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The following table describes all elements contained within input_list$pars, which are generated using the SPoRC::Setup_x functions. Note that when n_sexes > 1, the first dimension will always be females and the second dimension will always be males.

Fishery Removals

Parameter Dimension Description Default
ln_sigmaC n_regions × n_years × n_seas × n_fish_fleets Log-scale standard deviation for aggregated (regional) catch likelihood 0.01
ln_sigmaC_pop n_pop × n_regions × n_years × n_seas × n_fish_fleets Log-scale standard deviation for population-specific catch likelihood. Only used when UseCatch_pop[p,r,y,seas,f] == 1. 0.01
ln_sigmaF n_regions × n_seas × n_fish_fleets Log-scale standard deviation for fishing mortality process error (used under all three Fdev_model structures: "iid", "rw", "ar1") 0
Fdev_rho n_regions × n_seas × n_fish_fleets Unconstrained AR1 correlation parameter for fishing mortality deviations, transformed to (-1, 1) via 2 / (1 + exp(-2x)) - 1. Only used when Fdev_model = "ar1"; mapped to NA (unused) otherwise regardless of Fdev_rho_spec 0
ln_F_mean n_regions × n_seas × n_fish_fleets Log-scale mean fishing mortality rate 0.1
ln_F_devs n_regions × n_years × n_seas × n_fish_fleets Log-scale annual fishing mortality deviations (can be specified as random effects). Process error structure set via Fdev_model: "iid" (default), "rw", or "ar1"; catch-active years need not be contiguous for "rw"/"ar1" (see vignette("c_model_equations")) 0
ln_sigmaD n_regions × n_years × n_seas × n_fish_fleets Log-scale standard deviation for aggregated (regional) discard likelihood 0.01
ln_sigmaD_pop n_pop × n_regions × n_years × n_seas × n_fish_fleets Log-scale standard deviation for population-specific discard likelihood 0.01
ln_sigma_dmr n_regions × n_seas × n_fish_fleets Log-scale standard deviation for discard mortality rate process error 0
logit_dmr_mean n_regions × n_seas × n_fish_fleets Logit-scale mean discard mortality rate 0
logit_dmr_devs n_regions × n_years × n_seas × n_fish_fleets Logit-scale annual discard mortality rate deviations (can be specified as random effects). Fixed at zero by default; dmr_dev_spec = "est_all" estimates a deviation in every fished cell, whether or not discard data are present there (see vignette("t_model_options")) 0

Age-Disaggregated Observations

Estimated only when a fleet fits catch, discards or an index at age. Each is keyed by age and fleet through an integer matrix in which equal entries share a parameter and NA excludes one, so one structure gives a standard deviation per age, per age group, or per fleet. See vignette("t_model_options").

Parameter Dimension Description Default
ln_sigmaCAA n_ages × n_sexes × n_fish_fleets Log-scale standard deviation for the catch-at-age likelihood. Keyed by sigmaCAA_key log(0.5)
ln_sigmaCAA_pop n_pop × n_ages × n_sexes × n_fish_fleets Population-specific counterpart, keyed by sigmaCAA_pop_key log(0.5)
ln_sigmaDAA n_ages × n_sexes × n_fish_fleets Log-scale standard deviation for the discard-at-age likelihood, keyed by sigmaDAA_key log(0.5)
ln_sigmaDAA_pop n_pop × n_ages × n_sexes × n_fish_fleets Population-specific counterpart, keyed by sigmaDAA_pop_key log(0.5)
ln_sigmaSrvIdxAA n_ages × n_sexes × n_srv_fleets Log-scale standard deviation for the survey index at age, keyed by sigmaSrvIdxAA_key log(0.5)
ln_sigmaSrvIdxAA_pop n_pop × n_ages × n_sexes × n_srv_fleets Population-specific counterpart log(0.5)
trans_rho_catch n_regions × n_sexes × n_fish_fleets Unconstrained AR(1) correlation across ages for catch at age. Transformed to (-1, 1) by rho_trans(). Shared according to rho_catch_spec, one per fleet by default, and mapped to NA unless AgeObsCorr_catch is "1dar1" or "2dar1" 0
trans_rho_catch_year n_regions × n_sexes × n_fish_fleets Correlation across years, read only under AgeObsCorr_catch = "2dar1" 0
trans_rho_catch_us n_pairs × n_regions × n_sexes × n_fish_fleets Unconstrained entries of the unstructured across-age correlation, n_pairs = n_ages(n_ages - 1)/2. Read only under AgeObsCorr_catch = "us" 0
trans_rho_discard, trans_rho_discard_year, trans_rho_discard_us as above As above, for discards at age 0
trans_rho_srv_idx, trans_rho_srv_idx_year, trans_rho_srv_idx_us n_regions × n_sexes × n_srv_fleets (and n_pairs × …) As above, for the survey index at age 0
trans_rho_catch_pop, trans_rho_discard_pop, trans_rho_srv_idx_pop as above with a leading n_pop Population-specific counterparts, with their own _year and _us forms. The _pop data sources have their own correlation rather than borrowing the aggregated one 0

