# On the Distribution and Abundance of Species

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Science  26 Nov 1999:
Vol. 286, Issue 5445, pp. 1647
DOI: 10.1126/science.286.5445.1647a

Harte et al. (1) assumed the probability rule: if a species occurs in an areaA 0, then the probability that it occurs in half of that area is a constant, a, independent of areaA 0, 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 Sis 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 =cA i z, is equivalent to Si= (2 −i)z S 0. To complete their proof, it would be necessary to prove, for example, that the number of species occurring in an area (0.75)A 0 is (0.75)z S 0.

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 unlessa = 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 orc = 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 thatCancelling equal probabilities, we are left withThe probability rule also yieldsandhence, 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.E 2 is the number of species found only in one of the quarters. The rule implies, by Eq. 7. of Harte et al. (1), E 2= (1 − a )2 S 0. Therefore, the number of species in the remaining three quarters is By the power law,Therefore,which holds if S 0 = 0 =c. If S 0 ≠ 0, multiplying both sides by 22z/S 0 produceswhich 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).

## REFERENCES

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 2 (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.