Spawner-recruitment, time-varying biology, depletion

[Rick-Methot-NOAA]

This discussion is related to the development of issue #191 regarding spawner-recruitment relationships (SRR) that do not rely on steepness (h) parameterization. tagging: @chantelwetzel-noaa @shcaba

A key aspect is the way in which depletion is measured. Depletion (e.g. SSBt / SSB0) is an explicit quantity for the R0, h parameterization.
Here are 3 ways to measure depletion:

1. SSBt / SSB0: where SSB0 is the equilibrium spawning biomass using the start year biology.
2. SSBt / SSBd0t for dynamic B0, dB0t, which takes into account the time series of time-varying biology and recruitment deviations
3. SSBt / SSBe0t for equilibrium B0, eB0t, using the year t's biological parameters and SRR parameters

With time-varying biology, these will give different results. (1) and (2) are in SS3 today, but d0 is only used to describe depletion, not drive dependent quantities. (3) is not implemented in SS3.

The B-H SRR in SS3 with R0,h uses (1).
The MSY calculations in SS3 use a variation on (2) for biology (e.g. averaged wt-at-age over a range of years) and really should use (3).