The suggested citation for this analytic appendix is:
Thorley, J.L. and Amies-Galonski, E. (2021) Lower Columbia River Rainbow Trout Spawning 2020. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/626789785.
Each spring in the Lower Columbia River (LCR) below Hugh L. Keenleyside Dam (HLK) and in the Lower Kootenay River (LKR) below Brilliant Dam, thousands of Rainbow Trout spawn.
Since 1992, BC Hydro has stabilized the spring discharge releases from HLK to protect Rainbow Trout redds from dewatering.
The primary goal of the current analysis to estimate the abundance of spawners, egg survival and the stock-recruitment relationship. We also explore the use of drone imagery for redd mapping.
The aerial spawner counts were conducted by Mountain Water Research. The age-1 recruitment estimates were provided by the Fish Population Indexing Program. The drone surveys were conducted by Harrier Aerial Surveys. The remaining data were collected by Mountain Water Research, Poisson Consulting and Nuupqu.
The study area was divided into seven sections: Norns Creek Fan (NCF), NCF to LKR, LKR, LKR to Genelle, Genelle. Redd and spawner counts upstream of Norns Creek Fan and downstream of Genelle were excluded from the section totals because they constitute less than 0.1% of the total count and were not surveyed in all years. The redd and spawner counts for the Right Upstream Bank above Robson Bridge were also excluded as they appear to be primarily driven by viewing conditions (and constitute less than 2.5% of the total). Viewing conditions were classified as Good or Poor. If information on the viewing conditions was not available a decline in the redd count of more than one third of the cumulative maximum count for a particular section was assumed to be caused by poor viewing conditions.
The mapped peak redd and fish counts are the peak counts for that site.
The data were prepared for analysis using R version 4.0.4 (R Core Team 2017).
Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2015) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2016).
Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman, Simpson, and Betancourt 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61).
The parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% confidence/credible limits (CLs) and the p-value (Kery and Schaub 2011, 37, 42). The estimate is the median (50th percentile) of the MCMC samples, the z-score is \(\mathrm{mean}/\mathrm{sd}\) and the 95% CLs are the 2.5th and 97.5th percentiles. A p-value of 0.05 indicates that the lower or upper 95% CL is 0.
The results are displayed graphically by plotting the modeled relationships between particular variables and the response(s) with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% confidence/credible intervals (CIs, Bradford, Korman, and Higgins 2005).
The analyses were implemented using R version 4.0.4
(R Core Team 2019) and the
mbr family of packages.
The spawner abundance and spawn timings were estimated from the aerial fish and redd counts for the five sections (in three segments) using an Area-Under-The-Curve (AUC) model.
Key assumptions of the AUC model include:
The relationship between the number of spawners/eggs and the resultant number of age-1 fish was estimated using a Beverton-Holt stock-recruitment model (Walters and Martell 2004):
\[ R = \frac{\alpha \cdot S}{1 + \beta \cdot S}\]
where \(S\) is the stock (spawners or eggs), \(R\) is the recruits, \(\alpha\) is the recruits per stock at low density and \(\beta\) determines the density-dependence.
The carrying capacity is \(\alpha / \beta\).
Key assumptions of the stock-recruitment model include:
Key assumptions of the spawner stock-recruitment model include:
The mean of 90 for \(\alpha\) was based on an average of 2,900 eggs per female spawner, a 50:50 sex ratio, 50% egg survival, 50% post-emergence fall survival, 50% overwintering survival and 50% summer survival.
Key assumptions of the egg stock-recruitment model include:
The mean of 0.625 is based on a 50% egg survival, 50% post-emergence fall survival, 50% overwintering survival and 50% summer survival.
The spreadsheet Norn's Fan stage discharge curve.xls predicts the
stage at CNN (in m) from the discharge (in cms) from HLK and BRDs dam
based on the multiple regression equation
\[\text{CNN} = 417.133 + 0.001883826 \cdot \text{HLK} + 0.000559058 \cdot \text{BRD}\]
We predict the stage from the discharge using a multiple second-order polynomial regression with interactions. The main part model was of the following form:
\[\text{CNN} = \text{b0} + \text{bHLK} \cdot \text{HLK} + \text{bHLK2} \cdot \text{HLK}^2 + \text{bBRD} \cdot \text{BRD} + \text{bBRD2} \cdot \text{BRD}^2 + \text{bHLKBRD} \cdot \text{HLK} \cdot \text{BRD} + \text{bHLK2BRD} \cdot \text{HLK}^2 \cdot \text{BRD}\] To ensure model convergence \(\text{HLK}\) and \(\text{BRD}\) were the standardized discharges. The model also included a parameter to correct for an offset in \(\text{CNN}\) from May 1st 2017.
