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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