the density-dependence of spatial behaviour and the rarity ... · journal of animal ecology (1978),...

Journal of Animal Ecology (1978), 47, 383-406

THE DENSITY-DEPENDENCE OF SPATIAL BEHAVIOUR AND THE RARITY OF RANDOMNESS

By L. R. TAYLOR*, I. P. WOIWOD* AND J. N. PERRYt

Departments of Entomology* and Statisticst Rothamsted Experimental Station, Harpenden, Hertfordshire

SUMMARY

(1) Spatial distribution was analysed in 156 sets of field data comprising more than 200 000 sample units in 3840 samples from 102 species.

(2) Taylor's model, proposing that spatial variance is proportional to a fractional power of mean population density, provided an appropriate description of 147 sets of data.

(3) Iwao's mean crowding model provided a rather less adequate alternative but with marked limitations.

(4) The negative binomial with a common k was not satisfactory. (5) Two data sets only were judged random at all population densities. (6) In all other sets of samples, only at or near the lowest densities was disposition

indistinguishable from random and the range of spacing between individuals progressively and disproportionately changed as mean density increased.

(7) Spatial disposition is thus density-dependent and we deduce that spatial behaviour is also density-dependent as required by Taylor & Taylor's A-model for intrinsic population control by movement.

(8) The species examined range from Protista through annelids, molluscs, crustaceans, arachnids, insects, echinoderms, fish, birds and quadrupeds to men and orchids.

(9) The sampling scales range from ciliates on the surface of a flat-worm to the human population of the United States of America and include temporally stable and unstable populations.

(10) The sampling methods range from quadrat counts, through trap, net and grab samples to volumetric samples and complete censuses.

INTRODUCTION

In contrast to dispersion in the physical world, the dispersive processes of living organisms involve intrinsic behavioural responses that should make spatial randomness highly improbable. Nevertheless, randomness is often taken as a starting point for attempts to define spatial distributions in ecology because it appears to be the only condition, except evenness, capable of unequivocal definition. In fact, Kendall & Buck- land's (1957) statistical dictionary, like other statistical texts, does not define 'random' spatially but as 'representing an undefined idea, or if defined, (one that) must be expressed in terms of probability'. This is not an appropriate starting point for the ecologist who, although he is faced with statistical problems of comparing individual samples, is more concerned with the fundamental biological problem of grasping why organisms arrange

0021-8790/78/0600-0383$02.00 01978 Blackwell Scientific Publications

383

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themselves as they do in real space. A behavioural definition that can lead to the required probabilities is therefore needed and we define spatial randomness as that disposition of individuals in space which results from a lack of response of any one individual to any other or to its environment. In so far as it can be applied ecologically in a spatial context, this is what the Poisson law appears to us to say and it may arise in special circumstances as, for example, when such a superior randomizing process as turbulent atmospheric convection temporarily overrides the behaviour of small flying insects (Taylor 1974).

There have been many attempts to understand spatial distribution by analysis of the frequency distribution of counts of individuals per unit area, especially those based on the negative binomial using the parameter k as an index of aggregation, but the approach has severe limitations. The value of a fitted frequency distribution model lies largely in the graduation of data to stabilize statistics for use as population parameters and is corres- pondingly diminished if the model changes at different population densities or sample sizes. If k is to be used as a specific population parameter, the negative binomial must fit at all densities and the behaviour of k must be consistent. Also the vast amount of data needed to verify any distribution with confidence makes it difficult to confirm that these models are stable. For example, McGuire, Brindley & Bancroft (1957) collected corn borer larvae in up to 3205 units per sample for graduation by the negative binomial, Poisson binomial and Neyman type A distributions. In various samples at different densities, one, two or three of the distributions would fit, but not always the same ones. Even with these large samples there was no means of selecting the 'best' model and this is a common experience because we have no independent criterion for spatial behaviour. In practice, samples large enough for critical assessment are rare and it has never been shown that the same distribution always fits for the same species, even at the same population density. Especially with vertebrates, for example, individuals are often too thinly spread to obtain the necessary numbers. To seek common spatial properties in the behaviour of men and mites, sea urchins and orchids, birds and protozoons, and at all densities, it is necessary to be able to compare all sampling methods, all densities and all spatial scales.

It was the statistical requirement for transformation, rather than its behavioural interpretation, that first led to critical evaluation of field data. Bartlett (1936) implied that

s2=am+bm2 (1)

and Bliss (1941) used

s2 =amb (2)

for the relation between the sample variance (s2) and the sample mean (m), to transform particular sets of field counts. However, the variability in sample counts is not merely a result of unaccountable heterogeneity in the unique environment of the sample, to be removed by transformation based on ad hoc examination of that particular set of data. It also has an underlying dependence on the biology of the organism sampled and this can legitimately be sought as 'an empirical law' (Yates & Finney 1942); but the law must then be biological in concept although it will be expressed in statistical terms.

The power function used by Bliss (1941) for Fleming & Baker's (1936) Japanese beetle larva data, and later stated by Finney (1946) to fit the Yates & Finney (1942) wireworm data, was found to apply to a wide range of organisms by Taylor (1961) who suggested that it might be the empirical law sought by Yates & Finney. This relationship linking population variance V and mean ,u does not specify the frequency distribution at each sample point and so can, if necessary, incorporate any of the proposed distribution

384 Spatial behaviour



models at different population densities and to this extent is more flexible. It also includes, as a special case, the Poisson distribution for which

V=, (3)

when a = b= 1. This provides a means to investigate how often randomness occurs in nature without fitting distributions and it is in the relationship between mean and variance that an understanding of aggregation is sought. Although other non-random distributions may theoretically lead to s2 =m at particular densities, they are unlikely to persist over a wide range of densities in practice.

Our main interest lies in the change in aggregative behaviour as population density changes because this has been proposed as an intrinsic population control mechanism with an underlying behavioural model (Taylor & Taylor 1977). In this model, spatial disposition is seen to result from the movements of individuals acting under opposing ecological pressures; on the one hand to maximise living space by moving farther apart, whilst at the same time attempting to maximize environmental quality by moving closer together. Such pressures are universally applicable to all organisms and if both pressures are expressed as power functions of population density and summed to give

A = GpP -Hpq (4)

A becomes the resulting net displacement pressure, where p is population density and G, H,, p and q are specific behavioural parameters. Simulations from this model yield censuses which satisfy eqn (2) (Taylor & Taylor 1977). If eqn (2) is now found to apply universally to samples from the spatial disposition of organisms in nature, including complete censuses, it may reasonably be regarded as applying to the species from whence the samples came so that sample mean, m, estimates mean density, p/ , and

V= aCp. (5)

This would not verify the A-model but would demonstrate that its simulations generate a real spatial property not obvious from consideration of the familiar frequency distributions.

Bartlett's (1936) eqn (1) led, somewhat indirectly, to an alternative regression approach with a different background behavioural model. Iwao (1968) proposed fitting

i = a'+ b'm (6)

where A& is Monte Lloyd's (1967) mean-crowding statistic,

=m+ -1) .(7) m

Combining eqns (6) and (7) we return to Bartlett's (1) second-degree regression of variance on the mean;

s2= (a'+ l)m +(b'- l)m2 (Taylor 1971). (8)

Whether or not a quadratic is preferable in practice to a power function only time will tell, although polynomials have rarely been helpful in ecology because biological processes are inherently multiplicative, not additive. Iwao & Kuno (1971) claim a theoretical justification for eqn (6) but Monte Lloyd's 'mean crowding' behavioural model (7) has defined limitations which unfortunately are quite narrow. It applies by definition (see Lloyd 1967)

L. R. TAYLOR et al. 385



to comparatively rare, freely-moving animals that live in a relatively continuous, appar- ently uniform habitat, do not defend territories, and whose positions are random with

respect to each other and are constantly changing. Since it is necessary to include species that contravene all these restraints, this theoretical justification must be interpreted with caution.

A third regression approach to spatial variance can be derived from fitting the negative binomial with a common parameter k,

s2=m+b"m2, (9)

where b"= 1/k, which is a special case of Iwao's model with one of it's parameters constrained. It is not to be expected that this would be so flexible as the two-parameter equations, but it has the possible advantage of being directly based on the most widely used frequency distribution.

This paper, therefore, is concerned to establish three main points. First, to find whether or not the variance/mean power relation is universally applicable to organisms from all classes and at all spatial scales, as Taylor (1961) suggested, and if, on average, spacing is a

systematic non-linear function of density, i.e. density-dependent in the usual ecological sense. Secondly, to see whether or not Iwao's (1968) polynomial and the negative binomial provide equally valid alternative models, and third, whether randomness, as defined by the Poisson model (3) is as rare as our a priori reasoning might legitimately expect. We are not here concerned with the statistical justification of the power function for its use in transformation (Healy & Taylor 1962; Bliss 1967; Kendall & Stuart 1976), this investigation being essentially ecological.

We therefore examine the available sets of sample data for the fit of the four models, and consider the resulting parameter values.

DATA The data used here are given or analysed in the listed references which, for this reason, are numbered. Appendix B lists the sets of records and the derived parameters grouped into major taxa. There are 156 sets of data in all, totalling 3840 samples with more than 207 055 sample units of various sizes, mostly collected before 1970, and comprising about 109 individuals. But the literature has not been searched systematically nor with any limiting selection and new sets of data are now constantly appearing. It is therefore likely that the excess of insect records is partly due to our personal predilection, although more work has undoubtedly been done on spatial sampling in insects than on any other group except perhaps flowering plants. Nor have we searched the botanical literature carefully because the approach there has been somewhat different and there seems not to be much replication at a range of densities.

