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SAR & rarefaction curves

Is there a relationship between SAR theory and rarefaction curves? I see both applied in papers as a 'tool' but the theory appears to be primarily associated with SAR. Are rarefactions simply a subset of SAR? Also, does METE apply to other forms of macroecological patterns, ie to the accumulation of species in different communities?

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I don't think all rarefactions are a subset of SAR. I know what an underlying assumption of Coleman rarefaction (see Colwell et al 2012 doi: 10.1093/jpe/rtr044) is that species are randomly distributed in sample areas. Also sample based rarefaction undoubtedly relates to the underlying SAR, so in those instances of where samples are randomly taken from places with different SAR's you would expect different sample based rarefaction curves. I'm not sure how tightly coupled individual based rarefaction curves generated from the multinomial model are with the underlying SAR though.

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If you think that individuals occupy a fixed area as in the neutral theory, as you sample more individuals to do the rarefaction curve, you will sample bigger areas thus rarefaction curves and SAR could be the same. In [1] sections 3.3.7 and 7.7 is explained that:

if sampling is carried out without replacement, then the shape of the collector’s curve, S(N) is identical to the shape of the species–area relationship, S(A), that would be obtained if individuals are randomly distributed on a landscape. The reason is that sampling a larger area is the same as sampling a larger number of individuals.
and
The collector's curve describes the dependence on N of the number of species found in a random sample of N individuals
thus is equivalent to a rarefaction curve. But if the species are not randomly distributed the collector's curve could be calculated from the individual spatial abundance distributions, as derived in page 45 of the book.

The second part of your question about curves in different communities leads me to the concept of SAR scale collapse, it's another prediction of METE, that means that all SARs are scale displacements of a universal SAR shape (page 166), and that prediction is tested in page 190, but I am not sure if this is the answer that you are looking for.

1. Harte J (2011) Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics. Oxford University Press.

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Awesome. Thanks for these comments. At least with my peers and I, we use rarefaction a lot now and want to understand how it relates to both other measures and potential neutral processes.

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I have read the Harte's book and I didn't realize, up to now, that collector's curves were the same that rarefaction, I am glad you asked that question.

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I agree with Leonardo's great answer I only want to add that his answer relates to individual-based collector or rarefaction curves, there are also sample-based curves as well. The sample based rarefaction is also sometimes referred to as a non-spatially explicit species-area relationship (Scheiner 2003, see Type IIB and IIIB). The curves we tested in the paper were spatially explicit and contiguous (Type IIA). Also I just wanted to add that the section in Harte (2011) that discusses collector curves is 3.3.7 (not 3.7.7).

Harte, J. 2011. Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics. Oxford University Press Inc., New York.

Scheiner, S. M. 2003. Six types of species-area curves. Global Ecology & Biogeography 12:441–447.

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thanks. I had not seen the scheiner 2003 paper, only this one: Scheiner, S. M., Cox, S. B., Willig, M., Mittelbach, G. C., Osenberg, C. and Kaspari, M. 2000. Species richness, species-area curves and Simpson's paradox. - Evolutionary Ecology Research 2: 791-802.

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The two other answers addressed the similarity between the SAR and rarefaction. I wanted to address your second question:

Also, does METE apply to other forms of macroecological patterns, ie to the accumulation of species in different communities?

METE does predict more macroecological patterns than just the SAR, including the species-abundance distribution, community and species-level body-size abundance distributions, and distance-decay patterns (see Harte 2011 for a more complete list). As Leonardo pointed out, METE's predictions are invariant with respect to community type and only take the information on the measured constraints into consideration when making a prediction.

It is currently less clear if METE applies across community types or biomes. Harte (2011) suggests that METE should primarily apply within communities or biomes, and that failures of METE may be due to the fact that that samples capture too much environmental heterogeneity. That being said one can never truly define a completely homogeneous ecological sample or community (Palmer and White 1994), and the datasets we examined in the current study contain environmental variation and spatial species turnover at all scales yet METE still yielded accurate SAR predictions. Therefore, if we view METE as a practical tool for predicting the scaling of diversity from local to regional scales then we may not need to worry too much if our study sites where we are applying the inferences contain only one community type. Continental scales pose a special set of problems for predicting the scaling pattern of diversity because of species range restrictions (Allen and White 2003). Future iterations of METE may attempt to solve these problems though.

Allen, A. P., and E. P. White. 2003. Effects of range size on species-area relationships. Evolutionary Ecology Research 5:493–499.

Harte, J. 2011. Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics. Oxford University Press Inc., New York.

Palmer, M. W., and P. S. White. 1994. On the existence of ecological communities. Journal of Vegetation Science 5:279–282.

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"As Leonardo pointed out, METE's predictions are invariant with respect to community type and only take the information on the measured constraints into consideration when making a prediction."

This is a fascinating property. I have seen a renewed and sometime vigorous debate on the local versus regional drivers of change and even how we might more effectively contrast them. Perhaps appropriately applied null models such as METE can provide illumination with respect to this regard - particularly if we can use for relatively general sets or subsets of communities.

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One way to study patterns across communities is to use species turnover with distance or beta diversity or Community-level commonality, as defined in METE: the fraction of species found in a typical cell of area A that are expected to be in two such cells at distance D apart. This metric X(A,D) is not explicitly derived in Harte's book. A closely related function based on neutral theory (Chave J, Leigh EG (2002) Theor Popul Biol 62: 153–168.) was shown to be inconsistent with tropical forest data (Condit R, et al. (2002) Science 295: 666–669.) so it would be interesting to derive that function based on METE.

Another set of useful models with an explicit consideration of local vs regional processes are the ones used by Kraft NJB, et al. (2011) Science 333: 1755–1758. they observed a constant difference from random sampling the species pool, thus I suspect that a METE model or other macroecological model should work, but not the neutral model, and that could be an interesting route of investigation.

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I am in the process of writing a manuscript that tests METE's predictions for distance decay (i.e., X(A, D)) using the same dataset as examined in this SAR study. We could only examine the recursive approach as a non-recursive METE derivation of X(A,D) is still out of reach. I'll post a link to the pre-print here when it is posted.

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