2007). It has been suggested that biodiversity is not the primary driver of stability but rather a secondary driver and one that biodiversity and stability have a relationship that is bi-directional (Worm and Duffy, 2003).
To conclude, while the diversity-stability hypothesis is no longer untested, much remains to be understood. There are more studies that support this hypothesis than those that refute it, but there is also a recognition that we need to understand the exact mechanisms in operation and that much remains understood.
There is a good review article in Nature that may be of interest to those looking for more information. This article is somewhat dated given that a considerable amount of new research has since been published.