Reprinted from Nature, Vol. 323, No. 6085, pp. 255-257, 18 September 1986
© Macmillan Journals Ltd., 1986
Spatial heterogeneity: evolved
behaviour or mathematical artefact?
0.9-1.0(7.8%)
0.8-0.9(6 .1%)
0-0.1(34 %)
0.7-0.8(6.2%)
John A. Downing
Departement de Sciences Biologiques, Universite de Montreal,
CP 6128, Succursale 'A', Montreal, Quebec, Canada H3C 3J7
0.6-0.7(6.3%)
0.5-0.6(6.4%)
For more than a century ecologists have sought to explain the
spatial heterogeneity of plants and animals1, but progress has been
hampered by measurement bias2. A measure thought to be an
unbiased index of spatial heterogeneity3 is 6, the fitted exponent
in the empirical relationship s2 = aM\ where s2 is the variance
and M the average of randomly placed replicate population esti
mates4. This index is widely accepted because of the impressive
correlation between s2 and M and because it requires no interorganism distance measures. Theoretical models, based on migra
tory behaviour5 or demographic factors6, have been proposed to
account for the relationship between s2 and M. These models
disagree regarding the effect of environment on b and the diver
gence of b values shown by different species. Here I report data
showing that species-specific b varies among environments and
that different species often show similar b values, favouring the
demographic model. Analysis of these data shows that different
levels of replication and sampling coverage lead to biased b values,
casting doubt on the use of b for the deduction of cause of spatial
variance relationships or the comparative measurement of spatial
heterogeneity.
Plants and animals are not uniformly distributed in nature.
Two contrasting mechanisms5'6 have been proposed to explain
the empirical power-function relationship between s2 and M
0.4-0.5(6.
-0.2(10.9%)
0.3-0. 4(7
0.2-0.3(8.3%)
Fig. 1
Frequency of probability levels obtained in 24,310 pairwise
Mests of the hypothesis that species-specific b values are equal.
Comparisons are for 221 species of aquatic and terrestrial animals
(refs 17-37 and refs listed in ref. 29 as 3, 5,6,9, 10, 16, 18, 19, 21,
29-31, 38-40, 43, 44, 48, 55, 56, 59, 62, 67, 70, 74, 75, 81, 90, 94,
97 and 101). Variance functions were calculated on a per sampler
basis (unstandardized to area or volume), and include only organ
isms from a single sampling site. Analyses include only species for
which correlations between log s2 and log M were statistically
significant (P<0.01).
Frequency histograms of b values have been published4'5'15, but
nobody has examined these exponents to find the frequency
with which the null hypothesis of equality of b values can be
rejected.
I collected observations of M and s2 from several publications
(see Fig. 1 legend) and performed linear regression analyses of
log s2 on log M specific to species, site and design (sampler and
replication level) for 221 species of terrestrial and aquatic
animals (data available from me on request). Despite the
(ref. 7). One mechanism suggests that the fitted exponent b is
a measure of spatial heterogeneity that results from a specifically
strength of correlations (77% of r2 are >0.8) and the precision
evolved
of b values (78% of s.e.b/6 are <0.2), dissimilar b values are
combination
of
'migratory'
and
'congretory'
behaviours8'10 (the A model). The second suggests that b arises
from the stochastic interplay of demographic characteristics of
populations and environmental
heterogeneity6111"13. These
models disagree on at least three points. First, demographic
simulations suggest that b values are highly variable among
populations in different growth phases or environments6,
whereas the A model suggests that b is constant within species
regardless of environment5. Second, the A model suggests that
because b results from innate behavioural responses, b values
'segregate' species3'14. Therefore, b fitted for different species
should frequently be dissimilar. Third, demographic models
considering environmental heterogeneity show that b should
rarely lie outside the range 1-2 (ref. 6). The A model allows
greater latitude in b; it is supported by fitted values as high as
3.9 (ref. 15). Next I present tests of these controversial points.
Discussion surrounding one of the earliest estimations of a
s2: M power-function16 suggests that b is an intrinsic property
of the organisms concerned and therefore independent of
environmental influence. Previous tests of this hypothesis have
been anecdotal5 or confound temporal and spatial variability
(for example, ref. 14). Inter-habitat comparisons of b values
estimated by repeated sampling of several species of zooplankton, benthic invertebrates and terrestrial insects (Table 1) reveal
frequent dissimilarity of b values within species. I found sig
nificant differences among b values (P < 0.05) in 86% of species.
Of 19 possible intra-specific comparisons, 30% show different
values of b in replicate determinations. The spatial heterogeneity
of species, as measured by b, varies significantly among environ
ments.
rare. Figure 1 shows the distribution of t probability levels for
all possible pairwise tests of the null hypothesis. Only about
1/3 of the comparisons show significant differences (P<0.1)
and 30% of these would be expected by chance alone. The null
hypothesis of equal b values can therefore be accepted with
P>0.1 in at least 2/3 of the pairwise comparisons of b. Less
than 2% of the taxa examined yield b values different (P < 0.05)
from all other species. Either there is a high degree of conver
gence in the evolution of species behaviour or stochastic, mathematic or demographic factors are operating similarly on many
species.
Monte Carlo simulations of the stochastic/demographic
model of spatial distribution with moderate environmental
heterogeneity6 predict that b should vary only between 1 and
2, with exact values determined by population parameters and
environmental variation. This contrasts with empirical studies
that show values of b ranging between 0.4 and 3.9 (ref. 5). Such
studies use a wide range of replication both in calculation of
M and s2, and in the estimation of regression coefficients.
Greater replication in M and s2 calculations («„) would lead
to more accurate estimates of fi and a2 and less error in the fit
of regressions38'40. Inclusion of more (M, s2) pairs (nr) in
regressions would also lead to more stable estimates of b (ref.
