CEATTLE is an age-structured model that follows cohorts of
individuals of a species \(i\), age
\(a\), and sex \(s\) through time \(y\). Annual recruitment and the initial
age-structure are estimated and are treated as random effects or as
penalized deviates, with the variance in recruitment estimated or fixed,
respectively. Note that because the initial age-structure is estimated
(assuming the population is depleted prior to the first year of the
model), the mean recruitment parameter (\(\bar{R}_{i}\) will be biased low and
projected recruitment will have to account for that (e.g. resemble
recruitment deviates from hindcast). Similarly, biomass reference points
should use the mean of annual recruitment. Populations can be modelled
as 1- or 2-sexes via nsex in the data. The sex-ratio at
birth \(\rho_{i1}\) is set in the
sex_ratio data for the first age. Natural mortality is
partitioned into predation mortality (M2) and non-predation related
mortality (M1), the latter of which is either input or estimated
(estM1), and accounts for mortality related to causes other
than predation by the species included in the model. Predation can be
turned off by setting msmMode = 0 (see below for more
detail).
\[N_{is1,y} = R_{isy} = \bar{R}_{i }e^{\tau_{i,y}}*\rho_{i1}\] \[N_{isa,1} = \left \{ \begin{array}{} \bar{R}_ie^{ \left( -j M1_{isa} \right)}*\rho_{i1} \\ \frac{\bar{R}_ie^{ \left( -\sum^{a-1} M1_{isa} \right)}} {\left(1 - e^{\left( -\sum^{a-1} M1_{is A_i} \right)} \right)} *\rho_{i1} \end{array} \right.\] \[N_{is a+1, y+1} = N_{isa,y} e^{-Z_{isa,y}}\] \[N_{is A_i, y+1} = N_{is A_{i}-1,y} e^{-Z_{is A_{i}-1,y}} + N_{is A_{i},y} e^{-Z_{is A_{i},y}}\] \[B_{isa,y} = N_{isa,y} W_{isa,y}\] \[SSB_{isa,y} = B_{isa,y} \rho_{ia}\] \[Z_{isa,y} = M1_{isa} + M2_{isa,y} + F_{isa,y}\]
Multiple age-based predation parameterizations are supported by
Rceattle and are controlled by suitMode defining the
predator-prey preference (suitability/predator selectivity) and
msmMode defining the functional response of predation. The
data used to parameterize the predation model include the proportion of
prey-at-age in the diet of a predator-at-age (UobsAgeWt in
the data) and the ration/consumption data (defined below).
Suitability represents a predator’s prey and size preference and factors such as spatial overlap in the distribution of predators and prey.
Suitability is derived empirically from the ratio of mean proportion
(by weight) of prey-at-age in the stomach of predators-at-age \(\bar{U}_{pbjisa}\) (UobsAgeWt
in the data). Magnusson (1995) provides details on the derivation of
suitability coefficients. Magnusson (1995) provides details on the
derivation of suitability coefficients. suitMode = 0
Predator-prey suitability for predators of species \(p\), sex \(b\), age \(j\), and prey of species \(i\), sex \(s\) and age \(a\) \[\hat{S}_{pbjisa} = \frac{1}{n_y} \sum_y \left(
\frac{ \frac{\bar{U}_{pbjisa}} {B_{isa,y}} } {\sum_{isa} \left(
\frac{\bar{U}_{pbjisa}} {B_{isa,y}} \right) + \frac{1 + \sum_{isa}
\bar{U}_{pbjisa}} {B_p^{other}} } \right)\]
This is essentially multiple single-species age structured models
where all mortality is due to input or estimated natural mortality
(\(M = M1\)) and fishing.
msmMode = 0 \[M2_{isa,y} =
0\]
msmMode = 1 Predation mortality can be modelled
following previous MSCAA models (Curti et al., 2013; Holsman et al.,
2016; Tsehaye et al., 2014) based on MSVPA. Predation mortality due to a
predator-at-age in the model is derived from estimated or empirically
derived suitability coefficients \(\hat{S}_{pbjisa}\), input estimates of
annual ration/consumption (see below), estimated or input biomass, and
input biomass of other prey in the system for predator \(p\) (\(B_p^{other}\)). Estimated average biomass
is used to derive predation mortality and because of the interdependence
between predation and biomass, predation mortality and the underlying
age-structured dynamics are iteratively estimated until convergence.
