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Maximum Entropy and EcologyA Theory of Abundance, Distribution, and Energetics$

John Harte

Print publication date: 2011

Print ISBN-13: 9780199593415

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199593415.001.0001

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(p.240) Appendix C

(p.240) Appendix C

Source:
Maximum Entropy and Ecology
Publisher:
Oxford University Press

Predicting the SAR: An alternative approach

Here we start with the METE results for the spatial abundance distribution at scale A 0, and the spatial abundance distribution Π(n|A,n,A 0), and simply substitute into Eq. 3.13 to get the desired answer at any scale:

S ¯ ( A ) = S ¯ ( A 0 ) n = 1 N 0 e β 0 n n log ( β 0 1 ) ( 1 Π ( 0 | A , n , A 0 ) .
(C.1)

The subscript 0 on β is to remind you that it is evaluated at scale A 0, and β 0 is given by the solution to Eq. 7.27 as a function of the values of S̄ (A 0) and N(A 0). Π(0|A,n,A 0) in Eq. C.1 is determined from Eqs 7.48–7.50.

The predicted SAR can then be determined by numerical evaluation of Eq. C.1. For the case in which down-scaling of species' richness to determine S̄(A) is the goal, and the starting point is knowledge of S(A 0) and N(A 0), the procedure is straightforward. First solve for β 0 using Eq. 7.27 (or the simpler Eq. 7.30, if β 0 N(A 0) ≫ 1 and β 0 ≪ 1), solve for Π(0) using Eqs 7.48–7.50, and then substitute that information in to Eq. 7.48 to determine S̄(A). The summation in Eq. C.1 can be carried out with many software packages, including Excel, MathCad, Matlab, and Mathematica.

To use Eq. C.1 to up-scale species' richness to scale A 0, starting with an empirical estimate of S̄(A), the procedure is more complicated because Eq. C.1 contains two unknowns, β 0 and S̄(A 0), and so does Eq. 7.27. Now, just as with method 1, the two equations have to be solved simultaneously, rather than sequentially.

It might be objected that Eq. C.1 really contains three unknowns, β 0, S̄(A 0), and N(A 0), with only two equations (C.1 and 7.27) to determine them. As before, however, up-scaling abundance N from scale A to A 0 is not a problem. Because we are examining the complete nested SAR, an average of the measured N-values in each of the plots of area A is an estimate of the average density of individuals in A 0. Hence, we can assume:

N ( A 0 ) = A 0 A N ( A ) .
(C.2)

(p.241) C.1 A useful approximation for method 2

While Eqs 7.27, 7.48–7.50, and C.1 can be solved numerically to either up-scale or down-scale species' richness, it is useful to examine some simplifying approximations that will often be applicable to real datasets. With these simplifications we can obtain analytically tractable solutions that do not require numerical solutions and that provide considerable insight.

We assume that AA 0, so that 1 − Π(0) can be approximated by Eq. 7.54. In practice, this means that we are considering ratios A 0/A > 32. Particularly for up-scaling applications, this assumption is not very constraining because we are often interested in up-scaling to areas much larger than the area of censused plots. As always, we assume S 0≫ 1 so that exp(–β 0 N(A 0)) ≪ 1.

With A « A 0, we can replace Eq. C.1 with:

S ¯ ( A ) S ¯ ( A 0 ) n = 1 N ( A 0 ) e β 0 n n log ( β 0 1 ) ( n n + A 0 A ) = S ¯ ( A 0 ) n = 1 N ( A 0 ) e β 0 n log ( β 0 1 ) ( 1 n + A 0 A ) .
(C.3)

Now wo cases separately: β 0 A 0/A ≪ 1 and β 0 A 0/A ~ 1 (we will show that the case β 0 A 0/A ≫ 1 is unlikely to arise).

Box C.1 shows that if β 0 A 0/A ≪ 1, the summation in Eq. 7.63 results in:

S ¯ ( A ) S ¯ ( A 0 ) ( 1 log ( A 0 A ) + γ log ( β 0 1 ) ) ,
(C.4)

where γ is Euler's constant, 0.5772.…

If β 0 A 0/A is ~ 1, then a more complicated result, also shown in Box C.1, is obtained.

The SAR given in Eq. C.4 can be applied either to upscaling or down-scaling species' richness. As before, for down-scaling, Eq. 7.27 is used to determine β 0, which is then substituted into Eq. C.4 to determine S̄(A). For up-scaling, Eqs C.4 and 7.27 must be solved simultaneously. We see from the form of Eq. C.4 that for A 0A and β 0 A 0/A ≪ 1, the SAR is of the form:

S ¯ ( A ) S ¯ ( A 0 ) = a log ( A A 0 ) + b ,
(C.11)

where:

a = 1 / log ( β 0 1 )
(C.12)
and (p.242) (p.243)
b = 1  γ / log ( β 0 1 ) .
(C.13)

From Eqs C.11–C.13, we can extract further information about the shape of the species–area relationship. For the case of down-scaling, β 0 is a fixed parameter determined using Eq.7.27 from the values of S̄(A 0) and N 0 at scale A 0. Hence, S̄(A) depends logarithmically on area.

For up-scaling species' richness, we assume S̄(A) is known and express S̄(A 0), with A 0A, as:

S ¯ ( A 0 ) S ¯ ( A ) 1 log ( A 0 A ) + γ log ( β 0 1 ) .
(C.14)

Using Eq. 7.27, this can be rewritten as:

S ¯ ( A 0 ) S ¯ ( A ) 1 log ( A 0 A ) + γ S ¯ ( A 0 ) β 0 N ( A 0 ) = S ¯ ( A 0 ) S ¯ ( A ) S ¯ ( A 0 ) β 0 N ( A 0 ) [ log ( A 0 A ) + γ] .
(C.15)

Rearranging terms, we derive:

S ¯ ( A 0 ) S ¯ ( A ) + γ β 0 N ( A 0 ) + β 0 N ( A 0 ) log ( A 0 A ) .
(C.16)

At first glance, Eq. C.16 does not seem very helpful because the coefficient β 0 in front of the log(A 0) term itself depends on S̄(A 0). However, numerical evaluation of the up-scaling SAR from simultaneous solution of Eqs 7.27 and C.14 indicates that β 0 N(A 0) is nearly exactly constant over spatial scales for which our assumption A 0/A ≫ 1 holds. In particular, with each doubling of area, A 0 → 2A 0, N doubles and β is approximately halved. At anchor scale, β(A)N(A) is only slightly smaller than β 0 N(A 0) because while N exactly doubles with each area doubling, β decreases by slightly less than a factor of two for areas A 0 only a little bigger than A. The bottom line is that the coefficient β 0 N(A 0) in front of the log(A 0/A) term in Eq. C.16 is approximated by the value of βN at anchor scale. Numerical solutions to the method 2 up-scaling equations (7.27 and C.4) for a variety of anchor scale boundary conditions confirm the accuracy ot the approximations that lead to Eq. C.16.