## Age or Stage Based Models

The simple production model described in the previous sections assumes that all individuals in the population are more or less equals (e.g., the mean egg production per individual does not change over time, or the mean weight of each individual landed in the fishery does not change over time). In reality, there are additional demographic affects associated with the population age- or size-structure that could influence the dynamics of a given population. For example, older, larger fish contribute more eggs, or provide better parental care, etc., than younger, smaller fish. Furthermore, nearly all fishing gears are size-selective (i.e., small fish are less likely to be captured on large fish hooks, or the fishery operates in an area where only large fish reside) and it may take several years (even decades) for individual fish to grow to sufficient size that they become vulnerable to the fishing gear. These delays in production (production as seen by the fishery) are not sufficiently captured by simple production models and are better represented by age-structured or stage-structured models. The term stage-based model is used to represent specific life-history stages (e.g., larvae, juvenile, adult) or length classes rather than calendar ages, as it is not always possible to obtain age information (e.g., invertebrates are difficult to age because they lack permanent bony dY/dB = r(1 - B/K) - rB/K = 0 [6]

Figure 3 Three basic forms of the surplus production model where (a) production is maximized at less than % of the carrying capacity (m = 1.25 in eqn [4]), (b) production is maximized at (logistic growth model), and (c) where production is maximized at population densities that are greater than %K (m = 4 in eqn [4]).

Figure 3 Three basic forms of the surplus production model where (a) production is maximized at less than % of the carrying capacity (m = 1.25 in eqn [4]), (b) production is maximized at (logistic growth model), and (c) where production is maximized at population densities that are greater than %K (m = 4 in eqn [4]).

structures such as otoliths that are used to determine age). Population models based on life-history stages operate on the same basic principle as age-based models, where the principal difference is the time it takes to graduate from one stage to the next stage.

There are two basic approaches for the use of age-structured models in fisheries stock assessment: (1) a virtual population method that reconstructs the populations' num-ber-at-age backwards in time, and (2) a synthetic population that reconstructs the populations' numbers-at-age forward in time. These two approaches are usually referred to as virtual population analysis (VPA) and statistical catch-atage analysis (SCA), respectively. Both approaches rely extensively on catch-at-age data to estimate population parameters; however, it is not always necessary to have complete catch-age composition information for SCA approach. The basic equations that are used to describe the changes in numbers-at-age over time are given by

where Nat is the numbers-at-age in year t, S is the annual survival rate (assumed to be constant and independent of age in this case), Cat is the catch-at-age, vat is the propor-tion-at-age that are vulnerable to exploitation, and Ut is the overall exploitation rate in year t.

In the VPA case, the system of equations is actually solved backward in time (Figure 4) where the numbers-at-age in the last year and the numbers in the oldest age group for every year are unknown quantities that must be estimated from the data. For the unknown numbers in the oldest age group, it is usually assumed that no fish lives longer than the oldest age class, thus the number of fish in the oldest age class is simply the number caught in the fishery divided by the exploitation rate in that year. For the terminal numbers-at-age, estimates are usually obtained by solving eqn [10] for Nat and specifying and exploitation rate in the terminal year (Uterm). The VPA then proceeds to back-calculate the numbers at age by adding the Cat information to Na + 1t +1 and dividing by the survival rate. Therefore, it is essential that catch-at-age information be specified for each year. The vulnerability-at-age schedule (vat) in the terminal year must be specified in order to determine the abundance of incomplete cohorts (see Figure 4). These vulnerabilities are usually determined by examining the historical vulnerability-at-age patters in earlier years.

In the SCA case, the system of equations is solved forward in time (Figure 4) and the unknown quantities consist of the initial age-structure in the population and the new recruits each year (Figure 4). The unknowns in the SCA can be treated as unknown parameters that are estimated by fitting the model to observed catch-at-age information and time-series data on relative abundance. Alternatively, the new recruits each year may also be derived from a spawner-recruit relationship (e.g., a Beverton-Holt or Ricker type stock recruitment model), and the initial age-structure determined from an estimated initial recruitment and a survivorship curve defined by the initial total mortality rate. A major difference between the SCA and the VPA is that a vulnerability-at-age matrix (vat) must be specified or estimated from the available data in the SCA case and often this adds many additional parameters to the estimated parameter set and increases the uncertainty in the stock size estimate.

