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,NATI2!!~~!!:.:.2!~~I!:...~~~~. --.-.- LEnERSTONATURE---

awx,.Jtsh

.~~:I~r:~t~ ..; .~_r~....~~~.~i.onofthe excess manganese associated to the median vall,~.\ ;md does not spill over into the adjacent. . ""."", ;"."j,,'i, '~!dlOugn Il may leak out through fracture

. By~aki~g a con~ervative approach and grouping those plumes zones.tn whlr'J ~~~ "'!?""'!? ':'cc~:- ~~ ~h;:;::;~::-;;;~;:;;:~;, ~.;;: ::;;;~.~~~ . J'in:: anomaues n:'.>(I.-l sqllar.:s will he entered, hence the slop.0 1 ' the graph will be n= = 2. This is an artefacl of the melhod;consequently, in the prescnt study ali points that fell on fhe Iilley =2x (on a log/log pIo:) were omitted when estimating theslope 0 1 ' the line. At tht: other extreme of resolution, when avery large number 0 1 ' sqllares is used, the slope of the graphwill in general fali to I. This could be due 10 a lack of resolutionin the photograph or pccallse irregularities in the outline of the

plant no longer OCCI.'r

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1 0 0 1 J b '1 5 5l o i

j(1cw magnlflcallon)

lJ-:--ii 100 1,300No. of squares 00one

side of grid

flg. 1 a, Photographs of plants at various magnifications wereplaced under a grid. The number of squares entered by the outlineof the plant were counted, starting with a coursegrid of two largesquares on one side, then 2" squares, with n varying from 2 to 6or 7, depending on the grid size. For ease of representation, theplant's leaves in this figure are drawn fiat; in reality they areorientated at ali anglts with respeet to the grid. Also for c1arity,the progressivelyfiner divisions are only ilIustrated in one comerof the figure.The logarithm of the number of squares entered by

the outline of the plant was then plotted against the logarithmofthe number of squares along one side of the grid, as in b. Theslope of the line equals the fractal dimension, D. b. Data gatheredin this way for Virginiacreeper, photographed without leaves inearly spring. The twigswerephotographed at one scale, then partsof the samc twigs wererephotographed at a higher magnification,

permitting O to be estimated at two levclsof resolution,

dimension at different scales have been found in other naturalobjects 1,3.11.

Table 1 presents data on the fractal dimension of a selectionof pIants. Estimates ofD range from 1.28, for a close-up photo-graph of Virginia creeper, to 1.79 for cotoneaslet. Those plants

which one might, a priori, consider to have a more complexgrowth form have a higher fractal dimension. In those plantswhich were photographed at two scales, estimates of the fraetaldimension are lower at the higher magnific~tion, as would beexpeeted from the above argument.

The mean fractal dimension of the samp:es iu Table 1 is 1.44.For ease of calculation, assume that, typical1y, D= 1.5.It fol1owsfrom equation (1) that for an order-of-magnitude decrease in

Iuler ienglh, (he expected distance between two points on alinear transect (of dimension D=1.5) across a fraetal surfaeeincreases hy a factor ofjlo =3.16. The difficulties of measuringthe fractal dimension of a surface are considerable and no dataare yet available. However, squaring the increase in linear dis-tance (that is, adding the fractal dimensions of the orthogonaltransects; ref. I, p. 365) gives a heuristic upper bound estimatefor the expected increase in surface area. Therefore, themaximum expected increase in surface area is 3.162 = 10.0 foran order-of-magnitude decrease in ruler length. Bearing in mindthe disconnected character of the surface of vegetation, an

estimate for the lower bound ofthe fractal dimension is obtainedby adding I to the linear fructal dimension (for example, 1.5+ I;see rer. 1, p. 365). This holds exactly under some circumstances,and in particular when the surface is flat in a direetion transverseto the cross-section( which is dearly not the case for vegetation).This lower bound predicts a 3.l6-fold increase in surface areafor an order-of-magnitude decrease in ruler length, when D= 1.5.

