Estimating the relationship between abundance and distribution

Abstract

Numerous studies investigate the relationship between abundance and distribution using indices reflecting one of the three aspects of distribution: proportion of area occupied, aggregation, and geographical range. Using simulations and analytical derivations, we examine whether these indices provide unbiased estimates of the relationship when estimated from count data. The indices investigated include the proportion of empty samples, the proportion of structurally empty samples, Lloyds index of patchiness, measures derived from Lorenz curves (such as D95 and the Gini index), and measures based on Euclidean distance to the centre of gravity of the spatial distribution. Only the proportion of structurally empty areas, Lloyds index, and indices of the distance to the centre of gravity of the spatial distribution are unbiased at all levels of abundance. The remaining indices generate relationships between abundance and distribution even in cases where no underlying relationships exists, although the problem decreases for measures derived from Lorenz curves when samples contain more than four individuals on average. To illustrate the problem, the indices are applied to juvenile North Sea cod, Gadus morhua.Résumé : De nombreuses études examinent la relation entre l'abondance et la distribution à l'aide d'indices qui représentent l'un des trois aspects de la distribution, soit la proportion du territoire occupé, l'agrégation et la répartition géographique. Au moyen de simulations et de dérivations analytiques, nous vérifions si ces indices fournissent des estimations justes de la relation lorsqu'ils sont basés sur des données de dénombrements. Les indices examinés incluent la proportion d'échantillons vides, la proportion d'échantillons structurellement vides, le coefficient de répartition en taches de Lloyd, des mesures déri-vées des courbes de Lorenz, telles que D95 et le coefficient de Gini, ainsi que des mesures basées sur la distance euclidienne jusqu'au centre de gravité de la distribution spatiale. Seuls la proportion de sites structurellement vides, le coefficient de Lloyd et les coefficients de distance au centre de gravité de la distribution spatiale restent non biaisés à tous les niveaux d'abondance. Les autres indices génèrent des relations entre l'abondance et la distribution même lorsqu'il n'existe pas de relation sous-jacente; cependant, dans le cas de mesures dérivées du coefficient de Lorenz, le problème diminue lorsque les échantillons contiennent plus de quatre individus en moyenne. Nous utilisons les indices avec des données sur de jeunes morues, Gadus morhua, de la mer du Nord pour illustrer le problème.[Traduit par la Rédaction]

Effect of spatial differences in growth on distribution of seasonally co‐occurring herring Clupea harengus stocks

“…This index is proportional to the area covered by 95% of the biomass if the distribution is a two‐dimensional normal distribution in space and even when the distribution is skewed or in other ways deviates from normality, this indicator still reflects concentration of the stock (Rindorf & Lewy, ). The indicator is not responsive when the distribution is bimodal, but judging from the distribution, this was not a problem in the present analyses.…”

Spatial structure of a multi-species guild: the dominant biofilm-grazing microgastropods of seagrass

“…Following Hurlbert (1990), spatial dispersion of abundance was assessed by Morisita's original (1959Morisita's original ( , 1962) procedure, rather than by Smith-Gill's (1975) standardised version. This is effectively identical to the Lloyd (1967) index advocated by Payne et al (2005) and Rindorf & Lewy (2012). Statistically significant heterogeneity (patchiness) was tested by one-sided upper-tail χ 2 (Morisita, 1962).…”

Improving estimation of phytoplankton abundance and distribution in ballast water discharges

“…The Lorenz curve-the cumulative distribution of the samples ordered by ascending sizedescribes the aggregation as the difference between the observed distribution and a distribution where every sample contains the same number of individuals. Indices based on the Lorenz curve include the D y and the Gini Index (Rindorf and Lewy 2012). Both the D y and the Gini index are derived from socioeconomics, but are also applied to natural communities (Rindorf and Lewy 2012).…”

“…Indices based on the Lorenz curve include the D y and the Gini Index (Rindorf and Lewy 2012). Both the D y and the Gini index are derived from socioeconomics, but are also applied to natural communities (Rindorf and Lewy 2012). The Gini index (G) is defined as twice the area between the Lorenz curve and its diagonal, and has values ranging from 0 (samples are equal) to 1 (all individuals are recorded in a single sample) (Rindorf and Lewy 2012).…”

“…Both the D y and the Gini index are derived from socioeconomics, but are also applied to natural communities (Rindorf and Lewy 2012). The Gini index (G) is defined as twice the area between the Lorenz curve and its diagonal, and has values ranging from 0 (samples are equal) to 1 (all individuals are recorded in a single sample) (Rindorf and Lewy 2012). D y is the minimum area containing y% of the individuals and is calculated as D 50 , D 75 , and/or D 95 .…”