Fishery Selectivity

Parameter Dimension Description Default
fish_fixed_sel_pars n_regions × max_fish_pars × max_fishsel_blks × n_sexes × n_fish_fleets Log-scale selectivity curve parameters 0
fishsel_pe_pars n_regions × max(4, n_sel_pars) × n_sexes × n_fish_fleets Process error parameters for time-varying selectivity: sigma (iid/rw), or partial correlations + log-variance (semi-parametric forms), see [Fishery and Survey Selectivity] in vignette("c_model_equations"). Second dimension is 4 unless the selectivity form itself needs more parameters (e.g. double-normal, non-parametric) 0
ln_fishsel_devs n_regions × (n_years + n_proj_yrs_devs) × n_bins × n_sexes × n_fish_fleets Log-scale selectivity deviations w/ projection years (n_bins = n_ages or n_lens depending on selectivity type) (can be specified as random effects) 0
ln_fishsel_bin_devs n_regions × (n_years + n_proj_yrs_devs) × n_bins × n_sexes × n_fish_fleets Log-scale bin-override selectivity deviations. Bins named in fish_sel_bin_dev_bins take exp(deviation) as their selectivity outright, replacing the functional form for that bin; all other cells are fixed at zero and unused (can be specified as random effects) 0
fishsel_bin_devs_pe_pars n_regions × n_bins × n_sexes × n_fish_fleets Log-scale process error standard deviations for the bin-override deviations. Only estimated for overridden bins whose fleet requests iid or random walk structure via cont_tv_fishsel_bin_devs 0
ln_fishsel_sex_scale n_regions × n_blocks × n_sexes × n_fish_fleets Log-scale constant multiplying the realized selectivity curve of every sex beyond the first (fish_sel_sex_offset = "scale" or "par_scale"). The first sex is the reference and its slots are never estimated; only the blocks a fleet has are 0
ret_fixed_sel_pars n_regions × max_ret_pars × max_retsel_blks × n_sexes × n_fish_fleets Retention selectivity curve parameters 0
retsel_pe_pars n_regions × max(4, n_sel_pars) × n_sexes × n_fish_fleets Process error parameters for time-varying retention selectivity (see fishsel_pe_pars above) 0
ln_retsel_devs n_regions × (n_years + n_proj_yrs_devs) × n_bins × n_sexes × n_fish_fleets Log-scale retention selectivity deviations w/ projection years (can be specified as random effects) 0
ln_retsel_bin_devs n_regions × (n_years + n_proj_yrs_devs) × n_bins × n_sexes × n_fish_fleets Log-scale bin-override retention selectivity deviations (see ln_fishsel_bin_devs above) 0
ln_retsel_sex_scale n_regions × n_blocks × n_sexes × n_fish_fleets Log-scale curve multiplier for sexes beyond the first on retention selectivity (see ln_fishsel_sex_scale above; ret_sel_sex_offset) 0
retsel_bin_devs_pe_pars n_regions × n_bins × n_sexes × n_fish_fleets Log-scale process error standard deviations for the retention bin-override deviations 0

Note that fish_fixed_sel_pars holds parameters on the scale of the chosen functional form: log scale for most forms, logit scale for the "nonpar" form and for asymptote parameters, and log scale per bin for the "nonparlog" form (where each year’s curve is standardized to average one over the bins named in fish_sel_norm_bins, every bin by default).