data {
int<lower=0> nObs;
int<lower=0> nSection;
int<lower=0> nSegment;
int<lower=0> nYear;
int Year[nObs];
int Section[nObs];
int Segment[nObs];
real Doy[nObs];
int Fish[nObs];
int Redds[nObs];
parameters {
vector<lower=3,upper=9>[nSection] bFishAbundanceSection;
real<lower=0> sFishAbundanceYear;
vector[nYear] bFishAbundanceYear;
real<lower=0> sFishAbundanceSectionYear;
vector[nSection * nYear] bFishAbundanceSectionYear;
real<lower=0.8,upper=1.0> bFishObserverEfficiency;
real<lower=0.0,upper=2.0> bReddObserverEfficiency;
real<lower=1, upper=2> bReddPerFish;
real<lower=14, upper=21> bFishResidenceTime;
real<lower=30, upper=40> bReddResidenceTime;
real<lower=100, upper=150> bPeakFishArrivalTiming;
real<lower=0,upper=21> sPeakFishArrivalTimingYear;
vector[nYear] bPeakFishArrivalTimingYear;
real<lower=log(10), upper=log(50)> bFishArrivalDuration;
real<lower=0> sFishArrivalDurationSegmentYear;
vector[nSegment * nYear] bFishArrivalDurationSegmentYear;
real<lower=0, upper=2> bDispersionRedds;
real<lower=0, upper=2> bDispersionFish;
model {
vector[nObs] eFishAbundance;
vector[nObs] ePeakFishArrivalTiming;
vector[nObs] eFishArrivalDuration;
vector[nObs] eRedds;
vector[nObs] eFish;
sFishAbundanceYear ~ normal(0, 1);
bFishAbundanceYear ~ normal(0, sFishAbundanceYear);
sFishAbundanceSectionYear ~ normal(0, 1);
bFishAbundanceSectionYear ~ normal(0, sFishAbundanceSectionYear);
bPeakFishArrivalTimingYear ~ normal(0, sPeakFishArrivalTimingYear);
bFishArrivalDurationSegmentYear ~ normal(0, 1);
bFishArrivalDurationSegmentYear ~ normal(0, sFishArrivalDurationSegmentYear);
for (i in 1:nObs) {
eFishAbundance[i] = exp(bFishAbundanceSection[Section[i]] + bFishAbundanceYear[Year[i]] + bFishAbundanceSectionYear[((Section[i] - 1) * Year[i]) + Year[i]]);
ePeakFishArrivalTiming[i] = bPeakFishArrivalTiming + bPeakFishArrivalTimingYear[Year[i]];
eFishArrivalDuration[i] = exp(bFishArrivalDuration + bFishArrivalDurationSegmentYear[((Segment[i] - 1) * Year[i]) + Year[i]]);
eRedds[i] = eFishAbundance[i] * bReddPerFish * bReddObserverEfficiency * (normal_cdf(Doy[i], ePeakFishArrivalTiming[i], eFishArrivalDuration[i]) - normal_cdf(Doy[i], ePeakFishArrivalTiming[i] + bReddResidenceTime, eFishArrivalDuration[i]));
eFish[i] = eFishAbundance[i] * bFishObserverEfficiency * (normal_cdf(Doy[i], ePeakFishArrivalTiming[i], eFishArrivalDuration[i]) - normal_cdf(Doy[i], ePeakFishArrivalTiming[i] + bFishResidenceTime, eFishArrivalDuration[i]));
}
Redds ~ neg_binomial_2(eRedds, 1/bDispersionRedds);
Fish ~ neg_binomial_2(eFish, 1/bDispersionFish);Block 1. Model description.
.model {
bAlpha ~ dnorm(90, 50^-2) T(1,)
bBeta ~ dnorm(0, 1/10^-2) T(0,)
bDewatered ~ dnorm(0, 100^-2)
sRecruits ~ dnorm(0, 2^-2) T(0, )
for(i in 1:length(Stock)) {
log(eRecruits[i]) <- log(bAlpha * Stock[i] / (1 + bBeta * Stock[i])) + bDewatered * Dewatered[i]
Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
}Block 2. Model description.
.model {
bAlpha ~ dnorm(0.0625, 0.03^-2) T(0,)
bBeta ~ dnorm(0, 1/100000^-2) T(0, )
bEggLoss ~ dnorm(0, 100^-2)
sRecruits ~ dnorm(0, 2^-2) T(0,)
for(i in 1:length(Recruits)){
log(eRecruits[i]) <- log(bAlpha * Eggs[i] / (1 + bBeta * Eggs[i])) + bEggLoss * EggLoss[i]
Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
}Block 3.
.model{
bIntercept ~ dnorm(420, 2^-2)
bOffset[1] <- 0
for(i in 2:nOffset) {
bOffset[i] ~ dnorm(0, 1^-2)
}
bHLK ~ dnorm(0, 1^-2)
bHLK2 ~ dnorm(0, 1^-2)
bLKR ~ dnorm(0, 1^-2)
bLKR2 ~ dnorm(0, 1^-2)
bHLKLKR ~ dnorm(0, 1^-2)
bHLK2LKR ~ dnorm(0, 1^-2)
bCor <- 0.4
eCorStage[1] <- eStage[1]
for(i in 2:length(Stage)){
eCorStage[i] <- eStage[i] + bCor^(Days[i] - Days[i-1]) * (Stage[i-1] - eStage[i-1])
}
sStage ~ dnorm(0, 1^-2) T(0, )
for(i in 1:length(Stage)){
eStage[i] <- bIntercept + bOffset[Offset[i]] + bHLK * HLK[i] + bLKR * LKR[i] + bHLK2 * HLK[i]^2 + bLKR2 * LKR[i]^2 + bHLKLKR * HLK[i] * LKR[i] + bHLK2LKR * HLK[i]^2 * LKR[i]
Stage[i] ~ dnorm(eCorStage[i], sStage^-2)
}Block 4. The model description.
