## Fully Age Structured Methods

Population models that trace cohort abundance and mortality through time are widely popular for contemporary stock assessments. So-called catch-age analyses were made possible by large-scale research programs of aging fish through analysis of hard parts, usually scales or otoliths.

### Virtual population analysis and cohort analysis

The term 'virtual population' was used originally to mean the sum of catches at age from a single cohort through its lifespan. Thus, the virtual population provides a minimum estimate of initial cohort strength, but it neglects losses from natural mortality. That concept of virtual population has become obsolete.

Today, virtual population analysis (VPA) refers to a family of methods that account for losses to a cohort from fishing and natural mortality. In its simplest realization, VPA is a solution of the Baranov catch equation applied backward in time, starting from the oldest age (A) of each cohort. It is due independently to Murphy and Gulland, who pointed out that given catch at age, this procedure (under constant M) involves a system of A _ 1 equations with A + 1 unknowns, and that the need for two additional pieces of information could be filled by a value for the natural mortality rate M and an estimate or guess of the final population size NA or, more typically, the corresponding fishing mortality rate Fa. If M is thought to vary with age, Ma at each age must be provided.

Typically, a VPA applies the Baranov catch equation, rearranged to calculate abundance from catch, to the oldest age of a cohort:

zaca

From NA and the known CA 1, FA 1 is calculated:

where eqn [45] is derived from the ratio of catch at age (eqn [27]) to abundance at age (eqn [17]). With FA _ 1 computed, eqn [44] can be applied to the next younger age (A — 1) to calculate NA _ 1, and eqn [45] to estimate Fa _ 2. This procedure is repeated until reaching the youngest age for which catch data are available. Thus, Fa and Na are calculated for each age.

Equation [45] must be solved iteratively. To ease calculation, Pope provided an approximation for backwards calculation of population number,

whose application he termed 'cohort analysis'. MacCall later provided a slightly more accurate approximation.

CaMa

Although calculations for VPA or cohort analysis can also be done forward in time, when the backward algorithm is used, calculated values of F and N at young ages are relatively insensitive to choice of FA; that is, estimates of younger ages would be quite similar from a wide range of values for FA, with this insensitivity applying to older ages if the total mortality rate is high. This insensitivity (sometimes described as convergence) is a valuable property when estimates of recruitment are desired - as they frequently are. However, the convergence property applies only to cohorts that have been subject to high cumulative mortality; when incomplete cohorts are analyzed, calculated F for younger ages is sensitive to starting F.

Because the basic VPA algorithm treats each cohort independently, FA must be supplied for each cohort. That requirement can be eased by postulating, for example, that Fa,, — Fa _ 1,, for all A. Then after computations for the oldest cohort are completed, the calculated value of FA _ 1 can be used to initialize FA of the next youngest cohort.

In simple VPA, starting values and catch at age determine calculated values of Na,t and Fa,t entirely. Although this attribute removes the need for statistical optimization, it also means that simple VPA provides no estimates of variance. When errors occur in the catch at age, patterns of FA can fluctuate in unrealistic ways between adjacent cohorts or ages.

Although the convergence property of basic VPA provides stable values of initial year-class strength, the values of present-time cohort strengths are highly sensitive to assumptions of final-year F or N. This was one of the major motivations for the numerous generalizations of VPA that have been developed. Other major motivations are to allow for error in the observed data and to provide estimates of precision.