We know of three major sets of data, comprising about 100 species of aphids, 200 moths and 100 birds, from which only a small sample is included here, otherwise they would further emphasize the taxonomic imbalance. They will form the basis of another analysis relating to the parameters of equation (2). Much of the more recently published material lists only the variance/mean regression coefficients and is not included.

ANALYSIS The linear regression (6) proposed by Iwao for fitting his model is open to criticism from a statistical point of view since the mean appears strongly in both axes and this generates

Spatial behaviour 386



spuriously high correlation between them (Fig. 1). Also, his regressions were not weighted to allow for the scatter of regression points increasing with x, where x-m (Fig. 2 and Appendix A). To be valid, statistical comparison of Iwao's with other models requires them to have the same y-variate. Therefore, after a preliminary analysis using Iwao's method, the variance/mean formulation (1) of Iwao's model was used for more rigorous analysis and we are now concerned to find the best descriptive relationship for the four ecological models in s2 and p, or statistically in Vand u, which we estimate by s2 and x as follows:

Iwao's model s2 = a'x + b'x2; (1) Taylor's model s2= axh; (2) negative binomial with common k s2 = x + b"x2b" = I /k; (9) and the Poisson s2=x. (3)

a -b 100 -

40 -

' 10- /

" a ^~~ I ' A0^

^ ~0

0.1 I 10 100 0 20 40 m (log scale) m (arith. scale)

Sample mean

Fig. la. The log variance by log mean plot for a minimal set of samples of a bivalve (Tellina tenuis) (Holme 1950) shows a distribution of points, with large scatter, near to random, possibly tending towards regularity. b. Mean crowding is incapable of adequately describing such regular distributions, where s2 < m, and a spuriously strong regression appears because m > f > (m - I)

(eqn 7).

The distribution of s2 is highly skewed and V(s2) will almost certainly be approximately proportional to [E(s2)]2 where V is variance and E the expected value. Two approaches were therefore employed in comparing different aspects of the models.

First, after logarithmic transformation of s2 to stabilize V(s2) at different x's, the maximum likelihood estimates of the parameters of models (1), (2) and (9)'were calculated by the maximum likelihood computer program, MLP (Ross 1970). The residual mean squares were then compared to test the fit of the models. There was no weighting for the number of sample units in each sample. In the case of Taylor's model (2) this procedure was equivalent to his method of fitting a linear regression in the form

log (s2)=log a + b log x. (10)

Note that, in this instance the log transformation not only stabilizes variance but also gives a linear relationship. Iwao's model becomes

log (s2)= log (a'x + b'x2).


(I 1)




a b

400 - 100000- o0

0 :0 -/ o 10000- ? /

o *E3 200- o

c i00 10- /

o 0 / o % // :o - / / 0

0 0@ 0

N '. 1$1l l 200-l

1000- V

T '200- 0

O 0

L/ I0 100-

0 10 10 0 50 1 1 2 10 100 1000 0 50 100 150 200 250 mn (log scale) m (arith. scale)

Sample mean

FIG. 2a. Transformed to logs, the variance x mean scatter is parallel and the fitting efficient. b. In the Iwao plot of ih on m, untransformed, for the data on damage by the cocoa mirid bug (Taylor

1971), scatter widens as population density increases (see also Appendix A).

Secondly, in a more detailed analysis, equations (1), (2), (3) and (9) were treated as generalized linear models (Nelder & Wedderburn 1972) having Gamma error distributions, and the models fitted using the generalized linear interactive model computer program, GLIM, (Baker & Nelder 1978) which again stabilizes V(s2). The mean deviances were then compared to test the fit of the models.

In the GLIM analysis Taylor's model was fitted using a log link function in the form

s2 =exp{lna+b In x}. (12)

In order to test for curvature in models (1) and (2) a cubic and a quadratic term were added, respectively, giving

s2 =a'x+ b'2 + c'X3 (13)

for Iwao's model, and

s2 =exp {ln a+b ln x+c (ln x)2} (14)

for Taylor's. In each case the F-ratio, (difference of deviances)/(mean deviance of the fuller model), on 1:M-3 d.f. was used to detect curvature formally (M=number of sample points in the regression). A similar method was used to test the adequacy of the Poisson model (3) and to detect regularity.

Each of the M data points (x, s2) in each set is derived from a sample of N units, where N can differ from sample to sample; ideally each value of s2 should be weighted by N-1.



Where possible this was done for the GLIM analyses. Values of N were available for 113 sets of data, and of these N was constant within the data-set in sixty-three cases.

Parameters cannot be constrained to be positive using the GLIM program and in thirty sets of data the Iwao model gave negative values for a' or b' which is biologically as well as

statistically meaningless since it implies negative variance for some values of . Indeed for two sets of data, Aeneolamia varia (Evans 1972) and Ammonia beccarii (Buzas 1970) no fits were obtained for the Iwao model since at least one fitted value of s2 was negative. When the range of values of x is low there may be little difference in the fit of (1) whether b' is positive or negative. This is clearly a disadvantage since the forms of the model are then

totally different (Fig. 3). Had it been possible to constrain parameters to realistic positive values, the mean deviance of the Iwao model would have been at least as high as in the

Taylor model and curvature might have become more evident. In the MLP analysis a' and b' were constrained to be positive.

120- b

6 -

80-

4 -

O40-

2-

0

?l 20 406

/ ? i I -ve

2

0 1 2 3

Sample mean (m) Projected mean (m)

FIG. 3a. Iwao's quadratic fits short range data such as that for Ranunculus bulbosus (Blackman 1942), either (I) constrained (MLP) s2=2-569 x+0-00397 x2, or (II) unconstrained (GLIM) s2=2.699 x-0-0446 x2. b. Projected over a wider range of population densities the unconstrained (II) negative parameter (b) yields an unacceptable inverted parabola having negative

variances at high means.

In addition to the weighting by N, the quality of the data varies considerably from one set to another and their ability to discriminate between models varies accordingly. A subjective appraisal of each data set, in four classes from very good (1) to very poor (4), was made based on four criteria; number of sample points, M; average number of units

per sample, N; the range of means, x; and the scatter of s2 (see Appendix B). In a preliminary analysis, the Taylor and Iwao models were compared using linear

regression to fit eqns (6) and (10) in the way proposed by their authors, and a quadratic term was added to test for curvature. Although, as previously mentioned, the models cannot be rigorously compared by this method the results were in general agreement with those obtained using GLIM and MLP.




RESULTS The detailed results are given in Appendix B.

Best-fitting models (MLP and GLIM) Comparing first the Taylor and Iwao models in the MLP analysis, the residual mean

square is lower for Taylor's in 106 out of the 156 sets of data and in 50 for Iwao's. In the GLIM analysis we exclude eleven cases where Iwao's model gave better fit but with invalid parameters whereas the MLP analysis gave a better fit for Taylor's. Of the remaining 145 data-sets, the GLIM analysis then gave better fits (lower mean deviance) to Taylor's model for eighty-eight sets, and to Iwao's for fifty-six, with one tied. In most cases, both the Taylor and Iwao models fitted the data adequately using both MLP and GLIM (Appendix B).

When the negative binomial model is also included the MLP scores are, Taylor 99, Iwao 25, negative binomial 32. The GLIM scores are Taylor 77, Iwao 34, negative binomial 33 with one tied.

As expected, the negative binomial fits no better than the Iwao model, and is therefore pursued no further; the Taylor model fits markedly better than either.

The Poisson model gave the best fit in one case only. Models (3), (9) and (1) form an increasingly elaborate heirarchy, so their fits are expected to improve progressively, excluding consideration of degrees of freedom.

Four criteria test (GLIM) No consistent trends were found in the proportions of data-sets best fitted by the

models when classified into four quality groups using the four criteria listed. Therefore, the hypothesis that Taylor's model fits no better than Iwao's can be more formally tested using a two-tailed binomial test (r = 88k; n = 145; Ho: p = 2) and is rejected at the 1% level.

Curvature (GLIM)

Unconstrained, fifteen sets of data showed significant curvature only when using Iwao's model, six only with Taylor's and a further three with both. Apart from these three, curvature is generally low but markedly higher with Iwao's model than with Taylor's (Fig. 4). Curvature, if it exists, is more difficult to detect when M is small but is a more serious defect than poor fit if it occurs consistently.

Mean deviance (GLIM) and residual mean square (MLP) The mean deviance (GLIM) and residual mean squares (MLP) for the two main models

were compared on a log scale to investigate further the relative fits. Using the full unweighted data from MLP, there is some skewness in the residual mean squares (Fig. 5), in addition to the balance of data sets in favour of the Taylor model. This indicates that in some cases, when the Taylor model fits better, which it does more often than Iwao's, it also tends to fit much better. In the GLIM analysis, divided into unweighted and weighted batches, this skewness is more pronounced in the weighted than the unweighted batch suggesting that weighting increases the evidence in favour of the Taylor model.

Randomness and regularity Only two data sets adequately fitted the Poisson model, Poterium sanguisorba (Black-

man 1942) and Ips grandicollis (Mason 1970). Two further sets were significantly more regular than random; Dacus tryoni (Monro 1967) and Tellina tenuis (Holme 1950).

Spatial behaviour 390




6- a b o

" . ?0* 4

U) 0 a ~E

~~0?