Fishery Catchability

Parameter Dimension Description Default
ln_fish_q n_regions × max_fishq_blks × n_fish_fleets Log-scale fishery catchability coefficient 0

Fishery Composition Likelihoods

Region-aggregated and population-specific composition likelihoods have separate overdispersion and correlation parameters. The _pop_ variants mirror the structure of their region-aggregated counterparts with an added leading n_pop dimension.

Parameter Dimension Description Default
ln_FishAge_theta n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for region-aggregated fishery age compositions 0
ln_FishAge_theta_agg n_fish_fleets Log-scale overdispersion for region- and sex-aggregated fishery age compositions 0
FishAge_corr_pars n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal region-aggregated fishery age composition likelihood 0.01
FishAge_corr_pars_agg n_fish_fleets Correlation parameters for region- and sex-aggregated logistic-normal fishery age compositions 0.01
ln_FishAge_pop_theta n_pop × n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for population-specific fishery age compositions 0
ln_FishAge_pop_theta_agg n_pop × n_fish_fleets Log-scale overdispersion for population-specific sex-aggregated fishery age compositions 0
FishAge_pop_corr_pars n_pop × n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal population-specific fishery age composition likelihood 0.01
FishAge_pop_corr_pars_agg n_pop × n_fish_fleets Correlation parameters for population-specific sex-aggregated logistic-normal fishery age compositions 0.01
ln_FishLen_theta n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for region-aggregated fishery length compositions 0
ln_FishLen_theta_agg n_fish_fleets Log-scale overdispersion for region- and sex-aggregated fishery length compositions 0
FishLen_corr_pars n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal region-aggregated fishery length composition likelihood. The first parameter represents bin correlations, while the second represents sex correlations (if specified) 0.01
FishLen_corr_pars_agg n_fish_fleets Correlation parameters for region- and sex-aggregated logistic-normal fishery length compositions. The correlation represents bin correlations 0.01
ln_FishLen_pop_theta n_pop × n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for population-specific fishery length compositions 0
ln_FishLen_pop_theta_agg n_pop × n_fish_fleets Log-scale overdispersion for population-specific sex-aggregated fishery length compositions 0
FishLen_pop_corr_pars n_pop × n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal population-specific fishery length composition likelihood. The first parameter represents bin correlations, while the second represents sex correlations (if specified) 0.01
FishLen_pop_corr_pars_agg n_pop × n_fish_fleets Correlation parameters for population-specific sex-aggregated logistic-normal fishery length compositions. The correlation represents bin correlations 0.01
ln_FishAge_discard_theta n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for region-aggregated discard age compositions 0
ln_FishAge_discard_theta_agg n_fish_fleets Log-scale overdispersion for region- and sex-aggregated discard age compositions 0
FishAge_discard_corr_pars n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal region-aggregated discard age composition likelihood 0.01
FishAge_discard_corr_pars_agg n_fish_fleets Correlation parameters for region- and sex-aggregated logistic-normal discard age compositions 0.01
ln_FishAge_discard_pop_theta n_pop × n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for population-specific discard age compositions 0
ln_FishAge_discard_pop_theta_agg n_pop × n_fish_fleets Log-scale overdispersion for population-specific sex-aggregated discard age compositions 0
FishAge_discard_pop_corr_pars n_pop × n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal population-specific discard age composition likelihood 0.01
FishAge_discard_pop_corr_pars_agg n_pop × n_fish_fleets Correlation parameters for population-specific sex-aggregated logistic-normal discard age compositions 0.01
ln_FishLen_discard_theta n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for region-aggregated discard length compositions 0
ln_FishLen_discard_theta_agg n_fish_fleets Log-scale overdispersion for region- and sex-aggregated discard length compositions 0
FishLen_discard_corr_pars n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal region-aggregated discard length composition likelihood. The first parameter represents bin correlations, while the second represents sex correlations (if specified) 0.01
FishLen_discard_corr_pars_agg n_fish_fleets Correlation parameters for region- and sex-aggregated logistic-normal discard length compositions. The correlation represents bin correlations 0.01
ln_FishLen_discard_pop_theta n_pop × n_regions × n_fish_fleets × n_sexes Log-scale overdispersion parameter for population-specific discard length compositions 0
ln_FishLen_discard_pop_theta_agg n_pop × n_fish_fleets Log-scale overdispersion for population-specific sex-aggregated discard length compositions 0
FishLen_discard_pop_corr_pars n_pop × n_regions × n_sexes × n_fish_fleets × 2 Correlation parameters for logistic-normal population-specific discard length composition likelihood. The first parameter represents bin correlations, while the second represents sex correlations (if specified) 0.01
FishLen_discard_pop_corr_pars_agg n_pop × n_fish_fleets Correlation parameters for population-specific sex-aggregated logistic-normal discard length compositions. The correlation represents bin correlations 0.01

