Download n Z E (lr) =2+1(1+ 2n

COMMENT ON A REPORT BY J. H. VANDERMEER AND R. H. MACARTHUR
CONCERNING THE BROKEN STICK MODEL OF SPECIES ABUNDANCE
E. C. PIELou
Statistical Research Service, Canada Department of Agriculture, Ottawa
(Accepted for publication April 11, 1966)
Abstract. The correct formula is given for the relative abundanceof the rth rarest species
in MacArthur'soverlappingniche model. The correction given by Vandermeerand MacArthur
(1966) is shown to be erroneous.
problem the result has been given by Pielou and Arnason
(1966) who show that
MacArthur (1957, 1960) proposed two hypotheses to
account for the relative abundances of the species in a
many-species population. According to one of these (the
'overlapping niche' hypothesis), the abundance of each
species is proportional to the length of the line segment
lying between a pair of points placed at random on a line
of unit length. As shown by Pielou and Arnason (1966)
and also by Vandermeerand MacArthur (1966) the segment length, 1, is a random variate with distributionfunction F (l) _ 21 - 12. Its expectation is E (l) = 1/3.
It is now assumed that the abundances of the rarest,
E (lr) = 1-
second rarest, . . . species in an n-species population are
. . . segments respectively in a sample of it segments.
Suppose, then, we have a sample of n values of 1; we
may arrange them in increasing order of magnitude and
label them 11, 12, . . . , In. These are called the first,
second, . . ., nth order statistics. By the hypothesis, the
expected abundance of the rth rarest species is proportional to E(1r), the expectation of the rth order statistic.
We now wish to determine E(1,). The standard procedure for obtaining the expectations of order statistics
may be found in textbooks on Statistics (see, for example,
Kendall and Stuart 1958, p. 251). For this particular
_
E (nl) = n/3.
Z E (lr) =2+1(1+
also have 2 E (1,) = n/3.
r=
2n - 1
2n +1
-=1
1
(1?
Pielou and Arnason's expression for E (1,) meets this
requirementand that Vandermeer and MacArthur's does
not.
From Vandermeer and MacArthur's formula:
I
2n -
-2n+ 1
-
n
2n + 1
=
2,1/3
1
2'1-
2
3
2n - 4
2n-3
-E(l)2
1nl)-
2
3
4
2n - 2
2n-1
2n
2n + 1
6
2n-2
7 ....2n-lA
2n
2n+I
+
+
+
...
(+
2n -33
I
2n -2
1 - E(13)
(1 +...
1+
11
2n
It remains to show that
Adding these gives:
-
2
1
0.6062when n 2 insteadof 0.6667
0.9269when n = 3 insteadof 1.0000
+) =<
2n
Clearly this sum will be the same regard-
less of whether the l's are added in arbitrary order or in
order of increasing magnitude. It follows that we must
and so on.
To sum the terms of the correct formula we note that
(see Pielou and Arnason 1966):
2n
2n + 1
1E(11)
2nt
2n -2
1 - E(12) = 2n - 1
2n + 1
n-
(n +312)
0.2929 when n = 1 instead of 0.3333
(
1/
+ 3/2)
! X 1(n-r
P
-r)!
Vandermeerand MacArthur (1966) state that
E(lr) = 1-V1-r/(n
+ 1)
but this is incorrect. They reach this conclusion by
assuming, wrongly, that if we rank n values of I the
expected ranked values are those which partition the
distribution equally. That this argument leads to an absurd result may be shown as follows.
For an unranked sample we know that E (1) = 1/3.
Therefore the expectation of the sum of n values of I is
proportionalto the lengths of the shortest, second shortest,
n
_n_
r(n
+
4
2n- 46
3
(
)
7)
nI
/
So I E(lr)
=
n/? for all n and this provides the con-
formationrequired.
1074
Ecology, Vol. 47, No. 6
REPORTS
LITERATURE CITED
Kendall, M. G. and A. Stuart. 1958. The Advanced
Theory of Statistics. Vol. I. Hafner Publishing Co.,
New York. 433 p.