1.0

o

1-0 10 0 2 4 m (log scale) m (arith. scale)

Sample mean

FIG. 4. Data for wireworms in grassland by Yates & Finney (Yates & Finney 1942) and the fitted regressions; the mean-crowding regression using Iwao's model (b) is significantly curved. The

scatter has been reduced by taking means of variances. a. Log variance x log mean; s2 = 1-352 m '?82. b. Mean crowding x mean density; s2 = 1036

m+0-323 m2-0-037 m3.

40-

30-

20-

<-0-12 -0-10 -0-04 0-0 +0-04 + 0- 10>0.12 Difference of logs of mean squares (Iwao-Taylor)

FIG. 5. The log ratio of residual mean squares (Iwao/Taylor) shows a considerably greater number of better fits for the Taylor model ( + ) than the Iwao (-) and also a more skewed distribution ith

a longer tail (accumulated in figure to save space).

Taxonomic range, sampling scale and method Within Appendix B are listed species from a very wide range of major taxa. In Fig. 6 we

illustrate data for a ciliate protozoon (Urceolaria mitra von Sieb, Reynoldson 1950), a cattle tick (Boophilus microplus (Can.), Wilkinson 1964), two aphids (Aphis fabae Scop., Taylor 1970 and Myzus persicae Sulz., Taylor 1977), two moths (Euxoa nigricans L. and Spilosoma lutea Hufn., Taylor and Taylor 1977), a sea urchin (Echinocyamus pusillus, Ursin 1960), a fish (Melanogrammus aeglefinus (L.), Taylor 1953), two birds (Parus major L. and Streptopelia decaocto, Batten & Marchant 1976), a fox (Vulpesfulva, Elton 1942), modern man (Homo sapiens, Anon. 1972) and an orchid (Ophrys apifera Huds.,



Blackman 1942). It seems unlikely that, having successfully fitted this range of classes, other major taxa will now be found that do not conform, although there may well be occasionally species that are exceptions.

The aphid samples are at two scales, between stems within a field, and between regions throughout Great Britain (Fig. 6, VII & VIII). The protozoon, orchid, tick, sea urchin and haddock were sampled simultaneously at different places. The birds, moths, aphids, fox and man were sampled over the same area at successive time intervals. In some instances the population has cycled more than once during the sampling period, e.g. the garden dart moth (Euxoa nigricans) and the great tit (Parus major), whilst in man and the collared dove (Streptopelia decaocto) the population is still rising. The coloured fox (Vulpesfulva) data are from the classical Moravian Mission (later Hudson Bay Company) records and cycle about twenty-three times during the 91-year period of these samples. For man the

a

10 000000

1 000000 Ilooo ooo /

c/ 07 / 100000 I 0

17c 01980 o '0 0 o O

10 000

1967 1000?'// /1900 cr) 19660 s o ?"

o. 100 I/ /nsus

10 1 1/

1973 /

o307 1974 0 0

Men density (, ogscale) staggered at x 10 intervls

FIG. 6a. Variance x mean plots (log scale) for a range of taxa and sampling systems: I, Garden dart

19690 0 .1 0

I'0

?0'1

Mean density (m, log scale) staggered at x 0 intervals

FIG. 6a. Variance x mean plots (log scale) for a range of taxa and sampling systems: I, Garden dart moth (Taylor & Taylor 1977); II, Haddock (Taylor 1953); III, sea urchin (Ursin 1960); IV, cattle tick (Wilkinson 1964); V, population of man in U.S.A. (Anon. 1972); VI, Hudson Bay coloured

fox (Elton 1942).





uppermost point is a projection for 1980. In the two moth species the time period is the same, 1966-74, and the sampling sites, but the range of the mean density (Fig. 6,1 and X) is remarkably different, illustrating a very wide range of temporal stabilities. The range of scales is from the surface of a flatworm (protozoon) to the whole of the United States of America (man). There are few ecologically relevant scales not encompassed here.

The ciliate is epizooic and the tick an ectoparasite, the sampling unit being the real host for the ciliate and a pseudo-host for the tick, and the fox is a predator. The bird samples are census counts at selected sites made during the breeding season, the moths are sampled by attractant traps, the fish by nets, the orchid by quadrats, the sea urchin by grab sampling and the data for man is from complete censuses of the U.S.A. over a period of more than a century, the latter part at 10 years intervals. The fox data are not too well defined for various sampling reasons (see Elton 1942) and this is apparent in the scatter

b/

10000 000

I 000 000

100 000 ooo c/ 7 / Co-,

O/ / 10 000

0

100

? 0

100~~~X 0 10 r ,^~o/ ?/ / / ~o/

cola/0 X il

15)XIIbeorhd(lcmn14)Fofulrdsrpin/ eet0 0

.0

0.01 j 0.ol /

Mean density (m, log scale) staggered at x 10 intervals

FIG. 6b. VII, Peach aphid in Great Britain (Taylor 1977); VIII, bean aphid on plants (Taylor 1970, C.G.J.); IX, great tit (Batten & Marchant 1976); X, buff ermine moth (Taylor & Taylor 1977); XI, collared dove (Batten & Marchant 1976); XII, protozoon epizooic on flatworm (Reynoldson

1950); XIII, bee orchid (Blackman 1942); For fuller descriptions see text.



diagram. The Appendix includes all the various sampling methods in common usage and, in consequence, it seems that most samples that are systematically collected are likely to fit the model.

DISCUSSION

Many behavioural models have been proposed to account for the distributions resulting from sample counts. Most of these, including all those involving the negative binomial as well as mean crowding, lead to a polynomial relationship between variance and mean. So far as we are aware, the A-model (Taylor & Taylor 1977) is the only one at present leading to a power function.

The negative binomial distribution has admirable statistical properties when treated as a descriptive distribution; namely its flexibility and simple parameterization, and especially its convenience because so many hypothetical models lead to it. There is no doubt of its excellent fit to individual sets of data. Its biological meaning is less obvious. Some of the biological models that lead to it contradict each other and this leads to confusion and makes any general interpretation of its application suspect. We are aware of no good experimental evidence that k has a consistent biological meaning. The results obtained here show quite conclusively that, whilst some species are known to have a common k at different densities (Bliss & Fisher 1953; Bliss & Owen 1958), this is not a universal or even a widespread property. Any general theory incorporating the negative binomial therefore needs to explain why some species have a constant k and others have not and in what systematic manner k changes with m for these others. Until a better biological hypothesis is procured, the interpretation of k remains suspect and the negative binomial cannot be regarded as having any general or conistent fundamental biological significance over a range of densities.

It is difficult to produce a simple clear-cut, formal analysis for all the data presented here, collected by so many people from such different organisms for such different purposes. Nevertheless, Iwao's model has been fitted in three different ways, using Iwao's own method as well as two more rigorous and sophisticated ones, and the results are consistent. Iwao's model usually fits less well than Taylor's, more frequently requires curvature terms to account for systematic departures from the model and often, unless specifically constrained otherwise, predicts negative variances. These last two faults suggest that the functional structure of the model is unsound.

The power function behaves better on all counts and provides an adequate background model for the relationship between mean population density and its variance over almost all taxa, spatial scales and sampling methods (Fig. 6).

On an ecological time-scale, spatial disposition is a transient condition, an instantaneous aspect of a dynamic system which rarely repeats the identical geographical pattern in subsequent generations, even when the mean density returns to a former level (see Taylor & Taylor 1977, 1978). For these reasons frequency distribution models tend to be uninstructive unless the same distribution recurs each time the same density is restored. The regression model approach only requires a stable relationship between mean and variance involving continuity in both space and time. In Fig. 6 I, the population moves up and down the regression line in successive cycles. Alternatively in Fig. 6 XIII, the samples are scattered simultaneously in time at different points through space. This continuity may be of greater concern in population dynamics than a detailed model for the instantaneous density pattern in isolated demes or small areas.




Although it has been claimed that variance/mean relations are not good criteria for measuring aggregation (e.g. Morisita 1959; Lloyd 1967) we can find no valid biological justification for this view which seems to be based on the unverified assumption that aggregation is density independent. Indeed, in the early discussions of aggregation (see Waters 1959) it is clear that variance was used as a criterion for the behaviour of k. In the absence of an instantaneous set of co-ordinates for the spatial position of each individual, variance is here treated as a measure of the range of spacing, from complete isolation to high concentration, achieved by the individuals within a segment of a population at a given mean density for that segment. Standard deviation/mean ratio, s/m, could equally well be used as a measure but the Poisson equality (s2 =m) at the random density is a useful basis for comparison (Fig. 7). It is the rate of change of this condition as density changes that is our main concern and spatial variance isolates the element of behavioural responses that epitomizes the ever-changing geographical patterns. The A-model, which can generate the power function, specifies only that individuals respond to each other as part of their total environment in a density-dependent fashion. It proposes that the outcome in movement, deriving from the dynamic balance between opposing tendencies to move together or apart, is itself a power function. This is not unreasonable given the evidence presented above.

40-

E 30- a,

0

) 20-

0

0 0-4 0-8 1-2 1-6 2-0 2-4 2-8

Regression coefficient b

FIG. 7. Frequency distribution of the regression coefficient b from the equation log s2 = log a + b log m fitted to 156 sets of samples has a mean and S.D. of 1.45 + 0.39.