Survey Selectivity

Parameter Dimension Description Default
srv_fixed_sel_pars n_regions × max_srv_pars × max_srv_blks × n_sexes × n_srv_fleets Log-scale selectivity curve parameters 0
srvsel_pe_pars n_regions × max(4, n_sel_pars) × n_sexes × n_srv_fleets Process error parameters for time-varying survey selectivity (see fishsel_pe_pars in Fishery Selectivity) 0
ln_srvsel_devs n_regions × (n_years + n_proj_yrs_devs) × n_bins × n_sexes × n_srv_fleets Log-scale selectivity deviations w/ projection years (n_bins = n_ages or n_lens depending on selectivity type) (can be specified as random effects) 0
ln_srvsel_bin_devs n_regions × (n_years + n_proj_yrs_devs) × n_bins × n_sexes × n_srv_fleets Log-scale bin-override selectivity deviations (see ln_fishsel_bin_devs in Fishery Selectivity) 0
ln_srvsel_sex_scale n_regions × n_blocks × n_sexes × n_srv_fleets Log-scale curve multiplier for sexes beyond the first on survey selectivity (see ln_fishsel_sex_scale in Fishery Selectivity; srv_sel_sex_offset) 0
srvsel_bin_devs_pe_pars n_regions × n_bins × n_sexes × n_srv_fleets Log-scale process error standard deviations for the survey bin-override deviations 0

Survey Catchability

Parameter Dimension Description Default
ln_srv_q n_regions × max_srvq_blks × n_srv_fleets Log-scale survey catchability coefficient. Automatically fixed (not estimated) for fleets whose catchability is solved analytically via srv_q_type 0
srv_q_coeff n_regions × n_srv_fleets × n_covariates Regression coefficients for covariate-based survey catchability 0

Survey Composition Likelihoods

Region-aggregated and population-specific composition likelihoods have separate overdispersion and correlation parameters. The _pop_ variants mirror the structure of their region-aggregated counterparts with an added leading n_pop dimension.

