Resource Ecology and Ecosystem Modeling  Models, MSM
The multispecies statistical model: introducing predation interactions into a statistical catchatage model.
Singlespecies 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 predatorprey 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 MSVPAMSFOR 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 catchatage, s_{a} the agedependent gear selectivity (u and v are its parameters), F_{ t} the full fishing mortality.
In this model we assumed observation error in the catchatage and the relative indices of abundance. The loglikelihood 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 loglikelihood for the catchatage was assumed to be lognormal distributed:
(11) _{}
The sum of the loglikelihood 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 predatorprey species. The MSM estimates stock size and predation mortality based on catchatage data 19792002), 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 age0 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. NewtonRaphson) 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 catchatage (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 910) 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 loglikelihood 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 singlespecies 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 (pvalue ~ 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 singlesspecies stock assessment from AFSC, MSVPA (two species) and the MSVPA baserun (all predator species) were also similar.
References
Collie, J.S. and Gislason H. 2001. Biological reference points for fish stocks in a multispecies context. Can. J. Fish. Aquat. Sci. 58: 21672176.
Doubleday, W.G. 1976. A least squares approach to analyzing catch at age data. Res. Bull. Int. Comm. NW Atl. Fish. 12:6981.
JuradoMolina, J., and Livingston, P. A. 2002a. Multispecies perspectives on the Bering Sea groundfish fisheries management regime. N. Am. J. Fish. Manage. 22:11641175.
JuradoMolina, J., and Livingston, P. A. 2002b. Climate forcing effects on trophicallylinked groundfish populations: implications for fisheries management. Can. J. Aquat. Fish. Sci. 59: 19411951.
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. 6376.
Sissenwine, M. P., and Daan, N. 1991. An overview of multispecies models relevant to management of living resources.  ICES mar. Sci. Symp. 193: 611.
Sparre, P. 1991. Introduction to multispecies virtual population analysis. ICES mar. Sci. Symp. 193:1221.
For further information contact Kerim Aydin, Program Leader or Jesus JuradoMolina, 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)