Given this evidence, we must also accept that either ratio, s2/m or s/m, whether or not they are called the degree of aggregation, changes systematically and disproportionately with population density. In other words spatial behaviour is density-dependent in the majority of species, since increase in either ratio indicates different, and highly specific, extremes of both population concentration and individual isolation at high densities that could not be foretold by simple proportional projection from lower densities (Fig. 8). The most economical hypothesis is that the behaviour of the organism governs its spacing relative to its neighbours and that it is density-dependent, reflecting a fundamental life process and not, as Miller (1971) put it, 'some uninstructive element'.




Spatial behaviour

0

vno 0

'O /O /

2-

K I I , I l I 0 2 4 6

Sample mean (m)

FIG. 8. Yates & Finney's wireworm (Agriotes spp. mainly obscurus) in grassland data (Yates & Finney 1942), based on 45 440 sample units, show the accelerating rate of dependence of variance on the mean as population density increases on arithmetic scales. The variances are means at each

density interval.

Because b is only rarely equal to unity (Fig 7), the regression line for most species must cross the line s2 = m (a = b = 1) at some specific density. This density is usually so low that the whole sample contains only a few individuals, most sample units being empty. Variance and mean are there equal and the spatial disposition and statistical distribution are indistinguishable from randomness. This is not just a deficiency of the sampling system for, no matter how much sample size is increased, such a lower limit to sample sensitivity will always be reached. It is part of the familiar problem of defining population limits. Such a conclusion makes clear why a distribution-fitting procedure presents such a difficult approach to spatial analysis. The measurement needed to find the limiting density at which randomness occurs, and so fix the spatial structure of the population, cannot be obtained because the sample is too small to fit a distribution. Indeed one could legitimately say that, with very few exceptions, randomness only occurs when the density is so low that the one individual that can be found has no others with which to respond. Only two species out of 102 remain indistinguishable from randomness at all population densities. We conclude that true randomness is as biologically rare a property as our a priori expectation.

ACKNOWLEDGMENTS

It is a pleasure to thank Leo Batten and the British Trust for Ornithology, for access to the data for the Common Bird Census, and R. J. Baker, J. C. Gower, J. A. Nelder and G. J. S. Ross for helpful discussions.

396



L. R. TAYLOR et al.

REFERENCES

1. Anon. (1972). U.S. population data sheet. Population Profile, Nov. 1972, 1-4. Population Reference Bureau, Washington, U.S.A.

2. Baker, R. J. & Nelder, J. A. (1978). The GLIM System Manual, Release 3. Numerical Algorithms Group, Oxford.

3. Banks, C. J. (1954). A method for estimating populations and counting large numbers of Aphisfabae Scop. Bulletin of Entomological Research, 45, 751-6.

4. Bardner, R. & Lofty, J. R. (1971). The distribution of eggs, larvae and plants within crops attacked by wheat bulb fly, Leptohylemyia coarctata (Fall.). Journal of Applied Ecology, 8, 683-6.

5. Barnes, H. & Marshall, S. M. (1951). On the variability of replicate plankton samples and some applications of contagious series to the statistical distribution of catches over restricted periods. Journal of the Marine Biological Association of the United Kingdom, 30, 233-63.

6. Bartlett, M. S. (1936). Some notes on insecticide tests in the laboratory and in the field. Supplement to the Journal of the Royal Statistical Society, 3, 185-94.

7. Batten, L. A. & Marchant, J. H. (1976). Bird population changes for the years 1973-1974. Bird Study, 23, 11-20.

8. Beall, G. (1938). Methods of estimating the population of insects in a field. Biometrika, 30, 422-39. 9. Berthet, P. & Gerard, G. (1965). A statistical study of microdistribution of Oribatei (Acari) Part 1. The

distribution pattern. Oikos, 16, 214-27. 10. Blackman, G. E. (1942). Statistical and ecological studies in the distribution of species in plant communities

1. Dispersion as a factor in the study of changes in plant populations. Annals of Botany, London, N.S. 6, 351-70.

11. Bliss, C. I. (1941). Statistical problems in estimating populations of Japanese beetle larvae. Journal of Economic Entomology, 34, 221-32.

12. Bliss, C. I. (1967). Statistics in Biology, McGraw-Hill, New York, U.S.A. 13. Bliss, C. I. (1971). The aggregation of species within spatial units. Statistical Ecology (Ed. by G. P. Patil, E.

C. Pielou and W. E. Waters), 1, 311-335, Pennsylvania State University Press, University Park. 14. Bliss, C. I. & Fisher, R. A. (1953). Fitting the negative binomial distribution to biological data. Biometrics, 9,

176-200. 15. Bliss, C. I. & Owen, A. R. G. (1958). Negative binomial distribution with a common k. Biometrika, 45,

37-58. 16. Bowen, M. F. (1947). Population distribution of the beet leafhopper in relation to experimental field plot

layout. Journal of Agricultural Research, 75, 259-78. 17. Broadbent, L. (1947). The grouping and overwintering of Myzuspersicae Sulz. on Prunus species. Annals of

Applied Biology, 36, 334-40. 18. Burrage, R. H. & Gyrisco, G. G. (1956). The transformation of counts of European chafer larvae for analysis

of variance. Journal of Economic Entomology, 49, 179-82. 19. Buzas, M. A. (1970). Spatial homogeneity: statistical analysis of unispecies and multispecies populations of

Foraminifera. Ecology, 51, 874-9. 20. Calnaido, D. (1962). Studies on the abundance and dispersal offritflies. Unpublished Ph.D. thesis, University

of London. 21. Cancela da Fonseca, J. P. (1966). L'outil statistique en biologie du sol III. Indices d'interet ecologique. Revue

d'Ecologie et de Biologie du Sol, 3, 381-407. 22. Daum, R. J. & Dewey, J. E. (1960). Designing orchard experiments for European red mite control. Journal

of Economic Entomology, 53, 892-8. 23. Davidson, J. & Andrewartha, H. G. (1948). Annual trends in a natural population of Thrips imaginis

(Thysanoptera). Journal of Animal Ecology, 17, 193-9. 24. Dean, G. J. & Luuring, B. B. (1970). Distribution of aphids in cereal crops. Annals of Applied Biology, 66,

485-96. 25. Elton, C. (1942). Voles, Mice and Lemmings. Clarendon Press, Oxford. 26. Evans, D. E. (1972). The spatial distribution and sampling of Aeneolamia varia saccharina and A. postica

jugata (Homoptera: Cercopidae). Entomologia Experimentalis et Applicata, 15, 305-18. 27. Finney, D. J. (1946). Field sampling for the estimation of wireworm populations. Biometrics Bulletin, 2,

1-20. 28. Fleming, W. E. & Baker, F. E. (1936). A method for estimating populations of larvae of the Japanese beetle

in the field. Journal of Agricultural Research, 53, 319-31. 29. Forsythe, H. Y. & Gyrisco, G. G. (1961). Determining the appropriate transformation of data from insect

control experiments for use in the analysis of variance. Journal of Economic Entomology, 54, 859-61. 30. Forsythe, H. Y. & Gyrisco, G. G. (1963). The spatial pattern of the pea aphid in alfalfa fields. Journal of

Economic Entomology, 56, 104-7. 31. Fye, R. E., Kuehl, R. O. & Bonham, C. D. (1969). Distribution of insect pests in cottonfields. U.S.

Department of Agriculture, Agricultural Research Service Miscellaneous Publications No. 1140.

397



32. Gerard, B. M. (1960). The biology of certain British earthworms in relation to environmental conditions. Unpublished Ph.D. thesis, University of London.

33. Guppy, J. C. & Harcourt, D. G. (1970). Spatial pattern of the immature stages and teneral adults of Phyllophaga spp. (Coleoptera: Scarabaeidae) in a permanent meadow. Canadian Entomologist, 102, 1354-59.

34. Harcourt, D. G. (1960). Distribution of the immature stages of the diamondback moth, Plutella maculipennis (Curt.) (Lepidoptera: Plutellidae), on cabbage. Canadian Entomologist, 92, 517-21.

35. Harcourt, D. G. (1961). Spatial pattern of the imported cabbageworm, Pieris rapae (L.) (Lepidoptera: Peridae) on cultivated cruciferae. Canadian Entomologist, 93, 945-952.

36. Harcourt, D. G. (1963). Population dynamics of Leptinotarsa decemlineata (Say) in Eastern Ontario. I. Spatial pattern and transformation of field counts. Canadian Entomologist, 95, 813-20.

37. Harcourt, D. G. (1965). Spatial pattern of the cabbage looper Trichoplusia ni on crucifers. Annals of the Entomological Society of America, 58, 89-94.

38. Harcourt, D. G. (1967). Spatial arrangement of the eggs of Hylemya brassicae (Bouche), and a sequential sampling plan for use in control of the species. Canadian Journal of Plant Science, 47, 461-7.

39. Healey, M. C. (1971). the distribution and abundance of sand gobies, Gobius minutus, in the Ythan estuary. Journal of Zoology (London), 163, 117-229.

40. Healey, V. (1964). The density and distribution of two species of Aptinothrips (Thysanoptera) in the grass of a woodland. Entomologist, 97, 258-63.

41. Healy, M. J. R. & Taylor, L. R. (1962). Tables for power law transformations. Biometrika, 49, 557-9. 42. Heath, G. W. (1956). Biological studies in gall midges (Cecidomyidae). Unpublished Ph.D. thesis, University

of London. 43. Holme, N. A. (1950). Population-dispersion in Tellina tenuis da Costa. Journal of the Marine Biological

Association, Plymouth, 29, 267-80. 44. Howe, R. W. (1950). Studies on beetles of the family Ptinidae. III. A two-year study of the distribution and

abundance of Ptinus tectus Boieid in a warehouse. Bulletin of Entomological Research, 41, 371-94. 45. It6, Y., Gotoh, A. & Miyashita, K. (1960). On the spatial distribution of Pieris rapae crucivora population.