Preliminary analysis suggested that three to five iterations are
sufficient for convergence. Models are generally insensitive to a range
of \(B_p^{other}\), but historically
they have been derived from ecosystem-wide biomass estimates. Predation
mortality equations for predators of species \(p\), sex \(b\), age \(j\), and prey of species \(i\), sex \(s\) and age \(a\) \[M2_{isa,y}
= \sum_{pbj} \left( \frac{N_{pbj,y} \delta_{pbj,y} {S}_{pbjisa}}
{\sum_{isa} \left({S}_{pbjisa} B_{isa,y} \right) + B_p^{other} \left(1 -
\sum_{isa} \left({S}_{pbjisa} \right) \right)} \right)\]
msmMode = 2 A Type III functional response variant of
the MSVPA predation model. The available-food denominator replaces prey
biomass \(B_{isa,y} = N_{isa,y}
W_{isa,y}\) with \(N^2_{isa,y}
W_{isa,y}\), creating a sigmoidal (accelerating-then-saturating)
relationship between prey density and predation rate. This gives
low-density prey a refuge from predation and can stabilise multi-species
dynamics at low stock sizes. \[M2_{isa,y} =
\sum_{pbj} \left( \frac{N_{pbj,y} \delta_{pbj,y} {S}_{pbjisa}}
{\sum_{i'a's'} \left({S}_{pbji'a's'}
N^2_{i'a's',y} W_{i'a's',y} \right) +
B_p^{other} \left(1 - \sum_{i'a's'}
\hat{S}_{pbji'a's'} \right)} \right)\]
Multiple age-based consumption parameterizations are supported by
Rceattle. They include a environmentally driven bioenergetics
consumption and direction input of consumption-at-age. The consumption
model is controlled by the following objects in the data
(bioenergetics_control sheet of the excel data input):
Ceq Integer: switch for which bioenergetics equation to
use for each species for ft to scale max consumption: 1 = Exponential
(Stewart et al 1983), 2 = Temperature-dependence for warm-water species
(Kitchell et al 1977; sensu Holsman et al 2015), 3 = temperature
dependence for cool and cold-water species (Thornton and Lessem 1979); 4
= temperature function held constant at 1.0 — bypasses bioenergetics and
uses ration_data as direct empirical annual ration input
(set CA = 1, fday = 1,
CB = 0)Cindex Integer: which environmental index in env_data
to use to use for \(T_y\) for
bioenergetics based rationPvalue Tuning parameter \(\varphi_p\) that scales the maximum
consumption used for ration for each speciesfday Data \(Fday_p\)
number of foraging days per year for each speciesCA Parameter \(\alpha^{\delta}_p\) for slope of allometric
mass function for calculating maximum consumption: CA * Weight ^ CBCB Parameter \(\beta^{\delta}_p\) for slope of allometric
mass function for calculating maximum consumption: CA * Weight ^
1+CBQc Parameter \(Q^c_p\)
for temperature scaling function of maximum consumption specified by
CeqTco Parameter \(T^{co}_p\) for temperature scaling function
of maximum consumption specified by CeqTcm Parameter \(T^{cm}_p\) for temperature scaling function
of maximum consumption specified by CeqTcl Parameter \(T^{cl}_p\) for temperature scaling function
of maximum consumption specified by CeqCK1 Parameter \(CK1_p\) for temperature scaling function of
maximum consumption specified by CeqCK4 Parameter \(CK4_p\) for temperature scaling function of
maximum consumption specified by Ceqration_data \(RFR_{pbj,y}\) is the relative foraging rate
data used to modify ration. Essentially relative foraging rate and,
subsequently, ration can be tuned via fday and
pvalue.
The weight-at-age used for the consumption equations is specified by
pop_wt_index
Ceq %in% 1:3 Individual specific ration (\(\delta_{pbj,y}\) \((kg~kg^{-1}~yr^{-1})\)) of predator \(p\) of sex \(b\) and age \(j\) in year \(y\): \[\delta_{pbj,y} = RFR_{pbj,y}*Fday_p
*\alpha^{\delta}_p W_{pbj,y}^{\left(1+\beta^{\delta}_p\right)}
*f\left(T_y \right)_p\] Ration can be tuned by adjusting
Pvalue (\(\varphi_p\)) to
help convergence: \[\delta_{pbj,y} =
\varphi_p * \delta_{pbj,y}\]
Temperature scaling algorithms.
