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# `Mean Crowding'

Monte Lloyd
Journal of Animal Ecology
Vol. 36, No. 1 (Feb., 1967), pp. 1-30
DOI: 10.2307/3012
Stable URL: http://www.jstor.org/stable/3012
Page Count: 31
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## Abstract

Patchy distributions matter, from the point of view of the animals involved, because the individuals tend to find more others of their own kind right around them than would be the case in a random distribution. The animals are more 'crowded', in this sense, than their mean density would lead one to believe. For data from randomly placed quadrats, the proposed parameter 'mean crowding' ($\overset \ast \to{m}$) attempts to measure this effect by defining the mean number per individual of other individuals in the same quadrat. 'Mean crowding' is algebraically identical with mean density, augmented by the amount the ratio of variance to mean exceeds unity, i.e. $\overset \ast \to{m}=m+(\sigma ^{2}/m)-1$. It can be viewed as equal to that increased mean density which a patchily distributed population could have, and be no more 'crowded' on the average than it is now, if it had a random distribution. The contention that populations with differing mean densities but the same 'mean crowding' would suffer the same density-dependent effects on mortality, natality, and dispersal implies the untested assumption that the effect of crowding on each individual varies linearly with the number of others around it, which are responsible for the crowding. Also implied is that each local situation (quadrat) is equally good habitat for the animals, has an equal rate of supply of expendable resources, and provides an equal amount of 'room', or cover, which the animals can use to avoid each other. If this assumption is perfectly fulfilled, however, then the patchy distribution itself becomes an ecological enigma: if the effects of crowding are more severe in the locally densest patches, then the local density should decrease more rapidly and increase more slowly there than elsewhere. The overall patchiness should decrease, in fact, until the distribution eventually comes to resemble a random one. Undoubtedly, a great deal of the patchines that one finds in continuous habitats can be explained in terms of local differences in habitat suitability. On the other hand, there is often a large component that appears to be utterly capricious. Since all of the capricious component should long since have been eroded away by density-dependent effects of crowding-if these are important-I must conclude that they are not important in such species, or, if they are, that other population processes (e.g. interactions with other species) regenerate patchiness as rapidly as it is destroyed. This fits in well with a general conviction that patchiness has a great deal more to do with undercrowding than with overcrowding, since 'mean crowding' often appears to be as great in rare species as it is in common ones (of comparable size). A suitable measure of patchiness is the ratio of 'mean crowding' to mean density. Where the distribution can be adequately approximated by the negative binomial, with parameters m and k, we have $\overset \ast \to{m}/m=1+k^{-1}$. The sample estimate for 'mean crowding' is $\overset \ast \to{x}=\overline{x}(1+\hat{k}^{-1})$. Standard errors are given for the cases where k̂ is estimated by maximum likelihood, by moments, by the number of empty quadrats, and also for the truncated negative binomial. The idea is briefly extended to apply to 'mean crowding' between species and to 'subjective' species diversity, without attempting in these cases to develop standard errors. Most importantly, I have tried to specify the types of habitats and distributions to which the parameter 'mean crowding' is not intended to apply.

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