Japanese Journal of Applied Entomology and Zoology, 4, 141-5. 46. Iwao, Syun'iti (1968). A new regression method for analysing the aggregation pattern of animal popula-

tions. Researches on Population Ecology, 10, 1-20. 47. Iwao, Syun'iti & Kuno, Eizi (1971). An approach to the analysis of aggregation pattern in biological

populations. Statistical Ecology (Ed. by G. P. Patil, E. C. Pielou and W. E. Waters), 1, 461-513. Pennsylvania State University Press, University Park, U.S.A.

48. Jones, E. W. (1937). Practical field methods of sampling soil for wireworms. Journal of Agricultural Research, 54, 123-34.

49. Kanno, M. (1962). On the distribution pattern of the rice stem borer in a paddy-field. Japanese Journal of Applied Entomology and Zoology, 6, 85-9.

50. Kendall, M. G. & Buckland, W. R. (1957). A Dictionary of Statistical Terms. Oliver & Boyd, Edinburgh, Scotland.

51. Kendall, M. & Stuart, A. (1976). The Advanced Theory of Statistics (3rd edn). Griffin, London. 52. Kleczkowski, A. (1949). The transformation of local lesion counts for statistical analysis. Annals of Applied

Biology, 36, 139-52. 53. Kobayashi, S. (1965). Influence of parental density on the distribution pattern of eggs in the common

cabbage butterfly, Pieris rapae crucivora. Researches on Population Ecology, 7, 109-17. 54. Kono, T., Utida, S., Yoshida, T. & Watanabe, S. (1952). Pattern of spatial distribution of the rice-stem borer,

Chilo simplex, in a paddy field. Pattern of the spatial distribution of insects 2nd report. Researches on Population Ecology, 1, 65-82.

55. Kuno, E. (1968). Studies on the population dynamics of rice leafhoppers in a paddy field. Bulletin of the Kyushu Agricultural Experimental Station, 14, 131-246.

56. Littleford, R. A., Newcombe, C. L. & Shepherd, B. B. (1940). An experimental study of certain quantitative plankton methods. Ecology, 21, 309-22.

57. Lloyd, Monte (1967). Mean crowding. Journal of Animal Ecology, 36, 1-30. 58. Lyons, L. A. (1962). The effect of aggregation on egg and larval survival in Neodiprion swainei Midd.

(Hymenoptera: Diprionidae). Canadian Entomologist, 94,49-58. 59. Lyons, L. A. (1964). The spatial distribution of two pine sawflies and methods of sampling for the study of

population dynamics. Canadian Entomologist, 96, 1373-407. 60. Mason, R. R. (1970). Comparison of flight aggregation in two species of southern Ips (Coleoptera:

Scolytidae). Canadian Entomologist, 102, 1036-104. 61. McGuire, J. U., Brindley, T. A. & Bancroft, T. A. (1957). The distribution of European corn borer larvae

Pyrausta nubilalis (Hbn.) in field corn. Biometrics, 13, 65-78. 62. Menzies, R. J. & Widrig, T. M. (1955). Aggregation by the marine wood-boring Isopod Limnoria. Oikos, 6,

149-52. 63. Meyers, M. T. & Patch, L. H. (1937). A statistical study of sampling in field surveys of the fall population of

the European corn borer. Journal of Agricultural Research, 55, 849-71.





64. Miller, C. D. F., Mukerji, M. K. & Guppy, J. C. (1972). Notes on the spatial pattern of Hypera postica (Coleoptera: Curculionidae) on alfalfa. Canadian Entomologist, 104, 1995-9.

65. Miller, W. E. (1971). See discussion in Taylor (1971). 66. Milne, A. (1943). The comparison of sheep-tick populations (Ixodes ricinus L.). Annals of Applied Biology,

30, 240-50. 67. Monro, J. (1967). The exploitation and conservation of resources by populations of insects. Journal of

Animal Ecology, 36, 531-47. 68. Morisita, M. (1959). Measuring of the dispersion of individuals and analysis of the distributional patterns.

Memoirs of the Faculty of Science, Kyushu University, Series E. (Biology), 2, 215-35. 69. Mukerji, M. K. & Harcourt, D. G. (1970). Spatial pattern of the immature stages of Hylemya brassicae on

cabbage. Canadian Entomologist, 102, 1216-22. 70. Naylor, A. F. (1959). An experimental analysis of dispersal in the flour beetle Tribolium confusum. Ecology,

40, 453-65. 71. Nelder, J. A. & Wedderburn, R. W. M. (1972). Generalised linear models. Journal of the Royal Statistical

Society A, 135, 370-84. 72. Nielson, B. 0. (1963). The Biting Midges of Lyngby Aamose. Naturhistorisk Museum, Aarhus. 73. Nielson, C. 0. (1954). Studies on Enchytraeidae. 3. The micro-distribution of Enchytraeidae. Oikos, 5,

167-78. 74. Raw, F. (1956). The abundance and distribution of Protura in grassland. Journal of Animal Ecology, 25,

15-21. 75. Reynoldson, T. B. (1950). Natural population fluctuations of Urceolaria mitra (Protozoa, Peritricha) epizoic

on flatworms. Journal of Animal Ecology, 19, 106-18. 76. Ross, G. J. S. (1970). The efficient use of function minimisation in non-linear maximum likelihood

estimation. Applied Statistics, 19, 205-21. 77. Sevacherian, V. & Stern, V. M. (1972). Spatial distribution patterns of Lygus bugs in California cotton

fields. Environmental Entomology 1, 695-704. 78. Shibuya, M. & Ouchi, Y. (1955). Pattern of spatial distribution of the soy bean pod gall midge in a soy bean

field. Oyo-Kontyu, 11, 91-7. 79. Sylvester, E. S. & Cox, E. L. (1961). Sequential plans for sampling aphids on sugar beets in Kern County,

California. Journal of Economic Entomology, 54, 1080-5. 80. Symmons, P. M. (1963). The patterns of distributions of adults of the red locust (Nomadacris septem-fas-

ciata Serville) in an outbreak area. Entomologia Experimentalis et Applicata, 6, 123-32. 81. Takeda, S. & Hukusima, S. (1961). Spatial distribution of the pear lace bugs, Stephanitis nashi Esaki et

Takeya (Hemiptera: Tingitidae) in an apple tree and an attempt for estimating their populations. Research Bulletin of the Faculty of Agriculture, Gifu-ken prefect University, 14, 68-77.

82. Taylor, C. C. (1953). Nature of variability in trawl catches. U.S. Department of the Interior Fishery Bulletin, 83, 54, 145-66.

83. Taylor, L. R. (1961). Aggregation, variance and the mean. Nature, London, 189, 732-5. 84. Taylor, L. R. (1970). Aggregation and the transformation of counts of Aphisfabae Scop. on beans. Annals of

Applied Biology, 65, 181-9. 85. Taylor, L. R. (1971). Aggregation as a species characteristic. Statistical Ecology (Ed. by G. P. Patil, E. C.

Pielou and W. E. Waters) 1, 357-77. Pennsylvania State University Press, University Park, Pennsylvania, U.S.A.

86. Taylor, L. R. (1974). Monitoring change in the distribution and abundance of insects. Report Rothamsted Experimental Station for 1973 Part 2, 202-39.

87. Taylor, L. R. (1977). Migration and the spatial dynamics of an aphid, Myzus persicae. Journal of Animal Ecology, 46,411-23.

88. Taylor, L. R. & Taylor, R. A. J. (1977). Aggregation, migration and population mechanics. Nature, London, 265, 415-21.

89. Taylor, L. R. & Taylor, R. A. J. (1978). The dynamics of spatial behaviour. Behaviour and the Control of Population. Institute of Biology, London (in press).

90. Todd, C. D. (1978). Changes in spatial pattern of an intertidal population of the nudibranch mollusc. Onchidoris muricata in relation to life-cycle, mortality and environmental heterogeneity. Journal of Animal Ecology, 47, 189-203.

91. Ursin, E. (1960). A quantitative investigation of the echinoderm fauna of the central North Sea. Meddelelser fra Kommissionen for Danmarks Fiskeri og Havundersogelser, 2, (24), 1-204.

92. Utida, S., Kono, T., Watanabe, S. & Yoshida, T. (1952). Pattern of spatial distribution of the common cabbage butterfly Pieris rapae in a cabbage farm. Researches on Population Ecology, 1, 49-64.

93. Wada, Y. (1965). Population studies on Edmonton mosquitoes. Quaestiones Entomologicae, 1, 187-222. 94. Wadley, F. M. (1950). Notes on the form of distribution of insect and plant populations. Annals of the

Entomological Society of America, 43, 581-6. 95. Waters, W. E. (1955). Sequential sampling in forest insect surveys. Forest Science, 1, 68-79. 96. Waters, W. E. (1959). A quantitative measure of aggregation in insects. Journal of Economic Entomology,

52,1180-4.




97. Wilkinson, P. R. (1964). Pasture spelling as a control measure for cattle ticks in southern Queensland. Australian Journal of Agricultural Research, 15, 822-40.

98. Yates, F. & Finney, D. J. (1942). Statistical problems in field sampling for wireworms. Annals of Applied Biology, 29, 156-67.

