The multispecies statistical model: introducing predation interactions into a statistical catch-at-age model.

Single-species models play an important role in fisheries management allowing the establishment of target reference points and setting of allowable harvest limits.  Scientists involved in stock assessment have acknowledged the possibility that predator-prey interactions can influence population dynamics estimation and reference points, but the general belief is that the effects of predation and competition are subordinate to the direct effect of fishing (Sissenwine and Daan 1991).  Fishery managers are increasingly asked to consider multispecies interactions in their harvesting decisions because there is an increasing tendency to recognize that fish populations are not isolated entities.  Scientists acknowledge the fact that harvest limits of some prey species may depend on harvest limits of predators (Collie and Gislason 2001), thus there is a real need to link assessment models.  Therefore, several attempts have been made to include biological interactions in models.  The multispecies virtual population (MSVPA) and the multispecies forecasting model (MSFOR) have the potential to provide additional information to fisheries managers to improve management policies.  However, their lack of statistical assumptions impedes the inclusion of uncertainty into multispecies model parameter estimation. Thus, it is important to include assumptions on process and/or observation errors in multispecies models. A new approach requires the inclusion of the separable fishing mortality assumption (Doubleday 1976; Pope 1977), which allows for common statistical estimation procedures in a multispecies context, a task that is not possible with the current available MSVPA-MSFOR technology.  In the Alaska Fisheries Science Center we have developed a multispecies statistical model MSM.

In addition to the predation equations (see MSVPA), MSM incorporates the following equations:

(4)       

(5)       

(6)                                                                                                                     

(7)       

where   N_a,t,q represents the number of individuals of age a in year t in the quarter q, F_a,t the fishing mortality at age, M _a the natural mortality at age,  C_a,t the catch-at-age,  s_a the age-dependent gear selectivity (u and v are its parameters), F _t the full fishing mortality.

In this model we assumed observation error in the catch-at-age and the relative indices of abundance.  The log-likelihood function for each index was defined as:

(8)       

where CV is the coefficient of variation and I_t the observed relative index of abundance. The predicted relative index of abundance was estimated as:
(9)       

 The catchability coefficient was estimated using the following equation:

(10)     

Similarly, the log-likelihood for the catch-at-age was assumed to be log-normal distributed:

(11)     

The sum of the log-likelihood components associated with each species was used as the objective function. In the estimation process for both species a CV of 0.2 was used.

The MSM (quarterly form) for the Bering Sea include only walleye pollock (Theragra chalcogramma) and Pacific cod (Gadus macrocephalus) as predator-prey species. The MSM estimates stock size and predation mortality based on catch-at-age data 1979-2002), relative indexes of abundance (AFSC's bottom trawl BTS and the echo integration trawl EIT surveys), predator annual ration and predator stomach contents (data from AFSC food habits data base assumed to be measured without error) using estimation procedures for the statistical part and the predation mortality. The MSM statistically estimates parameters (the initial age structure in 1979 N_a,0, 12 parameters; yearly age-0 recruitment R_t, 24 parameters; yearly full fishing mortality F_t, 24 parameters; and selectivity coefficients, 2 parameters) using either an optimization algorithm (e.g. Newton-Raphson) or Bayesian methods. In both cases, for a given set of parameter values, MSM projects population trajectories (for each species) over the specified time frame. These trajectories are computed based on catch-at-age (assumed known) and on predation mortality. The predation mortality requires an iterative solution since its value in each year is confounded with the abundance of other species in that year.  This iterative process is referred to as the "predation algorithm" (Sparre 1991). Given fixed values for the parameters (N_a,0, R_t, F_>t and selectivity coefficients) and an initial guess for M2, the population trajectories are computed. These population estimates together with given values of the suitability coefficients allows the estimation of the predation mortality. The predation algorithm then adjusts the M2 values and updates population estimates until two consecutive iterations converge to marginally different M2 and suitability coefficients values according to established criteria (Sparre 1991). Once the criteria are reached and the estimates of predation mortality, suitability coefficients and population have converged, the likelihood (Equations 9-10) is used in the criteria of the main data fitting routine. This procedure (the main part and the internal estimation for M2) is repeated until the negative log-likelihood is minimized (by adjusting main parameters) or in the Bayesian methods until the posterior distribution is adequately represented. We also updated the MSVPA model for comparison purposes. Results from these models were also compared with results with the single-species statistical stock assessments carried out in the Alaska Fisheries Science Center (AFSC). 