41).
Contrary to current opinion3, b is biased by mathematical
artefact. The range and size of b vary with the number of samples
taken and the range of M considered. The range of b is largest
where nm nr or the range of M covered by the s2: M relationship
is small (Fig. 2A-C): b outside the range of 1-2 is rare where
The view that b arises because of species-innate behavioural
responses is essential to the A model and is inconsistent with a
stochastic/demographic approach. The evolutionary origin of
behavioural responses suggests both their constancy within
M and s2 are calculated on >60 replicates (Fig. 2A)\ more than
20 (M, s2) pairs are included in the regression (Fig. 2B)\ or
believe that, given enough statistical power, most species could
mates tend to 1< b < 2 as suggested by the demographic model6.
species and their individuality5'8. Proponents of the A model
be demonstrated to have significantly divergent b values14.
more than one order of magnitude is covered by M (Fig. 2C).
Extreme values of b (6 < 1 or b> 2) appear frequently at low
replication or narrow density coverage. Stable and robust esti
I found that b varies systematically with degree of replication.
A
:
Table 1
Student's f-tests of replicate species-fitted b values
Species/sampling site
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Michigan/Ontario
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Michigan/Ontario
1.15
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1.69
Bosmina coregoni
Donk Lake, Belgium
Belgian reservoir
Leptinotarsa decemlineata
Field near Chatham,
Ontario
Plots near Merivale,
Ontario
Limnodrilus hoffmeisteri
Lake Norman, North
Carolina
Lake Norman, North
Carolina
Lake St Clair,
Monodiamesa bathyphila
Lake St Clair,
Michigan/Ontario
Harding Lake, Alaska
Papillia japonicum
Plots near Jacobstown,
New Jersey
Plots near Jacobstown,
New Jersey
Plots near Jacobstown,
New Jersey
Pieris rapae
1
2
Range of m
Fig. 2 Site, species and design-specific b values from published
sources (see Fig. 1 legend), plotted against: A, the log10 of the
number of replicate population estimates included m M and s2
(nu); B, the number of (Af, s2) pairs included in each variance
regression (nr); and C, the range of Af in logarithmic form
(log,0Mraax-log,0 Mmia) covered by each variance regression.
Two negative values of b have been omitted from the extreme left
of each plot. Variance functions were calculated on a per sampler
basis (unstandardized to area or volume), and include only organ
Plots near Merivale,
Ontario
Field near Sendai,
Japan
Pyrausta nubilalis
Field in north-east
Iowa
Field in north-east
Iowa
Fields in Wood
County, Ohio
Fields in Wood
County, Ohio
s.e.6
1
0.608 L
<o.ooi"l
<0.00lJ
*
L
0.01 lj
0.240
0.327
Significant differences between b values at: P < 0.05 (*), P < 0.01 (**).
nv, number of replicate estimates included in each calculation of M
and s2; nr, number of (M, s2) pairs in each regression analysis; s.e.fr,
standard error of the estimate of b.
isms from a single sampling site. Analyses include only species for
which correlations between log s2 and log M are statistically sig
nificant (P<0.01).
A multiple regression analysis (three negative b values omitted
where M spanned 0.0001 of an order of magnitude) shows that
seem most realistic. Unfortunately, b values are biased and
cannot be used as support or criticism of any causal model of
spatial distribution.
I thank E. McCauley, R. H. Peters, W. L. Downing, A. Morin,
b = 2.057-0.111oglonu-0.371og,0 nr (218 species; F=8.2; P«
H. Cyr and C. Plante for helpful discussions. Supported by the
Natural Sciences and Engineering Research Council of Canada
and the Minister of Education of the province of Quebec
(FCAR).
Concrete examples of the bias in b are shown in Table 1.
Papillia japonicum and Pyrausta nubilalis have significantly
Received 27 February; accepted IS June 1986.
0.001) despite heteroscedasticity. Bias in b is a major technical
problem, suggesting that comparisons of b should only be made
where identical levels of replication are used and equivalent
ranges of M are examined.
different b values although determinations of b were made on
the same sites and dates. The lower value of b is always found
where more samples are analysed. Hence, competing theoretical
models are arguing about the ecological meaning of mathemati
cally biased estimators of spatial heterogeneity, rather than
about spatial heterogeneity itself.
Environmental differences, demographic factors, evolved
behaviour and statistical artefact may each play a part in the
perceived spatial distribution of natural populations. Analysis
of b values of spatial variance relationships suggest that spatial
heterogeneity is no less variable within species than among
species. The assumptions of the demographic model therefore
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Reprinted from Nature, Vol. 323, No. 6085, pp. 255-257, 18 September 1986
© Macmitlan Journals Ltd., 19S6
ERRATUM
0.9-1.0(7.8%)
0.8-0.9(6
-0.1(34 %)
0.7-0.8(6.2%)
0.6-0.7(6.3%)
0.5-0.6(6.4%)
0.4-0.5(6.6%)'
-0.2(10.9%)
0.3-0.4(7.1%)
0.2-0.3(8.3%)
Fig. 1
Frequency of probability levels obtained in 24,310 pairwise
f-tests of the hypothesis that species-specific b values are equal.
Comparisons are for 221 species of aquatic and terrestrial animals
(refs 17-37 and refs listed in ref. 4 as 3, 5, 6, 9,10,16, 18, 19 21
29-31, 38-40, 43, 44, 48, 55, 56, 59, 62, 67, 70, 74, 75, 81, 90,' 94^
97 and 101). Variance functions were calculated on a per sampler
basis (unstandardized to area or volume), and include only organ
isms from a single sampling site. Analyses include only species for
which correlations between logs2 and logM were statistically
significant (P<0.01).