99. Yoshida, T. (1954). The relation between the population density and the pattern of distribution of the rice-plant skipper Parnara guttata Bremer et Grey. Pattern of the spatial distribution of insects, 5th report. Oyo-Kontyu, 9, 129-34.

100. Yoshida, T., Utida, S., Kono, T. & Watanabe, S. (1952). On the distribution pattern of the 28-spotted lady-beetle Epilachna sparsa orientalis. Pattern of the spatial distribution of insects, 3. Researches on Population Ecology, 1, 83-93.

101. Youdeowei, A. (1965). A note on the spatial distribution of the cocoa mirid Sahlbergella singularis Hagl. in a cocoa farm in Western Nigeria. Nigerian Agricultural Journal, 2, 66-7.

(Received 3 April 1977)

APPENDIX A

In this appendix we compare theoretically two regressions by examining the variance of their y-variates, and in one case recommend a better procedure for fitting the data.

The regressions are:

log s2 =log a+b log x, proposed by Taylor (1961), s2

and ,i = a' + b'x, where A = +-?- 1, proposed by Iwao (1968). x s2 and x are derived fron N samples of organisms.

The variance of r,

v(h) = V(x) + V + 2Cov[ x, . (Al)

However, 2

_V(s2) 2E(s2)Cov(x,s2) [E(s2)]2 V(X- v - - [E(X-)12 3 + 4 (A2)

L7_X [E(xD]2 [E(x )] E(4

from Taylor's theorem in two variables ignoring second and higher order terms, where E is the expected value, and

Cov ,2 = E(s2) - E(x-) , and (A3)

rS2[ E(s2) Cov(x,s2) V(x)E(s2) Lx- E() [E() )]2 [E(x-)]3

from Taylor's theorem in two variables ignoring third and higher order terms.Now,

V(S2) tN (S)83 2

E(s2) = 2; Cov(x,s2) =

N'x)=N; E(x=N; V(x.)= . -2 E(x-) = u1 ; N'

where lk = E{[X- E(X)k} k > 1 ,1 = E(X)



and X is the random variable denoting counts of organisms. So, using (A2), (A3) and (A4), (Al) becomes

V() + - + N-11 1 I2I + 2[P3-13 91 (A5)

The variance of the y-variate for the other regression is, from Taylor's theorem in one variable ignoring second and higher order terms,

V(log S2)2[E(2)]2 = N 114 -( )2 . (A6)

A standard unweighted regression model requires constant variance of the y-variate for validity, otherwise the assumptions under which the model is fitted are false. However, the term #U2/N in eqn (A5), (the other terms forming a positive sum), suggests that V(m) will increase with increasing x since, for the field data normally encountered, 12 increases and N does not change systematically with x. Equation (A6) implies that, due to the divisor Nu22, for distributions generally involved V(log s2) will stabilize to a constant value with

increasing log x. Although theoretical distributions never adequately represent field data we shall compare V(m) and V(log s2) by assuming that counts of the organism have (a) Poisson distributions, and (b) Negative Binomial distributions with common parameter k.

(a) For a Poisson distribution with mean i, #i =12 =13 = ; 14-= +3 32,

Al 2 hence, V() - - + from eqn (A5) (A7) N N-l

2 1 and, V(log s2) N l +N-, from eqn (A6). (A8) N- I NA

Equations (A7) and (A8) show that while V(m) increases with increasing A, V(log s2) stabilizes to 2/(N- 1), a constant.

(b) A more realistic and contagious distribution to investigate is the Negative Binomial with parameters m and k, for which

ui =m, m(m + k)

12 = k

m(m + k) (2m + k) 3=

k2

m(m + k) (6m2 + 6km + k2 + 3k2m + 3km2) 14 -

k3

Then, V() N ( l)k{m[k2(N- 1)+ 2k(2N- 1)+ 3(N- 1)] +

m[k3(N- 1) + 2k2(3N- 1)+ 5k(N- 1) +

[2k2(N- 1 +kN)]}, (A9)

from eqn (A5) after some algebra and,





2 6 k V(log)s2) -+- + -

(A10) N- IkN Nm(m+k) '

from eqn (A6).

Equations (A9) and (A 10) show that for constant k, V(m) increases as m2, i.e. very quickly with increasing m, while V(log s2) stabilizes rapidly to [2/(N-1)] + [6/kN], a constant. There is no reason to suppose that more contagious distributions will not yield similar or even more extreme results.

Some form of weighting must be applied to the observations in order to fit the regression proposed by Iwao. m = a' + b'x can be reparameterized as s2 = yx + x-2, where y = a' + 1 and 6 = b'- 1. The two models can now be seen to be similar and discrimination between them may only be possible for large numbers of observations, over a wide range of x. The stability of the variance of the y-variate in the regression proposed by Taylor enables simpler techniques to be used in fitting the model than the regression proposed by Iwao for the mean crowding model.



APPENDIX B

Mean crowding model

Survey

PROTOZOA Urceolaria mitra von Sieb Ammonia beccarii Elphidium clavatum E. tisburburyensis Elphidium sp.

ANNELIDA Allolobophora chlorotica (Sav.) A. caliginosa (Sav.) Enchytraeidae

ARTHROPODA Crustacea

Limnoria tripunctata (Menzies) Microcalanuspygmaeus G.O. Sars.

Macrozooplankton Insecta Protura Protura spp. Orthoptera

Nomadacris septemfasciata Serville Hemiptera

Myzus persicae (Sulz.) on Prunus sp. M. persicae (Sulz.) on sugar beet M. persicae (Sulz.) in aerial samples Acyrthosiphon pisum (Harris) Aphisfabae Scop. (C.J.B.) A. fabae Scop. (C.G.J.) A. fabae Scop. (L.R.T.) Sitobion avenae F. field perimeter in wheat S. avenae F. field centre in wheat S. avenae F. field perimeter in barley S. avenae F. field perimeter in oats S. avenae F. field perimeter all crops S. fragariae Wlk. all crops Metopolophium dirhodum Wlk. all crops Nilaparvata lugens Stal. Eutettix tenellus (Baker) Aeneolamia varia (Distant) adults A. varia (Distant) nymphs Sahlbergella singularis Hagl. Stephanitis naski Esaki at Takeya Lygus spp. Sogatellafurcifera Horvath Nephotettix cinciteps Uhler Laodelphax striatellus Fall.

(MLP) Ref. Mean no. b +S.E. logloa + S.E. deviance

75 1-31 0-08 0-41 0-10 1-3416 19 2-63 0-28 -1.27 0-36 0-8471 19 2-02 0-16 -0-25 0-10 0-5011 19 2-30 0-26 -0-58 0-24 0'6076 19 1-44 0-24 0-24 0-07 0-7667

32 1-59 0-16 -0-18 0-21 0-9358 32 1-62 0-35 -0-31 0-39 1-2886 73 1-77 0-29 -0-27 0-66 15-59

62 1-32 0-07 0-45 0-04 3-9178 5 1-18 0-06 0-13 0-04 1-445

56 1-59 0-17 -0-43 0-43 3-5572

74 2-14 0-23 -0-37 0-31 0-9525

80 1-57 0.14 1.08 0-09 0-5133

17 1-36 0-03 0-72 0-02 3-3455 79 1-15 0-05 0-44 0-08 11-717 87 1-91 0-10 0-24 0-15 1-9542 30 1-42 0-03 0-48 0-03 3-473

3,84 1-75 0-06 -0-26 0-18 58-375 84 1-72 0-05 0-83 0.11 10-52 84 1-63 0-04 1.20 0-04 9-8523 24 1-43 0-13 0-61 0.10 0-1960 24 1-06 0-11 0-48 0-07 0-09452 24 1-50 0-28 0-44 0-18 0-02425 24 1-67 0-29 0-26 0-30 0-2214 24 1-43 0-10 0.54 0-08 0-1640 24 1-74 0-14 0-77 0-06 0-0959 24 1-41 0-40 0-78 0-12 0-6963 55 1-75 0-06 0-39 0-05 11-475 16 1-71 0-05 -0-29 0-04 0-2662 26 1-19 0-39 0-07 0-53 0-3833 26 1-83 0-22 -0-20 0-30 4-510

101 0-85 0-21 0-53 0-09 24-1375 81 1-48 0-12 0-43 0-03 11-642 77 1-60 0-12 -0-12 0-10 7-617 55 1-67 0-08 0-17 0-05 7-125 55 1-42 0-03 0-09 0-02 3-826 55 1-63 0-04 0-12 0-02 2-529

(GLIM) Significance of curvature

N.S. N.S. N.S. N.S. N.S.

(GLIM) Mean Significance

deviance of curvature

1-3535 No fit obtainabl

0-4846 0-6314t 0-7908

N.S. 0-9274 N.S. 1-2568 N.S. 14-00

N.S. 4-7214 N.S. 1-146 N.S. 3-5289

N.S. 1-004t

N.S.

N.S. N.S. N.S. N.S. N.S. N.S.

N** N.S. N.S. N.S. N.S. N.S.

N.S. N.S.

N.S. N.S. N.S.