Now consider the implications of the fractal nature of plantsurfaces for the animais living on them. As a first approximation,substitute animai body length (L) for step-Iength (A ) in equation(1). Ifthe way in which animais perceive and use their environ-ment is proportional to their body length12, then for ahomogeneous fractal surface having transects with D= 1.5, the

area perceived by animais 3 mm long may be up to an order ofmagnitude greater than the area perceived by animais 30 mmlong, for the same reference area. This increase in availablespace for animais of smaller body length may be combined witha consideration of the way Jn which metabolic rate scales withbody length 13,14 to make predictions about the distribution ofbody lengths of animais living on vegetation.

The metabolic rate of individual animais 13,1~ scales approxi-w"'di dS lle 0.75 power of body weight, W,that is, as (L';tI~.Next, suppose that population densities are approximately pro-p;):1ional tothe rcdprucal of individual rates of resource utiliz-ation (that is, to metabolic rate-I; see, for example, ref. 13).Then, if use of resources per individual is proportional to WO.75,it fol1ows that population density, N, scales as (e)-0.75. Hence,

ali other things being equal (especial1y the rate of appearanceof new resources), a 10-fold decrease in body length results ina (103).75 =178-fold increase in the density of individuais. Thisincrease in density may be combined with the expeeted increasein the available surface area, outlined above, for.a given decreasein body length, to prediet reiative numbers of individual animaisof different body lengths Iiving on the surface of vegetation. Foran order-of-magnitude decrease in body length, such calcula-

No.ofSpecies Magnification estimates Mean D s.d.

Barberry, Berberis vulgaris L.(evergreen) High 3 1.46 0.018

Low 3 1.43 0.042Virginiaereeper, Parlhenocissus Iricuspidala (Sieb. and Zucc.) Planch. High 6 1.28 0.078(twigsand buds) Low 3 1.S5 0.009

Weepingelm, Ulmus glabra forma pendula (Loud.) Rehd. (twigs and Low 6 1.41 0.111buds)

Cotoneaster, Coloneasler horizonJalis De~aisne(twigs and leaves) High 3 1.35 0.019Low 6 1.79 0.093

Ivy, Hedera helix L. (evergreen) Low 18 1.39 0.050Yew, Taxus baccala L. (evergreen) High 3 1.47 0.042

Low 3 1.68 0.099Silverbireh, Belula pendula Roth (lwigsand (eaves) Medium 3 1.40 0.040Downy birch, Relula pubescens Ehrh. (twigs and feaves) Medium 3 1.40 0.035Ash, Fraxinus excelsior L. (twigs anJ !eaves) Medium 3 1.42 0.083Sycamore, Acer pseudoplatanus L. (twigsand leaves) Medium 3 1.31 0.023

Photographs or branehes and twigs of a selection of woody plants, with or without leaves, were taken from the Universityof York eampus or

SkipwithCommon, North Yorkshire.High-magnificationestimateswere derivl'

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Fia. 2 Data on the number of individual arthropods (mainlyinsects) of different body lengths col-lected from vegetation. a, Under-storey foliage in primary forest,Finca Tabogo, Costa Rical5; b,Osasecondary vegetation' (.) andKansas secondary vegetation (O); c,Tabogo primary riparian vegetation(.) and Icacos vegetation (0)'6; d,understorey foliage in cacao planta-

tions in Dominica (.) and at FincaLa Lola, Costa Rica (O)"; e, Birch( Betula fluhptN'n (Freeman, New York. 19)13).2. Avnir, D., Farin, D. & . I'feifu. f', N,ullrr ~O". 7111-2".1 IIQH1. r\h,"lrlhl'II.1I 11 " p"u,lt. I: '-.\ 1 ' ;I U Il & l \, : \ , I N"h"~ . tUN, '.!I I!~ ll~N~I..t Matltttlh ., lI. H. S"lt"tll'f.1~6. \b r.JH (l"011.5. Ihnh)ughs, 1'. A. NUIII'~ 2'1.1. ~,.q!"2(~gl).6. Ilradhury, D. O., Rri