Harte et al. (1)
assumed the probability rule: if a species occurs in an area
A0, then the probability that it occurs in half
of that area is a constant, a, independent of area
A0, satisfying 0.5 
a 1. From
this rule, Harte et al. (1) give a mathematical
proof of the power law form of the species-area curve: if S
is the number of species in A, then S = cAz, where (0.5)z = a, 0 z 1, and c is constant.
Harte et al. (1) do not justify the last step in
their proof. Their final equation, Si = cAiz, is equivalent to Si = (2
i)zS0.
To complete their proof, it would be necessary to prove, for example,
that the number of species occurring in an area
(0.75)A0 is
(0.75)zS0.
The distribution of a given species in a habitat can be thought of as a
random point distribution over that habitat. Harte et al.
(1) need to provide at least one nontrivial example of such
a distribution that satisfies their probability rule. As far as we can
see, no random point distribution satisfies the rule unless
a = 0.5, in which case the only example known to us is the
uniform distribution of a single, randomly chosen, point.
We shall demonstrate from the probability rule of Harte et
al. (1) that a = 1 or a = 0.5 or
c = 0. The power law happens to hold for these values, but
in all other cases the rule and the power-law are in conflict.
Let x
o mean "species occurs only in right half,"
The probability rule of Harte et al.
(1) yields
Applying 6. to 1. and 1. to 6. gives
The first equality is derived in
(2). Notice that
Cancelling equal probabilities, we are left
with
The probability rule also yields
and
hence, a = 1 or a = 0.5.
Another way of arriving at this conclusion is to apply the probability
rule and the power law to an area consisting of three of the quarters
created by subdividing the habitat into four quarters. E2 is the number of species found only in one of
the quarters. The rule implies, by Eq. 7. of Harte et al.
(1), E2= (1 
a)2S0. Therefore, the
number of species in the remaining three quarters is
By the power law,
Therefore,
which holds if S0 = 0 = c. If S0 
0, multiplying both
sides by 22z/S0 produces
which implies z = 0 or z = 1, hence, a is either 0.5 or 1.
The rule proposed by Harte et al. (1)
implies that species are distributed in one of three trivial ways. In
general, the equivalence of the probability rule with the power law is invalid, as are all conclusions that rely upon it, such as the "endemics-area relationship" (1, 2).
Roger D. Maddux
Department of Mathematics
Iowa State University
Ames, IA
50011-2066, USA
E-mail: maddux{at}iastate.edu
Krishna Athreya
Department of Mathematics and
Department of Statistics
Iowa State
University
Ames, IA 50011-2066, USA
REFERENCES
-
J. Harte,
A. Kinzig,
J. Green,
Science
284,
334
(1999)
[Abstract/Free Full Text]
.
-
J. Harte and
A. Kinzig,
Oikos
80,
417
(1997)
.
10 May 1999; accepted 18 November
1999
Response: In our derivation of the
familiar species-area relationship (SAR), of a new endemics-area
relationship, and of a new abundance distribution from
self-similarity, we explicitly made use of successive shape-preserving
bisections of a biome that is taken initially to be a golden rectangle
with length-to-width ratio of 
(1). We also
stated in our report that species richness per unit area is dependent on patch shape. Odd-shaped patches of habitat will contain a different number of species than do squares or golden rectangles of the same
area, and thus the SAR only holds across scales when applied to
"well-shaped" patches (golden rectangles or squares). Maddux and Athreya seem not to have noted this relationship, examine species
richness in an odd shaped patch (the L-shaped patch that is left when
they go immediately from the whole biome to a quadrant), and conclude
that the fundamental self-similarity parameter in our theory,
a, can take on only particular values. There is plenty
of evidence cited in our report that shape does matter, and that
the specific prediction made by our self-similarity theory about the
dependence of species richness on patch shape is reasonable.
Maddux and Athreya also refer to the example of a random placement
model, in which the parameter a is indeed restricted to the
value 1/2. In our theory, the fraction of the species in a rectangle
that is also found in a particular half of that rectangle is not given
by random placement but rather is governed by the parameter,
a, which is independent of scale and not restricted to 1/2.
Whereas the assumption of independence is necessary to our
findings, our statement that there averages 1 species per unit square
was unnecessary and overly restrictive; if Sm
differs from 1, the distribution plotted on a ln(n) scale is
displaced horizontally but its shape is unchanged.
John Harte
Energy and Resources Group
310 Barrows
Hall
University of California
Berkeley, CA 94720, USA
E-mail:
jharte{at}socrates.berkeley.edu
Ann Kinzig
Department of Biology
Arizona State
University
Tempe, AZ 85287, USA
Jessica Green
Department of Nuclear Engineering
University of
California
Berkeley, CA 94720, USA
REFERENCES
-
J. Harte,
A. Kinzig,
J. Green,
Science
284,
334
(1999)
.
27 May 1999; accepted 8 November 1999