MacArthur, R. H. 1957. On the relative abundance of
bird species. Proc. Nat. Acad. Sci. U.S. 43: 293-295.
1. 960. On the relative abundance of species.
Amer. Naturalist 94: 25-36.
Pielou, E. C. and A. N. Arnason. 1966. Correction to
one of MacArthur's species-abundance formulas. Science 151: 592.
Vandermeer, J. H. and R. H. MacArthur. 1966. A
reformulation of alternative (b) of the broken stick
model of species abundance. Ecology 47: 139-140.
NOTE ON MRS. PIELOU'S COMMENTS
ROBERT MACARTHUR
Department of Biology, Princeton University
I am happy to agree that Mrs. Pielou's formula is
literally correct, and ours only a very good approximation.
We took what amounts to an average niche subdivision
and then ranked abundances; Mrs. Pielou ranked abundances and then took expectations. The distinction between these processes when applied to a single census is
somewhat vague biologically and almost undetectably
small numerically. Hence a graph with Mrs. Pielou's
correct formula will superimposealmost perfectly on ours.
DIFFERENTIAL
Let us hope these comments do not draw additional
attention to what is now an obsolete approach to community ecology, which should be allowed to die a natural
death. To forestall future waste of time, I shall add that
there is also an error in my relative abundance paper
(Amer. Nat. 94: 25-36) in the first page of the appendix,
as Mr. Joel Cohen of Harvard University has kindly
pointed out to me. Anyone interested may consult him
for the correct formulation.
FEEDING AND NICHE RELATIONSHIPS
AMONG ORTHOPTERA
ELIZABETH
BRUNING
CAPLAN
Department of Biology, University of Colorado,Boulder, Colorado
(Accepted for publication April 19, 1966)
Abstract. The acridian species MelaMnoplusbivittattus (Say), Melanoplus differentials
(Thomas) and Melanoplus lakinus (Scudder) occur in the same habitat, utilizing the same
foods. Such coexistence suggests that these species are in the same niche and in competition
for a common food supply. Their food usages were investigated by offering samples of dominant and semi-dominant vegetation from the common habitat to caged populations of each
species, then estimating the amount consumed. The overall usage of foods of each species
formed a preferential pattern sufficiently different from the patterns of the other two species
to indicate that the three grasshopper populations occupy separate niches in the community
and are not in complete competition for food.
INTRODUCTION
Gause's Principle holds that two or more similar species cannot coexist indefinitely in the same niche without
diverging in their ecological requirements. If ecological
divergence does not take place, the species with a slight
advantage in competition for any limited factor of the
environmentwill eventually displace others.
Although the niche of a species includes more than its
use of foods, food choice is a significant criterion upon
which to base niche relationships. Biologists have noted
that closely related species of grasshoppers feed upon the
same grasses and forbs in given habitats, thus appearing
to co-occupy a food niche. Such co-occupancy would be
a refutationof Gause's Principle. However, Isely (1946),
Gangwere (1961) and others have pointed out that apparent non-differentiationin food selection may be superficial; their studies indicate that grasshopperspecies feeding together exhibit distinct and separate food preferences.
In this paper the coexistence of Melanoplus bivittatws
(Say), M. differentialis (Thomas) and M. lakinus (Scudder) is considered.
These congeneric species of the
family Acrididae are found feeding together in the grassland habitats of the Boulder, Colorado region. While
one species, M. ltakinus, is distinct in size and other features, the other two are closely related in all characters.
Do these species occupy one niche in the habitat where
they are found, or are they segregated into separate
niches by their preference for different foods?
MATERIALSAND METHODS
Over a 3 month period in the summer of 1964, plant
foods eaten by Melanoplus bivittatus, M. differentialis
and M. lakinus under controlled conditions were observed.
Initial collections of grasshoppers were made in July by
net and supplemented with similarly captured specimens
throughout August and September. The insects were
placed in wooden feeding cages which measured 7?4 X 7?/4
in. by 9 in. high. The cages were supported by 4 in. legs.
Three sides of the feeding cage were covered with wire