Calculating delta diversity

     William James Tychonievich's recent post "Calculating beta diversity" presents a fascinating problem of trying to find a formula relating three types of ecological diversity: gamma diversity, alpha diversity, and beta diversity.  He presents these concepts lucidly, so read the above post before reading on.   

    Tychonievich presents five ways (there are four approaches, but Approach 2 contains two formulas) of calculating beta.  In the second and third, beta is expressed in terms of gamma and alpha, so it is not necessary to have an independent idea of between forest diversity to calculate beta.  The fourth and fifth methods give methods for calculating beta independent of alpha and gamma, so it is not necessary to know alpha or gamme before calculating beta.  For the reasons given in the post, William James Tychonievich shows that the first four methods fail to capture the intuitive idea of between forest diversity.  The fifth method best captures this concept.  

    At the end of the post, Tychonievich postulates that it should be possible to derive beta from alpha and gamma, possibly with a fourth variable.  

    Let us call the type of diversity in Approach 3 delta diversity, represented by the greek letter δ.  Also, let N represent the total number of trees in the environment, S the size of each forest (the number of trees it contains), and F the total number of forests.  If we assume that each forest has the same size, then it is possible to find a formula relating gamma, alpha, delta, and F.  The formula is: 

γ = α/F + δ(F-1)/F.  

Here is a link to a pdf with the details of the derivation of this formula.   Perhaps there is a way to derive it conceptually using minimal algebra.    

    Even though this is not what was asked for, since delta does not express the intuitive idea of between forest diversity, it is interesting that William James Tychonievich's hunch was borne out in that there is a relatively simple formula relating alpha, gamma, the number of forests, and a third measure relating to picking trees from different forests.  

    Relating beta, alpha, and gamma is a different type of problem.  It might be solved by finding a formula for beta first, rather than trying to derive beta from a formula involving alpha and gamma.  

    One relationship is that when the forests are the same size and beta is 0, then alpha must be equal to gamma.  We can see this because if beta = 0, this means that every forest is the same in terms of the proportion of different species of trees.  So, the macrocosom completely reflects the microcosm.  It seems like beta is a parameter that tells us to what extent we can expect that the microcosm of diversity of individual forests will reflect the macrocosm.