N.S. N.S. N.S. N.S. *** *** ***

0-5250

6-1999 21-992

2-0184t

Appraisal Weighting M of data used

N.S. 51 2 D

let 16 3 S N.S. 16 3 S N.S. 16 3 S N.S. 14 4 S

N.S. 52 3 S N.S. 30 3 S N.S. 4 2 D

N.S. 30 2 S N.S. 8 3 D N.S. 20 4 S

N.S. 8 3 S

* 42 3 N.A. O

*** 24 1 S N.S. 14 3 D N.S. 21 2 D

4-2648 * 25 57-125 N.S. 26 24-1 * 16 12-1585 *** 67 0-1769 N.S. 10 0-09345 N.S. 8 0-02657 N.S. 5 0-2306 N.S. 7 0-1422 N.S. 22 0-1121 N.S. 11 0-7193 *** 12

11-280 N.S. 87 0-1999 N.S. II

No fit obtainablet 15 4-801 N.S. 11

24-2125t N.S. 10 12-284 N.S. 21 7-337 N.S. 37 5-279 N.S. 30 3-469 N.S. 120 1-893 N.S. 58

1 S 1 D 1 D 1 D 3 N.A. 3 N.A. 4 N.A. 3 N.A. 2 N.A. 3 N.A. 4 N.A. 1 D 2 S 4 N.A. 2 S 3 S 3 D 3 D 2 D 2 D t 2 D w

Power law model

I



Survey

Thysanoptera Thrips imaginis Bagnall Aptinothrips rufus (Gmelin) A. stylifer Trybom

Lepidoptera Pieris rapae L. eggs P. rapae L. eggs from captive females P. rapae L. P. rapae L. I st instar larvae P. rapae L. 2nd instar larvae P. rapae L. 3rd instar larvae Parnara gutta (Bremer et Grey) Euxoa nigricans (L.) Spilosoma luteum (Hufnagel) Trichoplusia ni (Hbn.) T. ni (Hbn.) larvae Plutella maculipennis (Curt.) Pyrausta nubilalis Hbn. European corn borer P. nubilalis Hbn. European corn borer Chilo suppressalis W. young larvae C. suppressalis W. full grown larvae C. suppressalis W. infant C. simplex Butler Christoneurafumiferana (Clem.) Cactoblastis cactorum Berg.

Diptera Dacus tryoni Frogg. 'stings' per fruit Culiseta inornata (Williston) Culicoides impunctatus Goet. Jaapiella medicaginis (Rub.) Asphondylia sp. midges per plant Asphondylia sp. midges per pod Tipula spp. Leptohylemyia coarctata (Fall.) larvae total L. coarctata (Fall.) larvae group C L. coarctata (Fall.) eggs group D L. coarctata (Fall.) eggs A Oscinellafrit L. Hylemya brassicae (Bouche) eggs H. brassicae (Bouche) larvae

Power law model

(MLP) Ref. no. b +S.E. logloa + S.E.

23 1-52 0-27 40 1-49 0-16 40 2-10 0-27

35 2-75 0-74 53 1-23 0-09 45 1'91 0-04 92 1.18 0-29 92 1-22 0-14 92 1-16 0-03 99 1-03 0-03 88 2-36 0-14 88 1-71 0-17 37 1-27 0-06 31 1-18 0-02 34 1-39 0-15 63 1 63 0-03 61 1-25 0.01 49 1-13 0.19 49 1-44 0-12 49 1-32 0-15 54 1-48 0-09 95 1-37 0-06 67 1-49 0-21

67 0-47 0-22 93 1-60 0-05 72 1-49 0-16 42 1-24 0-13 78 1-37 0-05 78 0-98 0.01 6 1'58 0-20 4 1-21 0-04 4 1-01 0-08 4 1-58 0.15 4 1-24 0.05

20 2-04 0-06 38 2-37 0-34 69 1-43 0-09

nymenoptera Neodiprion swainei Midd. cocoons 59 1-35 0-06 N. swainei Midd. egg parasites 58 1-58 0-24 N. swainei Midd. egg clusters 59 1-81 0-15 N. sertifer (Geoff.) cocoons 59 1-22 0-02

Coleoptera Phyllophagaspp. 33 1-21 0-17 Popilliajaponica New. Japanese beetle larvae 11 1.05 0-14

0-43 0-48 0-77 0-20 0-44 0-19

-1 06 0-62 0-29 0-07 0-32 0-04 0-21 0.10 0-49 0.10 0-41 0-02 0-03 0-02 1.08 0-06 0-68 0-27 0-21 0-02 0-30 0-02

-0-02 0.05 0-55 0-06 0-15 0.01 1-04 0-06 0-41 0-05 1-46 0-06 0-83 0-04

-0-03 0-04 0-64 0-08

-0-43 0-04 0-71 0-06 0-66 0-07 0-18 0-05 0.38 0-04

-0-01 0-01 -0-41 0-26

0-12 0-03 0-12 0-03

-0-03 0-12 0-15 0-02 0-10 0-08

-0-84 0-42 0-44 0-04

0-66 0-04 0-78 0-13 0-15 0-10 0-26 0-01

Mean crowding model

(GLIM) Mean Significance

deviance of curvature

2-8664 N.S. 2-8614 1 6286 N.S. 2-003 1-942 N.S. 1-904

13-721 5-633 0-02351 0-01302 0-0762 0-00235 4-969 5-08 0-4400 0-03163 4-8402 0-7142 0-3083 0-2609 0-0167 0-01055 0-00641 3-7285 1-577

10-907

5-3675 1-2360 3-0682 0-9421 0-0643 0-001713 0-9324 0-1310 0-1301 0-03872 0-1145 0-6505

24-08 0-0701

1-6280 1-3611 0-1077 1-628

N.S. 12-778t N.S. 5-112 N.S. 0-02430 N.S. 0-01368 N.S. 0-0664 N.S. 0-00500 N.S. 4-267 N.S. 9-023t N.S. 0-4387 N.S. 0-03165 *** 3-2017 N.S. 0-7208 *** 0-1269 N.S. 2-3845 N.S. 0-0170 N.S. 0-00877 N.S. 0-00574 N.S. 3-7523 N.S. 1-571 N.S. 12-778

N.S. 6-9088t * 1-1214

N.S. 2-9509 N.S. 1-044 N.S. 0-0735

* 0-001314t N.S. 0-7254 N.S. 0-1381 N.S. 0-1305 N.S. 0-03817 N.S. 0-1094 N.S. 0-5832t N.S. 24-07t N.S. 0-0669

N.S. 1-9232 N.S. 1-4667 N.S. 0-1053 N.S. 1-9232

0-18 0'05 0-03361 N.S. 0-03476 0-06 0-18 0-01336 N.S. 0-01363

(GLIM) Mean Significcance

deviance of curvature M Appraisal Weighting

of data used

N.S. 16 4 S N.S. 9 3 D N.S. 9 4 D

N.S. 16 4 S N.S. 11 2 S N.S. 36 1 N.A. N.S. 5 4 N.A. N.S. 5 4 N.A. N.S. 5 3 N.A. N.S. 16 3 D N.S. 9 2 D N.S. 9 3 D N.S. 26 3 D N.S. 119 2 S N.S. 28 4 S *** 18 1 D N.S. 4 2 D N.S. 5 4 N.A. N.S. 5 4 N.A. N.S. 5 4 N.A. N.S. 15 3 D N.S. 50 2 S N.S. 8 4 D

N.S. 10 4 D N.S. 16 2 D N.S. 13 4 D N.S. 12 4 D N.S. 24 2 N.A. N.S. 24 2 N.A. N.S. 7 4 D

* 75 2 N.A. N.S. 84 4 N.A. N.S. 8 4 N.A. N.S. 83 4 N.A. N.S. 21 2 D N.S. 16 3 S N.S. 30 3 N.A.

N.S. 27 1 D N.S. 11 3 D N.S. 12 2 N.A. N.S. 27 1 D

N.S. 20 4 N.A. N.S. 11 3 N.A.

0



Power law model

(MLP) (GLIM) Mean Significance

b +S.E. logloa + S.E. deviance of curvature

Mean crowding model

(GLIM) Mean Significance Appraisal

deviance of curvature M of data

P. japonica New. P. japonica New. Amphimallon najalis Raz. European

chafer larvae Agriotes sp. wireworms in grass Agriotes sp. wireworms in arable Limnonius sp. wirewtrms Ptinus tectus Boield Tribolium confusum Jacquelin du Val. Epilachna sparsa orientalis Dieke Leptinotarsa decemlineata (Say)

Colorado beetle L. decemlineara (Say) eggs L. decemlineata (Say) Hypera postica (Gyll.) Brachyrhinus ligustica (L.)-larvae Ips grandicollis (Eichh.) I I. grandicollis (Eichh.) 2 I. grandicollis (Eichh.) 3 I. avulsus (Eichh.) I I. avulsus (Eichh.) 2 I. avulsus (Eichh.) 3

Arachnida Eniochthonius minutissintus (Berl.) Nothrus palustris (Koch) Platynothrus peltifer (Koch) Suctobelba subtrigona (Oudms.) Oppia quadricarinata (Mich.) O. nova (Oudms.) 0. ornata (Oudms.) Ceratoppis bipilis (Herm.) Tectocepheus velatus (Mich.) Odontocepheus elongatus (Mich.) Oribatula tibialis (Nic.) Chamobates incisus v.d. Ham Minunthozetes semirufus (c.l. Koch) Metatetranychus ulmi (Koch)-red spider mite Boophilus inicroplus-cattle ticks Punctoribates punctum Ixodes ricinus L.

