There are many different indices for calculating the degree of individual diet specialization. Most measures of niche breadth compare the frequency distribution of the individuals' resource use with that of the average population resource use. The TNW of a population can be broken down into the average variance of resources found within individuals' (WIC) and the variance between individuals (BIC), so that TNW = WIC + BIC. Assuming that variation in diet parameters can be expressed along a single continuous dimension x (e.g., prey size) and that xj is the size for the jth prey item in individuals i 's diet,
The relative degree of IS can be measured as the proportion of TNW that is explained by the within-individual variation, WIC/TNW, or its converse BIC/TNW. The latter is more intuitive, since it is larger in populations with greater diet variation, but the former has been used more frequently.
This index is, however, limited to continuous diet data. An alternative is to use the Shannon index as a proxy for variance for discrete data, such as the frequency with which individuals use a set of prey taxa. In this index, ny represents the number (or mass, or volume) of diet items in individual i 's diet that fall into category j. nij is then transformed into pij, which describes the proportion of the jth resource category in individual i 's diet. Then
As before, TNWs = BICs + WICs and the relative degree of IS is also as before: WICs/TNWs. The Shannon-Weaver index has been widely used, but suffers serious disadvantages. First, the index increases both with the number of prey categories and/or with the evenness with which those categories are used, making its interpretation rather ambiguous. Second, the index can substantially overestimate diet variation (low WIC/ TNW) when the population as a whole relies heavily on a single prey taxon. In this case, individuals whose diets largely match the population as a whole may have WIC = 0. The resulting WIC/TNW would thus be close to zero, falsely implying that there is pronounced diet variation.
There are two distribution-overlap measures that provide discrete data alternatives to WICs/TNWs. Both measure the mean pairwise overlap between each individual and the population. The first is a proportion similarity index (PS), where the diet overlap between an individual i and the population is pS, = 1 - °.5X |pj- qj I = X min (pj j jj where pij and qj are the same as above. For individuals that consume resources in direct proportion to the population as a whole, PSi = 1. For an individual that specializes on a single diet item j, PS,- takes on the value qj. The population-wide prevalence of IS is then measured by the average of individuals' PSi values.
The second overlap measure uses a likelihood approach to measure individuals' niche width. This method determines the probability that any individual i's diet (the vector of njs or pjs) was drawn randomly from the population resource distribution (the vector of qj s). This probability is proportional to the likelihood function, which can be compared to the maximum likelihood in which the individual and the population have
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