Results from MSM fitted the walleye pollock and Pacific cod abundance indices well. The estimates of the suitability coefficients from both models (MSVPA and MSM) for both species were similar. The regression between both types of suitability estimates (suitability coefficients from MSVPA as the independent variable) was significant (p-value ~ 0) with a slope of 0.99 ± 0.01 and explained 99% of the variability observed.  

The MSVPA and MSM (MLE) estimates of average predation mortality of walleye pollock (ages 0 and 1) followed the same trend. The estimates of Pacific cod predation mortality were not as similar as the pollock case.

Estimates of N3+ walleye pollock from the MSM, the singles-species stock assessment from AFSC, MSVPA (two species) and the MSVPA base-run (all predator species) were also similar.

The MSVPA model is an important tool, which can provide advice to fisheries managers on potential indirect effects of fishing in a multispecies context. However, its lack of statistical assumptions does not allow the use of tools regularly used in age-structured single-species stock assessment such as Bayesian analysis, risk analysis and decision analysis. It is likely that the predation equations can be incorporated into most of the statistical-catch-at-age models currently used in fisheries management that have the separable fishing mortality assumption (Doubleday 1976; Pope 1977).     

Several improvements could be made in future versions of the MSM including the addition of more predator species.  In particular, the addition of Pacific herring and rock sole (prey), which seem to be the most sensitive species to indirect effects of fishing mortality (Jurado-Molina and Livingston 2002a; Jurado-Molina and Livingston 2002b). It is also necessary to include the complete set of predator stomach content data and new data in future versions of the model. It is also important to incorporate a statistical model for the diet composition data in the future. A formal evaluation of the model through simulation of the system (adding sampling error) given a known dynamic will be necessary to see if the model is able to generate good estimates of the parameters. The final goal would be to incorporate all the data sources used in the single-species stock assessments into the multispecies statistical model.  Forward simulations to test management scenarios also need to be implemented. In those simulations, future recruitment can be log-normal distributed or modeled as either a Ricker or Beverton and Holt spawner-recruit relationships to account for compensatory effects.

The main goal in the development of this model is to incorporate the predation equations into the actual stock assessment models used to provide groundfish management advice for the Bering Sea. The development of the multispecies statistical method is an important step in its adoption in providing advice to fisheries managers because it allows consideration of uncertainty in a multispecies predator-prey modeling context and it will help to establish useful scenarios for the management of groundfish resources in the Bering Sea.

References

Collie, J.S. and Gislason H. 2001. Biological reference points for fish stocks in a multispecies context. Can. J. Fish. Aquat. Sci. 58: 2167-2176.

Doubleday, W.G. 1976. A least squares approach to analyzing catch at age data. Res. Bull. Int. Comm. NW Atl. Fish. 12:69-81.

Jurado-Molina, J., and Livingston, P. A. 2002a. Multispecies perspectives on the Bering Sea groundfish fisheries management regime. N. Am. J. Fish. Manage. 22:1164-1175. 

Jurado-Molina, J., and Livingston, P. A. 2002b. Climate forcing effects on trophically-linked groundfish populations: implications for fisheries management. Can. J. Aquat. Fish. Sci.    59: 1941-1951.

Pope, J. G. 1977. Estimation of the fishing mortality, its precision and implications for the management of fisheries. In Fisheries mathematics. Edited by J.H. Steele. Academic Press,    New York, pp. 63-76.

Sissenwine, M. P., and Daan, N. 1991. An overview of multispecies models relevant to management of living resources. - ICES mar. Sci. Symp. 193: 6-11.

Sparre, P. 1991. Introduction to multispecies virtual population analysis. ICES mar. Sci. Symp. 193:12-21.

For further information contact Kerim Aydin, Program Leader or Jesus Jurado-Molina, Author

A portion of this research was supported by the North Pacific Research Board (NPRB). For more information on the NPRB and this project see the NPRB website (external link not affiliated with the AFSC website)