Insect and virus/plant interactions Virus lesion Mirid damage to cocoa Wheat, damaged shoots wheat bulb fly Wheat, undamaged shoots Wheat, total shoots Wheat, total plants

11 1-06 0-11 0-24 0-10 0-03212 28 1-73 0-06 -0-14 0-15 49-194

18 0-99 0-04 0-05 0-02 1-277 98 1-18 0-01 0-13 0-00 0-00143 98 1-24 0-03 0-15 0-01 0-01187 48 1-35 0-06 0-27 0-04 5-1818 44 1-49 0-11 0-46 0-07 0-4866 70 0-75 0-04 0-13 0-03 1-675

100 1-05 0-12 0-13 0-09 7-665

8 1-75 0-19 36 1-40 0-37 8 1-35 0-08

64 1-27 0-09 29 2-03 0-11 60 1-05 0-18 60 1.10 0-08 60 0.91 0-15 60 1-10 0-13 60 1-39 0-14 60 1 60 0-14

-0-06 -0-03

0-06 0-45

-0-09 0-01

-0-01 0-05 0-17 0-20 0-26

9 1-57 0-15 0-55 9 1-02 0-10 0-29 9 1-54 0-10 0-63 9 1-43 0-14 0-57 9 1-41 0-10 0-64 9 1-42 0-12 0-78 9 1-29 0-08 0-68 9 1-29 0-17 0-44 9 174 0-12 0-41 9 1-22 0-31 0-57 9 1-07 0-20 0-53 9 1-62 0-28 -0-04

21 1-85 0-24 0-12 22 2-06 0-13 -0-13 97 2-16 0-07 0-37 21 1-00 0-30 0-22 66 1-46 0-11 0-24

52 1-32 0-09 0-16 85 1-99 0-08 0-16 4 1-23 0-04 0-19 4 1-27 0-07 0-20 4 1-26 0-09 0-19 4 1-19 0-06 0-11

0-13 0-21 0-06 0-07 0-10 0-04 0-03 0-04 0-06 0-08 0-07

0-00548 0-1024 1-305 0-2119 0-2712 1-4955 0-7542 1-4626 1-845 2-772 2-5383

0-12 9-383 0:04 1-4475 0.08 2-4375 0-16 6-369 0-07 3-627 0-09 13-775 0-09 1-400 0-07 6-493 0-13 7-821 0-10 5-920 0-19 2-454 0-30 7-02 0-21 0-2853 0-25 2-198 0.11 1-59 0-17 0-4508 0-14 4-8875

0-17 0-13 0-03 0-09 0-13 0-07

0-8600 0-1310 0-1403 0-3000 0-1648 0-1743

N.S. N.S.

N.S. N.S. N.S. N.S. N.S. N.S. N.S.

0-03309 42-950

1-277 0-00249 0-01209 5-0180 0-7432 1-805t 8-510

N.S. 0-00584 N.S. 0-1026 N.S. 1-059 N.S. 0-2412 N.S. 0-2736t N.S. 1-5018 N.S. 0-7691 N.S. 1-4477t N.S. 1-870 N.S. 2-784 N.S. 2-8783

N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S.

11-977 1-4385 4-195 7-437 3-604

15-238 1-480 6-015 8-631 6-093 2-355 6-58 0-2818 2-213 2-038t 0-4484t 3-3838

N.S. 0-8027 N.S. 0-1299t N.S. 0-1464 N.S. 0-2959 N.S. 0-1620 N.S. 0-1761

N.S. 13 N.S. 20

N.S. 74 *** 35

N.S. 35 N.S. 24 N.S. 6

* 79 N.S. 8

N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. N.S. ***

5 16 16 20 7

40 35 37 34 29 30

3 N.A. I D

3 S 1 N.A. 1 N.A. 2 S 3 S 3 S 4 D

4 N.A. 4 N.A. 2 D 2 N.A. 4 S 4 S 4 S 4 S 4 S 4 S 4 S

N.S. 5 4 S N.S. 6 4 S N.S. 6 3 S N.S. 12 2 S N.S. 12 2 S N.S. 10 2 S N.S. 12 2 S N.S. 6 4 S N.S. 12 2 S N.S. 6 4 S N.S. 6 4 S N.S. 6 4 S N.S. 8 4 S N.S. 12 2 S ** 18 1 S ** 7 4 S

N.S. 10 2 D

N.S. 17 2 D N.S. 31 2 N.A.

* 78 2 N.A. N.S. 81 3 N.A. N.S. 82 3 N.A. N.S. 81 3 N.A.

Survey Ref. no. Weighting

used

t' l

Co

r 0

06

Ur



Power law model

(MLP) Ref. no. b + S.E. logloa + S.E.

0 Mean crowding model

(GLIM)

Mean Significance deviance of curvature

(GLIM)

Mean Significance deviance of curvature M

Appraisal of Weighting of data used

Wheat, plants in attacked plots Wheat, plants in unattacked plots Wheat, straws in unattacked plots Wheat, straws in attacked plots Wheat, damaged shoots Wheat, plants surviving

MOLLUSCA Tellina tenuis da Costa Onchidorus muricata (Miiller)

ECHINODERMATA Ophiura albida Echinocyamus pusillus Echinocardium cordatum Amphiurafiliformis Total Echinoderms Acrocnida brachiata

VERTEBRATA Fish

Melanogrammus aeglefinus (L.), haddock Merluccius bilinearis, whiting Gobius minutus, sand goby

Birds Parus major, great tit Streptopelia decaocto, collared dove

Mammals Vulpes?fulva, coloured fox in Labrador Martes americana, marten in Labrador U.S.A. human populations

PLANTS Gentiana acaulis Arnica montana Campanula barbata Senecio campestris Cirsium arvense Ophrys apifera Ranunculus bulbosus Gentiana amarelle Erythraea centaurium Poterium sanguisorba Ribes spp.

4 1-06 0-14 4 1-04 0-17 4 0-75 0-11 4 0-92 0-10 4 1-31 0-13 4 0-84 0-25

0-19 0-08 0-16 0-12 0-69 0.10 0-69 0-08 0-20 0-13 0-51 0-39

43 0-56 0-16 -0-18 0-19 90 1-42 0-09 0-44 0-03

91 1-53 0-24 0-52 0-18 91 2-27 0-21 0-34 0-14 91 1-56 0-20 0-30 0-10 91 1-87 0-15 0-32 0-12 91 2-19 0-20 -0-43 0-22 91 1-64 0.05 0-37 0-03

82 2-33 0-18 -0-14 0-30 82 2-09 0-25 0-08 0-53 39 1-63 0-08 0-07 0-08

7 1-79 0-04 7 1-32 0-06

25 1-69 0-06 25 1.91 0-08

1 2-04 0-01

0-1093 0-1182 0-1111 0-1484 0-0560 0-2154

0-900 0-2308

1-3469 0-4432 0-7895 0-6844 1-108 0-008215

3-9164 6-4375 0-9551

0-18 0-04 1-365 0-75 0-05 4-386

N.S. 0-1097t N.S. 0-1185 N.S. 0-1076t N.S. 0-1484t N.S. 0-0574 N.S. 0 1675t

N.S. 0-629t N.S. 0-2714

N.S. 1-3800 N.S. 0-4532t N.S. 0-8205 N.S. 0-6447 N.S. 1-116 *** 0-01655

N.S. 4-8950t N.S. 6-8313t N.S. 0-8892

N.S. 1-226 N.S. 7-475

0.19 0-10 6-5539 N.S. 6-5180 0-38 0.09 0-6805 N.S. 0.6727 0-26 0-02 0-01635 N.S. 0-01761t

10 1-04 0-05 0-12 0-02 10 0-88 0-06 0-35 0-04 10 1-07 0-05 0-17 0-03 10 1-12 0-08 0-11 0-04 10 1-27 0-06 0-23 0-02 10 1-15 0-11 0-26 0-04 10 0'99 0.08 0-41 0-02 10 1-14 0-12 0-29 0-03 10 1-48 0-14 0-74 0-06 10 0'99 0'10 0.00 0-02 94 1-41 0.16 0-65 0-08

1-6292 4-5775 1-0801 4.537 1-889 2-487 2-5878 4-8331 7-490 2-2672

14-483

N.S. 1-6738 N.S. 4-7083t N.S. 1-2108 N.S. 3-775 N.S. 1-707 N.S. 2-548 N.S. 2-5883t N.S. 4-9625 N.S. 8-874 N.S. 2-2856t N.S. 13-023

N.S. 32 4 N.S. 33 4 N.S. 92 4 N.S. 95 4 N.S. 10 3 N.S. 7 4

N.S. 5 4 N.S. 17 3

N.A. N.A. N.A. N.A. N.A. N.A.

D D

N.S. 15 4 S N.S. 11 3 S

* 22 4 S N.S. 16 3 S N.S. 27 4 S *** 4 4 S

N.S. 16 3 D N.S. 18 3 D b N.S. 51 3 D

N.S. 21 2 D O N.S. 12 2 D

N.S. 91 2 D N.S. 46 2 S N.S. 10 1 S

N.S. N.S. N.S.

N.S. N.S. N.S. N.S. N.S.

N.S. N.S.

15 3 D 14 3 D 10 3 D 15 3 S 30 3 S 12 3 S 20 3 S 16 3 S 12 4 S 20 3 S 6 4 D

* P<0-05, ** P<0-01, *** P<0-001. * P<0-05, ** P<0-01, *** P<0-001. N.S. not significant. t Data sets in which one of the estimated parameters of the mean crowding model was negative. N.A., Values of N not available; S, Values of N available and identical; D, Values of N available and different.

Survey



the density-dependence of spatial behaviour and the rarity ... · journal of animal ecology (1978),...

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