Nested species-area curves and stochastic sampling: A new theory

We have discovered a severe problem with the current theory of species-area curves (SPARs). This theory claims that we should expect SPARs with z-values of about 0.26. However, that is wrong. The correct prediction turns out to be approximately 0.77. To make this prediction, we used a stochastic sampling scheme, and constructed species-area curves from a lognormal abundance distribution, exactly as previous theory meant to do. We arrived at our prediction using two independent methods: we performed computer simulations of the scheme and we derived its analytical equation. SPARs that result from the simulations are the same as those from the equation, validating the logic of our analysis. We explain what went awry with the previous theory. However, although logically accurate, the new theory has an empirical problem: real SPARs do not have z-values near 0.77. Rather, they tend to lie in the interval 0.1-0.2. To obtain these, we added an assumption to the lognormal abundance distribution. We assumed that range size and abundance are positively correlated. This new assumption is qualitatively similar to Hanski's (1982; Oikos 38: 210-221) pattern. Finally, we derive a simple relation connecting average point diversity, average range size and species diversity for a province.