Ceq = 1: Exponential function from Stewart et al. 1983:
\[f \left(T_y \right)_p = e^{Q^c_p
*T_y}\]
Ceq = 2: Temperature dependence for warm-water-species
from Kitchell et al. 1977: \[f \left(T_y
\right)_p = V^X e^{\left(X \left(1-V \right) \right)}\] \[V = \left( T^{cm}_p - T_y \right)/ \left(
T^{cm}_p - T^{co}_p \right)\] \[X =
\left(Z^2 \left(1+ \left( 1 + 40 /Y \right)^{0.5} \right)^2
\right)/400\] \[Z = ln\left(Q^c_p
\right) \left( T^{cm}_p - T^{co}_p \right)\] \[Y = ln\left(Q^c_p \right) \left( T^{cm}_p -
T^{co}_p + 2\right)\]
Ceq = 3: Temperature dependence for cool and cold-water
species from Thornton and Lessem 1979: \[f
\left(T_y \right)_p = Ka * Kb\] \[Ka =
C^{K1}_p * L_1 / (1+C^{K1}_p * (L_1 - 1 ))\] \[L_1 = e^{G_1 * (T_y - Q^c_p)}\] \[G_1 =
\frac{1}{T^{co}_p-Q^c_p}*\log\left(\frac{0.98*(1-CK1_p)}{CK1_p*0.02}\right)\]
\[Kb = \frac{CK4_p *L_2}
{1+CK4_p*(L_2-1)}\] \[L_2 = e^{G_2
(T^{cl}_p - T_y)}\] \[G_2 =
\frac{1}{T^{cl}_p-T^{cm}_p}*\log\left(
\frac{0.98*(1-CK4_p)}{CK4_p*0.02}\right)\]
Ceq = 4 and for directly input of consumption-at-sex/age
in the ration_data data sheet for \(RFR_{pbj,y}\). Set CA = 1,
fday = 1, and CB = 0. \[\delta_{pbj,y} = RFR_{pbj,y}* W_{pbj,y}
\]
CEATTLE is fit to time series of fishery and survey biomass, sex-ratio, and length- or age-composition data. Log-fishery catch, log-survey biomass and sex-ratio are assumed to be normally distributed, while age- and length-composition data are fit assuming multinomial distributions. Variance parameters for the lognormal distributions and initial input sample size for the multinomial distributions can be assumed known or for surveys also estimated as a free parameter or estimated analytically following Walters and Ludwig (1994). Separate selectivity and catchability functions can be estimated or input for each survey or fishery.
Catch is estimated in the model following the Baranov catch equation: \[\hat{C}_{isa,y} = \frac{F_{isa,y}} {Z_{isa,y}} \left( 1 - e^{-Z_{isa,y}} \right) N_{isa,y}\] \[\hat{C}_{i,y} = \sum_s\sum_a^{A_i} \frac{F_isa,y} {Z_{isa,y}} \left( 1 - e^{-Z_{isa,y}} \right) N_{isa,y} W_{isa,y}\] \[F_{isa,y} = \bar{F}_{f_i}e^{\epsilon_{{f_i},y} s_{{f_i}sa}}\] CPUE data is estimated given catchability \(q_{f_iy}\), selectivity \(S_{f_isa,y}\), and fleet month \(Month_{f_i}\) and can either be in biomass \(\hat{CPUE:B}_{{f_i},y}\) or numbers \(\hat{CPUE:I}_{{f_i},y}\): \[\hat{CPUE:B}_{{f_i},y} = \sum_s\sum_a \left( N_{isa,y} e^{-Month_{f_i} Z_{isa,y}} W_{{f_i}sa,y} S_{{f_i}isa,y} * q_{{f_i}y} \right)\] \[\hat{CPUE:I}_{{f_i},y} = \sum_s\sum_a \left( N_{isa,y} e^{-Month_{f_i} Z_{isa,y}} S_{{f_i}isa,y} * q_{{f_i}y} \right)\]
Multiple age-based selectivity functions are supported by Rceattle.