Entire book chapters and several thousand primary publications have been devoted to VPA and SCA modeling approaches and it is impossible to discuss all of the pros and cons of either method in such a short article. However, there are two primary concerns in using either of these

\ \ \ \ \ \ \ y e Na t Na t Na t Na t Na t Na t Na t

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Figure 4 A graphical representation of how the numbers-at-age are updated in (a) a VPA, and (b) an SCA. The © symbols represent unknown parameters or quantities that must be estimated in each of the approaches. In a VPA, these unknowns correspond to abundances in the terminal year and terminal age in all years. The shaded region represents the incomplete cohorts (those cohorts that are still remaining in the fishery). In an SCA, the unknowns are the initial age composition and the initial recruits in each year.

approaches that should be considered: (1) effects of aging error and (2) assumptions about vulnerability-at-age. In the VPA case, aging errors are propagated backward with expanded effect over time. For example, a simple error in the estimating of proportion-at-age in an older age class is propagated backward in time and is amplified by successive division of the survival rate S. This effect is dampened in the SCA case because the error is successively reduced by multiplication of the survival rate over time. The vulnerability-at-age schedule (i.e., the fraction of fish of a certain age class that is vulnerable to exploitation) is usually assumed to be constant over time (or perhaps over blocks of time that represent periods of certain types of fishing activities). Temporal changes in the vulnerability-at-age associated with changes in fishing locations or gear types can be interpreted as just that, or they can also be interpreted as strong year classes that have entered or disappeared from the fishery. For example, if fishermen learn that smaller fish fetch a higher value, they begin to target smaller fish relative to larger fish (i.e., va,t increases for younger fish) and this information is not passed on to the stock assessment scientist, then the interpretation of the data is that a strong cohort is about to enter the fishery and the stock size will be severely overestimated. As a result, stock forecasts will be very optimistic. This scenario occurred in northern cod stocks of the east coast of Canada in the late 1970s and early 1980s (Figure 5).

### Stock-Recruitment Models

The stock-recruitment relationship is one of the most critical components in fisheries models. The role of the stock-recruitment model is twofold: (1) to provide short-

term forecasts about future recruitment based on estimates of current spawning abundance, and (2) to provide a biological basis for the underlying production function in the model. Ultimately the stock-recruitment relationship defines just how much we can safely harvest and at what rate this should occur. There are many different analytical stock-recruitment models in use, but the two most commonly used models are the Ricker and the Beverton-Holt models:

where Et is a measure of egg production (which is often assumed to be proportional to spawning stock biomass), a is the maximum juvenile survival rate, and b represents density-related effects on juvenile survival. The Ricker model is frequently used to describe the production dynamics of salmon populations, and the Beverton-Holt model more frequently used in the assessment of pelagic and benthic marine fishes. Both of these models predict a compensatory response or improvement in the juvenile survival rate from egg (Et) to recruit (Rt) as egg production or spawning stock biomass is reduced. Stock-recruitment models are often integrated into statistical catch-at-age models and the unknown parameters (a,b) of the model are estimated (or derived from other model parameters) simultaneously with all other model parameters.

### Multispecies or Multistock Models

There are many fisheries that engage in multispecies fisheries around the world, or single fishing fleets that target several distinct populations or stocks of fish of the same species. That is, a single fishing vessel is permitted

Actual stock NAFO 1980 NAFO 1981 NAFO 1982 NAFO 1983 NAFO 1984 NAFO 1986 CAFSAC 1989 CAFSAC 1991

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Figure 5 Estimated biomass of northern cod of the east coast of Canada (2J3KL) stocks and the biomass forecasts produced by the stock assessment models (VPA-based assessment models) in the late 1970s and early 1980s.

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Figure 5 Estimated biomass of northern cod of the east coast of Canada (2J3KL) stocks and the biomass forecasts produced by the stock assessment models (VPA-based assessment models) in the late 1970s and early 1980s.

to catch and land an array of different species or several different stocks of the same species. For example, the trawl fishery for groundfish off the west coast of British Columbia deliberately targets about 19 different species of fish. The Fraser River sockeye salmon fishery can harvest 20 or more distinct sockeye salmon populations that spawn in particular watersheds. Pelagic long-line fisheries that target tuna are also permitted to harvest billfish, sharks, and other small pelagics. In comparison to the single-species realm, there have been very few stock assessment applications that simultaneously model multiple species or multiple stocks.

There have been two general approaches to modeling the multispecies or multistock fisheries: (1) data aggregation approaches, and (2) explicit modeling of each species or stock. Data aggregation approaches involve the use of a single model to represent the dynamics of a collection of species or stocks of the same species. In these cases, the mean life-history characteristics and demographics parameters are represented by a simple population dynamics model (e.g., logistic production model). The principal problem with this approach is often the subjective nature of how the data are aggregated and interpreted. For example, the CPUE indices are assumed to be proportional to abundance, but the CPUE index itself cannot be used to track changes in community composition because the effort index lacks information about changes in targeting. The second approach involves integrating several population dynamics models into a single framework, where the fishing mortality rates for each species are explicitly modeled using information on targeting (usually inferred using assumptions about ideal free distribution and economic variables such as the relative price differences in each species and/or cost of fishing in certain areas).

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