Age Structured Population Model

The AGEPRO framework is based on a pooled-sex, age-structured population model that tracks changes in abundance through time due to recruitment, natural mortality, and fishing mortality from one or more fleets. The model represents an iteroparous fish population in which abundance at each age evolves continuously under the combined effects of natural and fishing mortality. Recruitment occurs at the start of each year (January 1) and represents the number of new age-1 fish entering the population (Table 1). The following sections describe the model’s treatment of population abundance, survival, spawning biomass, catch, harvest control, initial population abundance, and retrospective adjustment.

Population Abundance

AGEPRO computes the number of fish alive in each age class for every year of the projection. Let \(Y\) denote the total number of projection years, with \(t = 1,2, ... Y\) indexing time. The youngest age class (age 1) comprises recruits, and the oldest age (age-\(A\)) is a plus-group which aggregates all individuals age \(A\) or older. The total number of age classes is set by the user. For each age \(a\) and year \(t\), the number of fish alive at the start of the year (January 1) is \(N_a(t)\). The number of fish in the plus-group is \(N_A(t)\), which accounts for all fish of age 𝐴 or older.

Hence, population abundance at the start of year \(t\) is represented by

\[ \underline{N}(t) = \begin{bmatrix} N_1(t) \\ N_2(t) \\ N_3(t) \\ \vdots \\ N_A(t) \end{bmatrix} \tag{1} \label{eq:1} \] where \(N_a(t)\) is determined by a recruitment submodel (see section Stock-Recruitment Relationship)

Survival

Population survival from year \(t\) to \(t + 1\) is governed by instantaneous natural and fishing mortality at age. Let \(M_a(t)\) denote the instantaneous natural mortality rate for age \(a\) and and \(F_a(t)\) the instantaneous fishing mortality rate for age \(a\) in year \(t\), where \(F_a(t)\) is the sum of fishing mortalities at age \(a\) over fleets, \(F_a(t) = \sum\limits_{v}F_{a,v}(t)\). The expected number of survivors of age a fish from year \(t\) to year \(t + 1\) is

\[ N_{a+1}(t+1) = N_{a}(t)\cdot{e^{-M_{a}(t)-F_{a}(t)}} \tag{2}\label{eq:2} \]

For the plus-group, survival accounts for both \(A\) and \(A-1\) cohorts:

\[ N_A(t+1) = N_{A}(t)\cdot{e^{-M_{A}(t)-F_{A}(t)}} + N_{A-1}(t)\cdot{e^{-M_{A-1}(t)-F_{A-1}(t)}} \tag{3}\label{eq:3} \]

Recruitment \(N_a(t)\) is determined by a stochastic process that may depend on spawning biomass in year \(t\) or be independent of it.

Spawning Biomass

Spawning biomass represents the total weight of mature fish surviving to the midpoint of the spawning season. It is calculated from the abundance vector \(\underline{N}(t)\), mortality rates, maturity and weight at age.

Let \(p_a(t)\) be the fraction of age-\(a\) fish that are mature in year \(t\), \(w_{s,a}(t)\) be the average spawning weight of age-\(a\) fish in year \(t\), and \(\varphi(t)\) be the fraction of total mortality that occurs the spawning season midpoint in year \(t\). Population size at the midpoint of the spawning season \(\underline{N}_S(t)\) is

\[ \underline{N}_S(t) = \begin{bmatrix} N_1(t)\cdot e^{-\varphi(t)[M_1(t) + F_r(t)]} \\ N_2(t)\cdot e^{-\varphi(t)[M_2(t) + F_2(t)]} \\ N_3(t)\cdot e^{-\varphi(t)[M_3(t) + F_3(t)]} \\ \vdots \\ N_A(t)\cdot e^{-\varphi(t)[M_A(t) + F_A(t)]} \end{bmatrix} \tag{4} \label{eq:4} \]

Spawning Biomass in year \(t\) is then

\[ B_S(t) = \sum\nolimits_{a=1}^{A}w_{S,a}(t) \cdot p_{a}(t) \cdot N_{S,a}(t) \tag{5}\label{eq:5} \]