They are controlled by Selectivity_index,
Selectivity, N_sel_bins,
Time_varying_sel, Time_varying_sel_sd_prior,
and Bin_first_selected in fleet_control sheet of an
Rceattle data file. They are defined as follows:
Selectivity_index: index to use if selectivities of
different surveys are to be the sameSelectivity: Selectivity parameterization to use for
the species: 0 = empirical selectivity provided in
srv_emp_sel; 1 = logistic selectivity; 2 = non-parametric
selecitivty sensu Ianelli et al 2018; 3 = double logistic; 4 =
descending logistic; 5 = non-parametric selectivity sensu Taylor et al
2014 (Hake), 6 or “2DAR1” for age by year, 7 or “3DAR1” sensu Cheng et
al (2025).N_sel_bins: Number of ages to estimate non-parametric
selectivity. For example, if minage = 1 and selectivity
parameters are estimated up till age-6, N_sel_bins = 6 and
if minage = 0 and selectivity parameters are estimated up
till age-6, N_sel_bins = 7. Not used otherwiseTime_varying_sel: Whether a time-varying selectivity
should be estimated for logistic, double logistic selectivity,
descending logistic, or non-parametric selectivity
(Selectivity = 5 or "Hake"). 0 = no, 1 = penalized deviates
given sel_sd_prior, 2 = penalized deviates and estimate sel_sd_prior, 3
= time blocks with no penalty, 4 = random walk following Dorn, 5 =
random walk on ascending portion of double logistic only.Time_varying_sel_sd_prior: The sd to use for the random
walk of time varying selectivity if set to 1.Bin_first_selected: Age/length bin at which selectivity
is non-zero. Selectivity before this age will be set to 0.Input: Selectivity = 0 or "Fixed" and input
selectivity-at-age in the emp_sel data.
Input: Selectivity = 2 or "NonParametric",
N_sel_bins = 6, Sel_curve_pen1 = 12.5,
Sel_curve_pen2 = 200, and
Bin_first_selected = 1
Equation: \[sel_{f_i sa}=e^{\phi_{f_isa}}\]
Input: Selectivity = 1 or "Logistic",
N_sel_bins = NA, Time_varying_sel = 0, and
Time_varying_sel_sd_prior = NA
Equation: \[sel_{f_i sa}=1/(1+e^{-Slp_{f_i} (a-Inf_{f_i} ) } )\]
Input: Selectivity = 1 or "Logistic",
N_sel_bins = NA, and
Time_varying_sel_sd_prior = 0.1 or other desired value.
Equation: \[sel_{f_i
sa}=1/(1+e^{-(Slp_{f_i}+e^{\phi_{f_i sy}^{slp} } )
(a-(Inf_{f_i}+\phi_{f_i sy}^{Inf} ) ) } )\] Penalized likelihood:
Time_varying_sel = 1 \[\phi_{f_i
sy} \sim N(0,\sigma^{\phi}_{f_i}) \] Random effect:
Time_varying_sel = 2 \(\hat\sigma^{\phi}_{f_i}\) is estimated
\[\phi_{f_i sy} \sim
N(0,\hat\sigma^{\phi}_{f_i}) \] Block:
Time_varying_sel = 3 main parameters are set to 0 and the
following is specificed: \[\phi_{f_i sy} =
\phi_{f_i sy_{block}}\] Penalized likelihood random walk
Time_varying_sel = 4 \[\phi_{f_i
sy}-\phi_{f_i sy-1} \sim N(0,\sigma^{\phi}_{f_i}) \]
Input: Selectivity = 3 or "DoubleLogistic",
N_sel_bins = NA, Time_varying_sel = 0, and
Time_varying_sel_sd_prior = NA
Equation: \[sel_{f_i sa}=1/(1+e^{-Slp1_{f_i} (a-Inf1_{f_i} ) } )(1-1/(1+e^{-Slp2_{f_i} (a-Inf2_{f_i} ) } ))\]
\[sel_{f_i sa}=1/(1+e^{-(Slp1_{f_i}+e^{\phi_{f_i sy}^{slp1} } ) (a-(Inf1_{f_i}+\phi_{f_i sy}^{Inf1} ) ) } )(1-1/(1+e^{-(Slp2_{f_i}+e^{\phi_{f_i sy}^{slp2} } ) (a-(Inf2_{f_i}+\phi_{f_i sy}^{Inf2} ) ) } ))\]
