The Canonical Distribution of Commonness and Rarity: Part I F. W ...

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Spring 1962 700 CANONICAL and since, 0.262 is not far from 1/4, this may be written E
' synlmetrical distribution, of which we showed examples in Preston ( 1948). 17'e have shown above the mathelnatical consequences of 2 assumptions; viz, that abundance is typically distributed lognormally among species, and that this distribtuion is Canonical in the sense that not all lognormals meet the requirement. but only those in which a definite relation exists between the nunlber of species, (N), the number of species in the modal octave (yo), and the logarithnlic standard deviation (3). 71re now consider what degree of confirmation or refutation is available from observation. Thc relatiztt? corzstancy of o and "a" In any Gaussian distribution or. in our case, any lognormal distribution, we have from equation (S), That is, the total number of species in the "universe" is proportional to the product of the number of species in the 111odal octave and the logarithmic standard deviation. .As the number oi species increases, yo or o, or both, must increase. \\-hen the distribution is canonical both increase, but 3 increases only slowly while yo increases rapidly. In fact if we double N,J, increases by about 857c, but o by only 870 in the range of most interest to us. Similar-1~-, since "a," the "modulus of precision," is related to 3 by the formula u' = 1/(2a2), "a" also is relatively constant. This is what we found in Preston ( 1915). Tlli,tllt~iri~rical eullc~s of "a" aftd o Kot only are these values relatively constant over the accessible range of 7 alues of N,but Table 111 shows that, in this range, where S averages perhaps two or three hundred species, "a" is about 0.175 and 3 is about 3 octaves, or something like one and a fifth orders of magnitude. Xoxv, referring to Table I. vie see that for 311 species we should theoretically ha1.e an average value of "a" of about 0 169. This agreement is close, perhaps fortuitously so (see below on contagious distributions). but it warrants a few com- ~tlents. The only attempts to get a picture or' the cornplete ensemble by direct observation are those of Fisher (1952) and Xerikallio ( 1958 ). There are considerable difficulties with the experimental work and soine uncertainties, sotne of which the authors have indicated. The other results come PRESTOS Ecology, Vol. 43, No, 2 TABLE111. (Ohserved Relationships). N is the number of species estimated, on the basis of the sample, to be present in the total "universe" or "population": yo is the number of species in the modal octave, and "a" is the "n~odulusof precision" of the lognormal distribution I Reference Eaunders (birds) 91 10 Preston 1948 Diri.:mofhs~~ . . 410 48 Preston 1948 Dirks (female moths'] . .. 383 42 Preston 1948 WiUiam (nioths! . . . . 273 35 Preston 1948 King (moths! . . . . . . . . , 277 33 Preston 194s Seamans !moths; . . . 332 3U Preston 194.8 Maryland birds . . . . . . . 233 28 Preston 1957 Kation-aide bud count.. . 530 ; 38 Preston 1959 Nearctic estimate . ... . . . , 600 Praton 1948 Land Birds of England Flsher 1952 !vIerikalllo 1958 from estimates of what the "universe" is like as a result of studying a sample. Indeed neither Fisher nor hlerikallio was studying a perfect "isolate," though they approximated it. The sample theoretically has the same modal height and the same dispersion as the universe, and it has also a 3rd variable, the position of the "I'eilline," or what is the same thing in the end, the abscissa of the mode. This 3rd disposal~le variable makes our estimates of the other 2 tnore uncertain than they would otherwise be, and therefore agreement in our estimate of "a" or s \vithin abut 67% seems in part fortuitous. For a discussion of the fitting of truncated Gaussian distributions see Hald (1952,). Furthemore, u+en we are dealing with truncated ensembles, we do not directly observe the value of K, the total nunlber of species, but have to estinlate it from the sample. This throws a further strain npon the interpretation of our observations. Thi,rclatio~l l~etnvi.~~ !I, and To the extent that ~ve find the correct relationship bet\\-een o and N \x7e must necessarily find a correspondingly correct relationship between yo and N but, since 3 is SO nearly constant over the observable range of N while yo varies rather rapidly, yo may throw some further light on the matter. Table I11 gives the observed values of yo for various T alues of K,most of which are estimated from inconlplete or truncated distributions : Figure 7 shows the data in graphical for~n. It should be once more emphasized that the line is purel? theoretical. The s~l~all circles represent observed points from Tal~le 111. The line is not "fitted" to the points and then extrapolated; the line is from theory, the points fronl observation. But it will be observed that the line passes neatly among the observed points wl~icli lie in a narrow
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Page 11 and 12: given as 167,000: 290,000; and 49,0
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Page 27 and 28: FRANK W. PRESTON Ecology, vol.43, N
Page 29 and 30: logaritlin~ic and is indicated at t
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