Ecology: Indices to calculate species overlap in a community?

Question

Experimental design: I sampled the abundances of four species for each month of the year. I did this across 6 sites. I want to calculate the temporal niche overlap for all the species at each site. Ultimately, I want to compare overlap across sites.

I want to compare the extent of overlap of species abundances in the following two secanrios. In the first scenario, species barely overlap so the index or overlap value should be zero. in the second case species fairly overlap in their abundances as shown below (Note: My data has absolute abundances not relative. The graph is only for representation).

I found Pianka's index (from Gotelli 2000) to be one such index that estimates the extent of overlap. However, on closer reading, I find that takes the resource utilization matrix as input and returns the average pairwise Pianka's niche overlap index averaged over each species pair as so:

$$O_{kl} = O_{lk} = \frac {\sum\limits_i p_{ij}p_{ik} } { \sqrt{\sum\limits_i p_{il}^2 \sum\limits_i p_{ik}^2}}$$

I am not sure if I can use it to compute the overlap of species abundance distributions alone. I am wondering are there any other indices that I can use to do a similar computation? Is there a statistical route that I can take to capture a breath of overlap?

Specific questions (in response to Ben Bolker's comments):

1. I want to calculate a single number for the overall overlap. It has to be able to deal with 0s.

2. I would be curious to know the overlap coefficient for each species pair. My guess is that I can use a alpha-diversity index like Shannon’s diversity index.

3. Would it be incorrect to use Pianka's niche overlap here? Why can't I?

Answer 1

If you only care about direct overlap, this sounds like a case for classical correlation coefficients. These answer you whether a high abundance of Species A coincides with a high abundance of Species B. The known advantages and disadvantages of the respective correlation coefficients apply, e.g., ranked correlation coefficients being less sensitive to outliers.

Beware that when you want to determine the significance of your results, you need to take into account that your individual data points are correlated through time. For example, if Species A and B have a high abundance today, they probably also have a high abundance tomorrow. Therefore you cannot apply standard ways to obtain significance as those assume uncorrelated data points and you would commit pseudoreplictation. However, it is quite easy to implement a computational null model that preserves the temporal correlations while destroying the ecological correlations (i.e., between species): For example, you can randomly shift one time series if your data is cyclic (e.g., capturing a strictly yearly cycle) or sufficiently large that border effects do not matter. Or you can use time-series surrogates to create null-model data with an identical frequency structure.

Things become more complicated if you care about close overlaps, like Species A being abundant in the month after Species B was abundant. Here, you can use tools like cross-correlation, but this really depends on what exactly your constraints are and what you want to know.