Penalized likelihood: Time_varying_sel = 1 \[\phi_{f_i sy} \sim N(0,\sigma^{\phi}_{f_i})
\] Random effect: Time_varying_sel = 2 \(\hat\sigma^{\phi}_{f_i}\) is estimated
\[\phi_{f_i sy} \sim
N(0,\hat\sigma^{\phi}_{f_i}) \] Block:
Time_varying_sel = 3 main parameters are set to 0 and the
following is specificed: \[\phi_{f_i sy} =
\phi_{f_i sy_{block}}\] Penalized likelihood random walk
Time_varying_sel = 4 \[\phi_{f_i
sy}-\phi_{f_i sy-1} \sim N(0,\sigma^{\phi}_{f_i}) \]
Input: Selectivity = 3 or "DoubleLogistic",
N_sel_bins = NA, Time_varying_sel = 5, and
Time_varying_sel_sd_prior = NA
Equation: \[sel_{f_i sa}=1/(1+e^{-(Slp1_{f_i}+e^{\phi_{f_i sy}^{slp1} } ) (a-(Inf1_{f_i}+\phi_{f_i sy}^{Inf1} ) ) } )(1-1/(1+e^{-Slp2_{f_i} (a-Inf2_{f_i} ) } ))\]
\[\phi_{f_i sy} - \phi_{f_i sy-1} \sim N(0,\sigma^{\phi}_{f_i}) \]
Input: Selectivity = 4 or "DescendingLogistic",
N_sel_bins = NA, Time_varying_sel = 0, and
Time_varying_sel_sd_prior = NA
Equation: \[sel_{f_i sa}=1-1/(1+e^{-Slp2_{f_i} (a-Inf2_{f_i} ) } )\]
Input: Selectivity = 4 or "DescendingLogistic",
N_sel_bins = NA, Time_varying_sel = 1 or
2 , and Time_varying_sel_sd_prior = NA
Equation: \[sel_{f_i sa}=1-1/(1+e^{-(Slp2_{f_i}+e^{\phi_{f_i sy}^{slp2} } ) (a-(Inf2_{f_i}+\phi_{f_i sy}^{Inf2} ) ) } )\]
Penalized likelihood: Time_varying_sel = 1 \[\phi_{f_i sy} \sim N(0,\sigma^{\phi}_{f_i})
\] Random effect: Time_varying_sel = 2 \(\hat\sigma^{\phi}_{f_i}\) is estimated
\[\phi_{f_i sy} \sim
N(0,\hat\sigma^{\phi}_{f_i}) \] Block:
Time_varying_sel = 3 main parameters are set to 0 and the
following is specificed: \[\phi_{f_i sy} =
\phi_{f_i sy_{block}}\] Penalized likelihood random walk
Time_varying_sel = 4 \[\phi_{f_i
sy}-\phi_{f_i sy-1} \sim N(0,\sigma^{\phi}_{f_i}) \]
For each age \(a \geq A_{min}\) there is an incremental selectivity parameter \(p_{f_i say}\) for the fleet \(f_i\). Selectivity at age is calculated as follows:
\[sel_{f_i sa}=e^{sel^{`}_{f_i sa} -
max(sel^{`}_{f_i sa})}\] where \[
sel^{`}_{f_i sa} = \sum_{i=A_{min}}^a p_{f_i sa}\] Selectivity is
fixed at \(sel_{f_i sa} = 0\) for \(a < A_{min}\) and \(p_{f_i sa} = 0\) for ages above
minage + N_sel_bins, giving constant selectivity beyond the
last estimated value. minage is specified in the data and
is the minimum age of the modeled population.
Input: Selectivity = 5 or "Hake",
N_sel_bins = 6, Time_varying_sel = NA,
Time_varying_sel_sd_prior = NA, and
Bin_first_selected =\(A_{min}\)
As above, but a set of deviations on \(p_a\) is used to control annual changes in selectivity given a fixed or estimated standard deviation (\(\sigma^{\phi}_{f_i}\)): \[p_{f_i say}=p_{f_i sa}+\phi_{f_i say}\]
Input: Selectivity = 5 or "Hake",
N_sel_bins = 6, Time_varying_sel = NA,
Time_varying_sel_sd_prior = NA, and
Bin_first_selected =\(A_{min}\)
Penalized likelihood: Time_varying_sel = 1 \[\phi_{f_i sy} \sim N(0,\sigma^{\phi}_{f_i})
\] Random effect: Time_varying_sel = 2 \(\hat\sigma^{\phi}_{f_i}\) is estimated
\[\phi_{f_i sy} \sim
N(0,\hat\sigma^{\phi}_{f_i}) \]
Separate selectivity and catchability functions can be estimated for
each survey or fishery. Additionally, catchability can be set the same
for multiple surveys by setting Q_index to the same value.
