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For this reading, we approached two statistical principles by which stability in a community can be understood, the insurance hypothesis and the portfolio effect.  Here I will attempt to explain the insurance hypothesis, as I understood it from our discussion.  The idea behind stability is that a community is stable when its components do not vary.  Therefore, to quantify stability in a community, we measure the amount of variability in the community’s species, recognizing that there is a negative relationship between the two (high stability means low variability, and low stability means high variability).    However, because species can interact with one another, variability in one species (such as a predator or competitor) can directly cause variability in another (such as its prey or competitor).  For species that are not related in such a way, there is also the possibility that a change in the environment will cause different responses in different species, so that if a few species cannot handle the change, there will still be others that can take over their role in the system. 

To mathematically describe the variability in this system, we have two components: a term that increases the variability of the system, and a term that decreases the variability of the system.  The term that increases variability represents how much the individual species vary independently of each other and is given by the sum of the variance of the individual species.  For two species it would be the variance of species A plus the variance of species B: var(a) + var(b).  The term that decreases variability represents the variability associated with a relationship between the species, that is, if the species directly influence each other (the article gives competition as an example) or indirectly respond differently to change.  This component represents the covariance of species A’s effects on species B, and species B’s effects on species A, and is given by: 2cov(a,b).  To describe the variability of a community of two species, A and B, we put the two terms together and get: variance(a + b) = var(a) + var(b) +2cov(a,b).

The more species you add the mix, the greater the chance that some are related to each other or respond differently to change, increasing the chance that the covariant term will play a role in the equation, decreasing variability and increasing stability.  Therefore, the insurance hypothesis suggests that stability will increase with increasing biodiversity.