Initialize Numbers-at-Age (NAA) for a Population Model

Computes equilibrium initial numbers-at-age (NAA) for a structured population model across populations, regions, sexes, and ages. Several initialization methods are supported, ranging from numerical iteration to analytical geometric-series solutions that optionally incorporate seasonal movement.

Usage

Get_Init_NAA(
  init_age_strc,
  init_iter,
  n_regions,
  n_pop,
  n_sexes,
  n_ages,
  n_seas,
  n_fish_fleets,
  seasdur,
  rec_seas_prop,
  natmort,
  init_F,
  dmr,
  fish_sel,
  ret_sel,
  R0_r,
  sexratio,
  Movement,
  do_recruits_move,
  ln_InitDevs
)

Arguments

init_age_strc

Integer specifying the initialization method:

0 Iterative equilibrium solution
1 Scalar geometric-series solution (no movement in any age)
2 Matrix geometric-series solution (movement allowed)
3 Hybrid solution: movement in ages < plus group, scalar solution for plus group

init_iter

Integer; number of annual iterations used when init_age_strc = 0.

n_regions

Integer; number of spatial regions.

n_pop

Integer; number of populations.

n_sexes

Integer; number of sexes.

n_ages

Integer; number of age classes (including the plus group).

n_seas

Integer; number of seasons per year.

n_fish_fleets

Integer; number of fishing fleets.

seasdur

Numeric vector (n_seas) giving the fraction of the year represented by each season.

rec_seas_prop

Matrix (n_pop x n_seas) giving seasonal recruitment proportions.

natmort

Array (n_pop x n_regions x n_ages x n_sexes) of natural mortality rates.

init_F

Numeric array (n_regions x n_seas x n_fish_fleets) giving fishing mortality applied in each region, season, and fleet during initialization. Set to zero for an unfished population.

dmr

Numeric array (n_regions x n_seas x n_fish_fleets) giving discard mortality rate applied in each region, season, and fleet during initialization (first year). Set to zero for an unfished population.

fish_sel

Array (n_pop x n_regions x n_seas x n_ages x n_sexes x n_fish_fleets) of total fishery selectivity at age.

ret_sel

Array (n_pop x n_regions x n_seas x n_ages x n_sexes x n_fish_fleets) of retained fishery selectivity at age.

R0_r

Matrix (n_pop x n_regions) giving unfished recruitment allocated to each region.

sexratio

Array (n_pop x n_regions x n_sexes) giving the proportion of recruits by sex.

Movement

Array (n_pop x origin x destination x n_seas x n_ages x n_sexes) containing seasonal movement probabilities.

do_recruits_move

Integer indicator:

0 Recruits do not move during their first year
1 Recruits move according to the movement matrix

ln_InitDevs

Array (n_pop x n_regions x (n_ages - 1)) containing log-scale deviations applied to ages 2 through $A$.

Details

The resulting age structure represents an equilibrium population under constant recruitment ($R_0$), mortality, fishing mortality, and movement.

Initial numbers-at-age are derived assuming constant recruitment ($R_0$) and constant mortality and movement.

Let

$N_{p,r,a,s}$ denote numbers-at-age
$M_{p,r,a}$ denote natural mortality
$F_{r,a,s}$ denote total fishing mortality (retained + dead discards)
$Z = M + F$ denote total mortality

Recruitment at age 1 is

$$ N_{p,r,1,s} = R_{0,p,r} \times sexratio_{p,r,s} $$

Within-season survival follows

$$ N_{p,r,a,s+1} = N_{p,r,a,s}\exp(-Z_{p,r,a,s}) $$

Ages advance at the end of the final season of the year:

$$ N_{p,r,a+1,1} = N_{p,r,a,n_{seas}} \exp(-Z_{p,r,a,n_{seas}}) $$

The plus group accumulates survivors from the terminal age:

$$ N_{A^+} = N_{A-1} e^{-Z_{A-1}} + N_{A^+} e^{-Z_{A^+}} $$

Fishing mortality at age is decomposed into retained and dead discard components, summed across all fleets:

$$ F_{p,r,s,a} = \sum_{f=1}^{n_f} F^{init}_{r,s,f} \left[ sel_{p,r,s,a,f} \cdot ret_{p,r,s,a,f} + sel_{p,r,s,a,f} \cdot (1 - ret_{p,r,s,a,f}) \cdot dmr_{r,s,f} \right] $$

where $sel$ is total fishery selectivity, $ret$ is retention selectivity, and $dmr$ is the discard mortality rate.

### Scalar geometric-series solution

When movement is absent, equilibrium abundance follows

$$ N_a = N_1 \exp\left(-\sum_{i=1}^{a-1} Z_i \right) $$

The plus group has a closed-form solution

$$ N_{A^+} = \frac{N_{A-1} e^{-Z_{A-1}}} {1 - e^{-Z_{A^+}}} $$

### Matrix geometric-series solution

When movement occurs, survival and movement are combined into seasonal transition matrices:

$$ \mathbf{T}_a = \prod_{s=1}^{n_{seas}} \mathbf{M}_{a,s}\mathbf{S}_{a,s} $$

where

$\mathbf{M}$ is the movement transition matrix
$\mathbf{S}$ is a diagonal matrix of survival probabilities

The plus group equilibrium satisfies

$$ \mathbf{N}_{A^+} = (\mathbf{I} - \mathbf{T}_{A^+})^{-1} \mathbf{T}_{A-1} \mathbf{N}_{A-1} $$

The iterative method (init_age_strc = 0) numerically applies the full seasonal population dynamics repeatedly until the population converges to equilibrium.

After equilibrium is derived, log-scale initial age deviations (ln_InitDevs) are applied multiplicatively to ages $2,\dots,A$.