The Density-Dependence of Spatial Behaviour and the Rarity of Randomness

24 Pages·2007·2.47 MB·English

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) Spatiald istributionw as analysedi n 156 sets of field data comprisingm ore than
200 000 sampleu nits in 3840 samplesf rom 102 species.
(2) Taylor's model, proposingt hat spatial variancei s proportionalt o a fractional
power of mean populationd ensity,p rovideda n appropriated escriptiono f 147 sets of
data.
(3) Iwao'sm eanc rowdingm odel provideda ratherl ess adequatea lternativeb ut with
markedl imitations.
(4) The negativeb inomialw ith a commonk was not satisfactory.
(5) Two data sets only werej udgedr andoma t all populationd ensities.
(6) In all other sets of samples,o nly at or near the lowest densitiesw as disposition
indistinguishablfer om randoma nd the range of spacingb etweeni ndividualsp rogress-
ively and disproportionatelcyh angeda s mean densityi ncreased.
(7) Spatiald ispositioni s thusd ensity-dependenatn d we deducet hat spatialb ehaviour
is also density-dependenats requiredb y Taylor& Taylor'sA -modelf or intrinsicp opula-
tion controlb y movement.
(8) The speciese xaminedr angef romP rotistat hrougha nnelids,m olluscs,c rustaceans,
arachnidsi,n sects,e chinodermsf,i sh, birdsa nd quadrupedst o men and orchids.
(9) The samplings calesr angef romc iliateso n the surfaceo f a flat-wormt o the human
populationo f the United States of Americaa nd includet emporallys tablea nd unstable
populations.
(10) The samplingm ethodsr ange from quadratc ounts, throught rap, net and grab
samplest o volumetrics amplesa nd completec ensuses.
INTRODUCTION
In contrast to dispersion in the physical world, the dispersive processes of living
organismsi nvolvei ntrinsicb ehaviourarl esponsest hat shouldm ake spatialr andomness
highly improbable.N evertheless,r andomnessi s often taken as a starting point for
attempts to define spatial distributionsi n ecology because it appearst o be the only
condition,e xcepte venness,c apableo f unequivocald efinition.I n fact, Kendall& Buck-
land's (1957) statisticald ictionary,l ike other statisticalt exts, does not define' random'
spatiallyb ut as 'representinagn undefinedid ea, or if defined,( one that)m ustb e expressed
in termso f probability'T. his is not an appropriates tartingp oint for the ecologistw ho,
althoughh e is faced with statisticalp roblemso f comparingi ndividuals amples,i s more
concernedw ith the fundamentabl iologicalp roblemo f graspingw hy organismsa rrange
0021-8790/78/0600-0383$02.000 1978 BlackwellS cientificP ublications
383
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384 Spatial behaviour
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
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L. R. TAYLORe t al. 385
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 distribu-
tions.
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 justifica-
tion 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)
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386 Spatial behaviour
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
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L. R. TAYLORet al. 387
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,w hich 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 ^~A~0 ^I ^ ' ~0
0.1 I 10 100 0 20 40
m (log scale) m (arith.s cale)
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)w ill almost certainly be approximately
proportional to [E(s2)]2w here 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)'werec alculated
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)
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388 Spatial behaviour
a b
400 -
100000- o0
o 10000- ? / :00 -/
o *E3 200- o
o / o 0
0 0@ 0
% // :o
- 1000- V / / 0
N '1. $1l 2T0 0-'2l 00- 0 l
ci 00 10- /
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 distribu-
tions, 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 GLIMa nalysis 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 (lnx )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.
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L. R. TAYLOR et al. 389
Where possible this was done for the GLIMan alyses. 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
2
/? i I -ve
0 1 2 3
Sample mean (m) Projected mean (m)
FIG. 3a. Iwao's quadratic fits short range data such as that for Ranunculusb ulbosus (Blackman
1942), either (I) constrained (MLP)s 2=2-569 x+0-00397 x2, or (II) unconstrained (GLIM)
s2=2.699 x-0-0446 x2. b. Projectedo ver a wider rangeo f populationd ensitiest he uncon-
strained( II) negativep arameter( b) yields an unacceptablein vertedp arabolah avingn egative
variancesa t 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 obtainedu singG LIM and MLP.
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390 Spatial behaviour
RESULTS
The detailed results are given in Appendix B.
Best-fitting models (MLPa nd 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 GLIMa nalysis 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(A ppendix
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%l evel.
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)a nd residual mean square (MLP)
The mean deviance (GLIMa) nd 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 GLIMa nalysis, 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.
Randomnessa nd 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).
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L. R. TAYLORet al. 391
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 (Aphisf abae 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 (Echinocyamusp usillus,
Ursin 1960), a fish (Melanogrammusa eglefinus (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.,
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392 Spatial behaviour
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
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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).
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