CATÁLOGO DE NARRACIONES BREVES

In this chapter a non-equilibrium stock production model expressed by difference equations has been introduced. For the model estimation routine, process and ob- servation errors were considered and a nonlinear estimation method was employed. The population parameters natural mortality, stock resilience, and catchability, were generally set to fixed prior values but have also been treated as estimated parame- ters. The way these parameters affect the interpretation of the status of the exploited population was explored.

Process error was introduced in the production function as this is more consistent with the fundamental definition of process error (Rosenberg and Restrepo, 1994). The pro- duction function incorporates the population fluctuation in terms of recruitment and mortality variations, for instance. Despite the fact that process noise modelling has been applied to the dynamic equation by several authors (Walters and Hilborn, 1976, Schnute, 1977, Polacheck et al., 1993, Chen and Andrew, 1998), there are no clear biological mechanism leading to this assumption. When process error is accounted for in the dynamic equation, observation errors affecting the catch and biomass index data are also conflated into the process error, whichis an undesirable feature. The

The Conser Mixed Model Chapter 3 weighting of the error terms does not correspond to a clear separation between the fundamental sources of error.

For the re-implementation, the Conser model (Conser, 1998) had to be modified to avoid negative arguments during minimisation which would lead to crash of the log- normal objective function. The results generated by minimising the modified Conser objective function were consistently similar to the original work, suggesting the new objective function and minimisation algorithm are acceptable.

The use of weighted least squares minimisation is particularly advantageous due to the flexibility of the fitting procedure, demonstrating how changes in the relative weight of both sources of errors can vary the estimated results. If the data variances were known, then the approximate weights are given by w = 1/σ2 _{(Lassen and Medley,}

2001). In this study, process and observation error variances were unknown but were assumed to be similar (if approximately scaled), so a range of relative weight values were tested in order to find an equally balanced objective function. When only one error was admitted at one time, by setting the weight of the other term to zero, the estimation clearly “twisted” putting all the error in the other term, demonstrating that the model needs the right balancing between both errors. The relative weight applied should ideally transform the response variances to a constant value (Lassen and Medley, 2001) consistent with the known expected errors in the data and the process.

Conser (1998) pointed out the importance of assuming both errors, justifying this with the wide variation the parameters can have if just one of the uncertainties was considered. A highly variable output was identified when just one of the components of uncertainty was estimated showing the need of assuming both errors, especially since there have been several studies (Polacheck et al., 1993, Chen and Andrew, 1998, Punt, 2003) carried out considering either observation or process error, but not both. Clearly, the weighting ratio has been proved to be crucial for the model estimation. The estimated current biomass can be bigger or smaller than BM SY just by alter-

The Conser Mixed Model Chapter 3 beforehand is an indirect control of the sum of both observation errors squared and process errors squared. Neither observation error nor process error only models yield credible results. Another fundamental influence in the current biomass estimation is the setting value of pristine biomass. Under and overexploitation stock status can be reached by slightly varying the value of pristine biomass.

The process error is defined as log-normally distributed by Conser (1998) since its deviations are assumed to be dominated by recruitment fluctuations which are usually considered as log-normally distributed (Fogarty, 1993). However, there are other factors in addition to recruitment which affect the sequential evolution of biomass. Thus, considering the combined effect of these processes, it is not clear that the resultant should be log-normal.

When resilience is also estimated, the this parameters assumes unrealistic values. Therefore, resilience should probably be used as a set value but a limited range of it should also be tried. The estimation of pristine biomass seems to be more sensitive, probably because of being scaled with estimated biomass in the production function whereas resilience is just a multiplying factor, i.e. without constrain, in the same equation. Even though estimated pristine biomass are more realistic it was fairly optimistic and should be considered with caution.

The pointed out inconsistencies in Conser model assumptions, i.e. peculiar process errors formulation, neglecting the weighting balance for process and observation errors in the objective function and the lack of analysis of a range of resilience values were proved to be coherent. As a result of the inadequate process error incorporation and the fixation of certain parameters, this objective function will not be objective of further analyses since it will be proposed an improved estimation method.

The Conser Mixed Model Chapter 3