Estimation of natural mortality using catch-at-age and aged mark-recapture data: a simulation study comparing estimation for a model based on the Baranov equations versus a new mortality equation

Attempts to estimate natural mortality, as a single constant M, simultaneously with other model parameters in integrated assessments via CASAL for the Heard and McDonald Islands (HIMI, CCAMLR Division 58.5.2) Patagonian toothfish (Dissostichus eleginoides) fishery have been unsuccessful. An alternative estimation strategy was tested using simulation whereby catch-at-age and aged mark-recapture data were generated for 12 years of fishing. Two alternative estimation models were programmed in R. Both use the same Poisson likelihood for annual number of recaptures by age class, and both model the age-structured population for each fishing year by annual recursive use of difference equations for population numbers at age obtained by integrating a first order ordinary differential equation (ODE) for within-year population dynamics. The difference in the models derives from differences in their ODE. The BODE model is based on the well-known Baranov ODE and corresponding mortality equation. The model based on a new mortality equation (CCODE model) uses an ODE with constant within-year catch per unit time for each age class, and as a result allows catch to be removed directly from the population. Compared to the BODE model the CCODE model involves less parameters and does not have competing likelihood components avoiding the difficultly in appropriately weighting separate components. Also, for a fishery that is not substantially depleted within any single fishing season, given fishers tend to increase their effort if required to achieve their target catch for the year, the CCODE model is the more realistic. In simulation studies of multiple years of releases, both the BODE and CCODE models gave accurate estimates of M when all other parameters were fixed at their simulation values. When all parameters were jointly estimated and selectivity is low for older age classes there was a problem of substantial positive bias in estimation of M for both the BODE and CCODE models. This problem was reduced so that bias in CCODE model estimates of M became progressively smaller as less “severe” selectivity was imposed in the simulation model. If selectivity for older age classes is close to 1 then the CCODE model estimates M with small bias and reasonable precision assuming 500,000 fish caught and 1,000 released per year.