According to Steve Munch (and based on results in two Sugeno and Munch 2013 papers in Ecology and Ecol Apps), calculating slopes of the paths of the GPs introduces a lot of numerical error.

Alternative approach: semi-analytic method to evaluate the derivative of the GP at a point specified by a parameter (e.g., the replacement threshold)...uses convenient property of Gaussians (see method described in appendix C of Sugeno and Munch 2013). This requires specifying a GP prior function conditioned on a parameter of interest. Thus, we could specify the GP prior to be of the form of a flexible stock-recruit function like the Shepherd model, which has an additional parameter describing the slope of the function at high N_t or adult abundance.

i.e., is the slope at high densities different from the slope at low densities?

Criteria at high density:

• If slope is negative, overcompensation.
• If slope is zero, perfect compensation
• If slope is positive, partial compensation
• If slope is R_0 (or r or whatever), no density dependence at all
• If slope is greater than R_0 (or whatever), then positive density dependence

The easiest way to do this would be to randomize the ordering of the time series.

This would force the model to avoid overcompensation and allow for some really interesting comparisons.

Might be important for count data & if mean and variance are related.