Parameter Dimension Description Default
ln_SrvAge_theta n_regions × n_srv_fleets × n_sexes Log-scale overdispersion parameter for region-aggregated survey age compositions 0
ln_SrvAge_theta_agg n_srv_fleets Log-scale overdispersion for region- and sex-aggregated survey age compositions 0
SrvAge_corr_pars n_regions × n_sexes × n_srv_fleets × 2 Correlation parameters for logistic-normal region-aggregated survey age composition likelihood 0.01
SrvAge_corr_pars_agg n_srv_fleets Correlation parameters for region-aggregated logistic-normal survey age compositions 0.01
ln_SrvAge_pop_theta n_pop × n_regions × n_srv_fleets × n_sexes Log-scale overdispersion parameter for population-specific survey age compositions 0
ln_SrvAge_pop_theta_agg n_pop × n_srv_fleets Log-scale overdispersion for population-specific sex-aggregated survey age compositions 0
SrvAge_pop_corr_pars n_pop × n_regions × n_sexes × n_srv_fleets × 2 Correlation parameters for logistic-normal population-specific survey age composition likelihood 0.01
SrvAge_pop_corr_pars_agg n_pop × n_srv_fleets Correlation parameters for population-specific sex-aggregated logistic-normal survey age compositions 0.01
ln_SrvLen_theta n_regions × n_srv_fleets × n_sexes Log-scale overdispersion parameter for region-aggregated survey length compositions 0
ln_SrvLen_theta_agg n_srv_fleets Log-scale overdispersion for region- and sex-aggregated survey length compositions 0
SrvLen_corr_pars n_regions × n_sexes × n_srv_fleets × 2 Correlation parameters for logistic-normal region-aggregated survey length composition likelihood. The first parameter represents bin correlations, while the second represents sex correlations (if specified) 0.01
SrvLen_corr_pars_agg n_srv_fleets Correlation parameters for region- and sex-aggregated logistic-normal survey length compositions. The correlation represents bin correlations 0.01
ln_SrvLen_pop_theta n_pop × n_regions × n_srv_fleets × n_sexes Log-scale overdispersion parameter for population-specific survey length compositions 0
ln_SrvLen_pop_theta_agg n_pop × n_srv_fleets Log-scale overdispersion for population-specific sex-aggregated survey length compositions 0
SrvLen_pop_corr_pars n_pop × n_regions × n_sexes × n_srv_fleets × 2 Correlation parameters for logistic-normal population-specific survey length composition likelihood. The first parameter represents bin correlations, while the second represents sex correlations (if specified) 0.01
SrvLen_pop_corr_pars_agg n_pop × n_srv_fleets Correlation parameters for population-specific sex-aggregated logistic-normal survey length compositions. The correlation represents bin correlations 0.01

Natural Mortality

Parameter Dimension Description Default
ln_M n_pop_block × n_regions_block × n_years_block × n_ages_block × n_sexes_block Log-scale natural mortality rate (dimensioned by user-defined blocks) 0.5

