See https://rdrr.io/rforge/FLCore/man/SRModels.html

library(FLCore)
library(FLBRP)
library(ggplotFL)
library(ggplot2)
library(plyr)
library(dplyr)

Introduction

References

Summary

If you fit a Beverton and Holt stock recruitment relationship there is a strong residual pattern, i.e. low recruitment up to 2000, after which recruitment increases. The estimate of steepness (\(h\)) is low at 0.4.

If two regimes are assumed by fitting a three parameter stock recruitment relationship with two value of virgin biomass (\(V_0\)) then the resisual patterns improve and steepness increases to 0.6. However, the fits still fail the residual runs test.

When fitting a stock recruitment relationship assuming steepness and virgin biomass you also have to specify the spawner per recruit of the unexploited stock (\(SPR_0\)). This however can vary overtime, in which case the fts pass the runs tests. This provides a way of linking the biological assumptions and recruitment.

Results

Fitting a Beverton and Holt stock recruitment relationship to the NEA mackerel SSB and recruitment pairs (Figure 1), results in strong residual patterns (Figure 2). There appears to be two regimes, i.e. the early period upto 2000 and the more recent period (Figure 3).

A jackknife where individual pairs are removed (Figure 4) showed that some points were influential, i.e. the most recent years and the strong year-classes of 2005 and 2006

Fitting the Bevertion and Holt stock recruitment relationship parameterised as steepness and virgin biomass requires an estimate of \(SPR_0\). However, \(SPR_0\) depends on natural mortality, growth and maturity and in the case of NEA mackerel these quantities vary over time (Figure 5).

Therefore a Beverton and Holt stock recruitment relationships were fitted where it was assumed there were two regimes that affected virgin biomass (\(V_0\)) and that steepness (\(h\)) was unaffected. \(h\) depends on the early life history mortality rate (\(M_0\)), which is known to exhibit high year-to-year variations like natural mortality of the older ages (\(M\)). Since \(M\) was assumed constant in the ICES assessment, although this assumption may not be valid it is consistent with the assessment.

Figure 6,7,8 and 9 show the two regime fits, where i) two values of \(V_0\) were estimated ii) two values of \(V_0\) were estimated with variable \(SPR_0\) iii) two values of \(V_0\) were estimated with autocorrelated residuals iv) two values of \(V_0\) were estimated with variable \(SPR_0\) and autocorrelated residuals

The residuals are summarised in Figure 10 and residual runs tests are summarised in Figure 11

When two regimes are assummed and \(SPR_0\) allowed to vary the stock recruitment relationship fits past the runs test.

Figure 1. Time series of stock and recruitment.

Figure 2. Stock recruitment fit for Beverton and Holt

Figure 3. Recruitment residuals, with regimes.

Figure 4. Sensitivity steepness and virgin biomass to observations from jackknife

Figure 5. Time series of \(SPR_0\)

Figure 6. Stock recruitment relationship for two regimes.

Figure 7. Stock recruitment relationship for two regimes and variable \(SPR_0\).

Figure 8. Stock recruitment relationship for two regimes and AR1.

Figure 9. Stock recruitment relationship for two regimes, variable \(SPR_0\) and AR1.

Figure 10. Residuals by two regime fits, red are the Beverton and Holt fits.

Figure 11. Runs tests of residuals.

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Software Versions

  • R version 4.0.3 (2020-10-10)
  • FLCore: 2.6.16
  • FLPKG:
  • Compiled: Tue Apr 27 09:15:03 2021
  • Git Hash: f68fcf6

Author information

Laurence KELL.

Acknowledgements

References

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