Catchability is controlled by the following parameters in the
fleet_control sheet of the data
Q_index index to use if catchability coefficients are
to be set the sameCatchability Text or integer switch specifying
catchability. 0 or “Fixed” = fixed at prior; 1 or “Estimated” = Estimate
single parameter; 2 or “Estimated-with-prior” = Estimate single
parameter with prior; 3 or “Analytical” = Estimate analytical q from
Ludwig and Walters 1994; 4 or “PowerEquation” = Estimate power equation;
- 5 or “Environmental” = Linear equation log(q_y) = q_mu + beta *
index_y); 6 or “AR1” = annual AR1 catchability deviates are fit to
environmental index sensu Rogers et al 2025.Q_prior Starting value or fixed value for
catchabilityQ_sd_prior Variance of q prior: dnorm (log_q,
log_q_prior, q_sd_prior)Time_varying_q Whether a time-varying q should be
estimated. 0 = no, 1 = penalized deviate, 2 = random effect, 3 = time
blocks with no penalty; 4 = random walk from mean following Dorn 2018
(dnorm(q_y - q_y-1, 0, sigma). If Catchability = 5, this determines the
environmental index to be used in the equation log(q_y) = q_mu + beta *
index_yTime_varying_q_sd_prior The sd to use for the random
walk of time varying q if set to 1Time_varying_q = 0 for all formulations
Catchability = 0
\(q_{f_i} =\) Q_prior
from data
Catchability = 1 A freely estimated catchability
parameter \(\hat{q}_{f_i}\) is
estimated for survey/index fleet \(f_i\)
\[q_{f_i} = \hat{q}_{f_i}\]
Catchability = 2 A freely estimated catchability
parameter \(\hat{q}_{f_i}\) is
estimated for survey/index fleet \(f_i\) assuming a lognormal prior
\[ \hat{q}_{f_i} \sim lognormal(log(Q\ prior), Q\ sd\ prior) \]
Catchability = 3 Catchability \(\hat{q}_{f_i}\) is analytically derived for
survey/index fleet \(f_i\) following
Walters and Ludwig (1994). * For CPUE in weight with time-invariant
standard error of the survey: \[\hat{q}_{f_i}
= \exp\left( \sum_y log \left[ \frac{CPUE:B_{{f_i},y}} {\sum_s\sum_a
\left( N_{isa,y} e^{-Month_{f_i} Z_{isa,y}} W_{{f_i}sa,y} S_{{f_i}isa,y}
\right)} \right]/n \right)\] * For CPUE in numbers with
time-invariant standard error of the survey \[\hat{q}_{f_i} = \exp\left( \sum_y log \left[
\frac{CPUE:I_{{f_i},y}} {\sum_s\sum_a \left( N_{isa,y} e^{-Month_{f_i}
Z_{isa,y}} S_{{f_i}isa,y} \right)} \right]/n \right)\] * For CPUE
in weight with time-varying standard error of the survey \[\hat{q}_{f_i} = \exp\left( \sum_y \left( log
\left[ \frac{CPUE:B_{{f_i},y}} {\sum_s\sum_a N_{isa,y} e^{-Month_{f_i}
Z_{isa,y}} W_{{f_i}sa,y} S_{{f_i}isa,y}} \right]/\sigma^{2}_{f_iy}
\right)/ \sum_y{\frac{1}{\sigma^{2}_{f_iy}}} \right)\] * For CPUE
in numbers with time-varying standard error of the survey \[\hat{q}_{f_i} = \exp\left( \sum_y \left( log
\left[ \frac{CPUE:I_{{f_i},y}} {\sum_s\sum_a N_{isa,y} e^{-Month_{f_i}
Z_{isa,y}} S_{{f_i}isa,y}} \right]/\sigma^{2}_{f_iy} \right)/
\sum_y{\frac{1}{\sigma^{2}_{f_iy}}} \right)\]
Annual catchability estimated via penalized likelihood or as random effects
NOTE: If Catchability = 2 a prior will be put on mean
catchability as above. Catchability %in% c(1:2) and
Time_varying_q %in% c(1:2) \[\hat{q}_{f_iy} = e^{\bar{q}_{f_i}+
\omega_{f_iy}}\] \(\omega_{f_iy} \sim
N(0,\)Time_varying_q_sd_prior\()\) where
Time_varying_q_sd_prior is either fixed
Time_varying_q = 1 or estimated
Time_varying_q = 2 & random_q = TRUE in
Catchability time-blocks
Catchability %in% c(1:2),
Time_varying_q = 3 where the blocks are based on the blocks
in index_data data sheet. \[\hat{q}_{f_iy} = e^{\omega_{f_i
block_y}}\]
Annual catchability via random walk
Catchability %in% c(1:2),
Time_varying_q = 4 \[\hat{q}_{f_iy} = e^{\bar{q}_{f_i}+
\omega_{f_iy}}\] \(\omega_{f_iy} \sim
N(\omega_{f_iy-1},\)Time_varying_q_sd_prior\()\)