Recruitment

Parameter Dimension Description Default
ln_global_R0 n_pop x n_R0_blocks Log-scale unfished/mean recruitment per population and R0 time block. One column unless R0_blocks asks for more, so a model without R0 blocks is unchanged. Only the recruitment computed each year reads that year’s block; the initial age structure, the regional apportionment, the R0 prior, the ln_rinit penalty and the stock-recruit scale all read R0_ref_block. Also supplies the scale of the stock-recruit penalty curve when sr_R0_spec = "shared" (the default) 15
ln_rinit n_pop Log-scale initial recruitment scalar per population. Initializes the equilibrium age structure when use_rinit = 1, and additionally supplies the scale of the stock-recruit penalty curve when sr_R0_spec = "rinit", so one parameter sets both; ignored when use_rinit = 0. 15
ln_sr_R0 n_pop Log-scale asymptote of the stock-recruit curve fitted as a penalty under rec_model = "mean_rec". Estimated only when sr_R0_spec = "est" and sr_penalty is not "none"; mapped off in every other configuration, so it never enters the parameter vector by accident ln_global_R0
rec_region_prop_pars n_pop × (n_regions - 1) Logit-scale regional recruitment proportion parameters (simplex parameterization; omitted when rec_region_prop_spec == 1) 0
rec_seas_prop_pars n_pop × (n_seas - 1) Logit-scale seasonal recruitment proportion parameters (simplex parameterization; omitted when use_fixed_rec_seas_prop == 1) 0
steepness_h n_pop × n_regions Steepness of the stock-recruitment relationship (logit scale, bounded 0.2 to 1). Used by both "bh_rec" and "ricker_rec", and by the curve fitted as a penalty under rec_model = "mean_rec" with sr_penalty; mapped off under mean recruitment with no such curve. Steepness values are not interchangeable between the two curves (see vignette("c_model_equations")) 0
ln_sigmaR 2 × n_pop × n_regions Log-scale standard deviations for early (index 1) and late (index 2) recruitment periods 0
ln_InitDevs n_pop × n_regions × (n_ages - 1) × n_sexes Log-scale initial age structure deviations (can be specified as random effects). Under InitDevs_sex_spec = "est_shared_s" (default) every sex reads one shared curve; under "est_all" each sex has its own 0
ln_RecDevs n_pop × n_regions × (n_recdev_years + n_proj_yrs_devs) Log-scale annual recruitment deviations w/ projection years (can be specified as random effects) 0
ln_NAA n_pop × n_regions × n_years × n_seasons × n_ages × n_sexes Log-scale numbers at the start of a season when NAA_re is on. A centered state, the log numbers themselves rather than a deviation, estimated for ages two and older in years two onward including the plus group; cells outside NAA_re_ages, NAA_re_years and NAA_re_seasons are mapped off. Season one is the year boundary, so the default NAA_re_seasons = "annual" leaves the numbers deterministic within a year. Specified as random effects. Starts from an equilibrium decay from R0R_0 at the mean MM, split over regions and sexes, with the plus group accumulated; supply ln_NAA in starting_values to override equilibrium
ln_sigmaNAA n_pop_block × n_regions_block × n_years_block × n_seasons_block × n_ages_block × n_sexes_block Log-scale process error standard deviation for ln_NAA, dimensioned by the NAA_sigma_*blk_spec blocks as ln_M is. Only NAA_re = "iid" admits variation over the year and age dims; the season dim varies freely unless NAA_re_season = "us" log(0.3)
NAA_pe_pars n_pop × n_regions × 3 × n_sexes Unconstrained correlations over the age-year grid for ln_NAA, read as age, year and cohort. Only the slots the chosen NAA_re form uses are estimated 0
NAA_region_corr_pars n_pop × n_regions(n_regions - 1)/2 × n_sexes Unconstrained parameters for an unstructured correlation across regions, shared according to NAA_re_region_spec 0
NAA_season_corr_pars n_pop × n_k(n_k - 1)/2 × n_sexes Unconstrained parameters for an unstructured correlation across the nkn_k active seasons, shared according to NAA_re_season_spec 0
NAA_pop_corr_pars n_pop(n_pop - 1)/2 Unconstrained parameters for an unstructured correlation across populations, global to the model 0
NAA_sex_corr_pars n_sexes(n_sexes - 1)/2 Unconstrained parameters for an unstructured correlation across sexes, global to the model. One parameter with two sexes, and weakly identified unless sex-structured compositions hold the contrast 0
sexratio_pars n_pop × n_regions × n_blocks Logit-scale sex ratio parameters (defaults to 50:50 when n_sexes = 1) 0
stray_rate_pars n_pop × n_stray_blocks Logit-scale stray rate parameters controlling the proportion of population p’s spawning biomass in a non-natal region that contributes to other populations’ effective SSB at their natal regions. Only estimated when use_fixed_stray_rate == 0. Note: stray rate is likely unidentifiable from fishery and survey data alone and should be fixed or given an informative prior. 0

Movement

Parameter Dimension Description Default
move_pars n_pop × n_regions × (n_regions - 1) × n_years × n_seas × n_ages × n_sexes Logit-scale movement probabilities (unstructured Markov; move_type = 0) 0
log_move_diffusion_pars (Vector; depends on formula) Log-scale diffusion parameters for CTMC movement (move_type = 1) log(0.1)
move_preference_pars (Vector; depends on formula) Preference parameters for CTMC movement (move_type = 1) 0
move_devs n_pop × n_regions × (n_regions - 1) × (n_years + n_proj_yrs_devs) × n_seas × n_ages × n_sexes Movement deviations w/ projection years (can be specified as random effects); IID process error only, sharing structure set via cont_vary_movement 0
move_pe_pars n_pop × n_regions × n_seas × n_ages × n_sexes Standard deviations for continuously varying movement (IID). Which dimensions are actively estimated vs. kept fixed at index 1 is determined by cont_vary_movement (e.g. "iid_y" only varies over year) 0

Tagging

Parameter Dimension Description Default
ln_init_conv_tag_mort 1 Log-scale initial tagging mortality rate -1000
ln_conv_tag_shed 1 Log-scale chronic tag shedding rate -1000
conv_tag_fish_reporting_pars n_regions × n_blocks × n_fish_fleets Logit-scale tag reporting rates 0
ln_conv_fish_tag_theta 1 Log-scale overdispersion parameter for tag recapture likelihood 0