The suggested citation for this analytic appendix is:
Thorley, J.L. and Amies-Galonski, E. (2020) Lower Columbia River Rainbow Trout Spawning 2019. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/949693135.
Each spring in the Lower Columbia River (LCR) below Hugh L. Keenleyside Dam (HLK) and in the Lower Kootenay River (LKR) below Brilliant Dam, thousands of Rainbow Trout spawn.
Since 1992, BC Hydro has stabilized the spring discharge releases from HLK to protect Rainbow Trout redds from dewatering.
The primary goal of the current analysis to estimate the abundance of spawners, egg survival and the stock-recruitment relationship. We also examine the survival of dewatered eggs and the temperatures in the gravels.
The aerial spawner counts were conducted by Mountain Water Research. The age-1 recruitment estimates were provided by the Fish Population Indexing Program. The remaining data were collected by Mountain Water Research, Poisson Consulting and Nuupqu.
The study area was divided into seven sections: Norns Creek Fan (NCF), NCF to LKR, LKR, LKR to Genelle, Genelle. Redd and spawner counts upstream of Norns Creek Fan and downstream of Genelle were excluded from the section totals because they constitute less than 0.1% of the total count and were not surveyed in all years. The redd and spawner counts for the Right Upstream Bank above Robson Bridge were also excluded as they appear to be primarily driven by viewing conditions (and constitute less than 2.5% of the total). Viewing conditions were classified as Good or Poor. If information on the viewing conditions was not available a decline in the redd count of more than one third of the cumulative maximum count for a particular section was assumed to be caused by poor viewing conditions.
The mapped peak redd and fish counts are the peak counts for that site.
The data were prepared for analysis using R version 3.6.2 (R Core Team 2017).
Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2015) and STAN (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2016).
Unless stated otherwise, the Bayesian analyses used weakly informative normal and half-normal prior distributions (Gelman, Simpson, and Betancourt 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61).
The parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% confidence/credible limits (CLs) and the p-value (Kery and Schaub 2011, 37, 42). The estimate is the median (50th percentile) of the MCMC samples, the z-score is \(\mathrm{mean}/\mathrm{sd}\) and the 95% CLs are the 2.5th and 97.5th percentiles. A p-value of 0.05 indicates that the lower or upper 95% CL is 0.
The results are displayed graphically by plotting the modeled relationships between particular variables and the response(s) with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% confidence/credible intervals (CIs, Bradford, Korman, and Higgins 2005).
The analyses were implemented using R version 3.6.2
(R Core Team 2019) and the
mbr family of packages.
The spawner abundance and spawn timings were estimated from the aerial fish and redd counts for the five sections (in three segments) using an Area-Under-The-Curve (AUC) model.
Key assumptions of the AUC model include:
The relationship between the number of spawners and the resultant number of age-1 fish was estimated using a Beverton-Holt stock-recruitment model (Walters and Martell 2004):
\[ R = \frac{\alpha \cdot S}{1 + \beta \cdot S} \quad,\]
where \(S\) is the spawners (stock), \(R\) is the recruits, \(\alpha\) is the recruits per spawner at low density and \(\beta\) determines the density-dependence.
Key assumptions of the stock-recruitment model include:
The mean of 90 for \(\alpha\) was based on an average of 2,900 eggs per female spawner, a 50:50 sex ratio, 50% egg survival, 50% post-emergence fall survival, 50% overwintering survival and 50% summer survival.
The carrying capacity is \(\alpha / \beta\).
data {
int<lower=0> nObs;
int<lower=0> nSection;
int<lower=0> nSegment;
int<lower=0> nYear;
int Year[nObs];
int Section[nObs];
int Segment[nObs];
real Doy[nObs];
int Fish[nObs];
int Redds[nObs];
parameters {
vector<lower=3,upper=9>[nSection] bFishAbundanceSection;
real<lower=0> sFishAbundanceYear;
vector[nYear] bFishAbundanceYear;
real<lower=0> sFishAbundanceSectionYear;
vector[nSection * nYear] bFishAbundanceSectionYear;
real<lower=0.8,upper=1.0> bFishObserverEfficiency;
real<lower=0.0,upper=2.0> bReddObserverEfficiency;
real<lower=1, upper=2> bReddPerFish;
real<lower=14, upper=21> bFishResidenceTime;
real<lower=30, upper=40> bReddResidenceTime;
real<lower=100, upper=150> bPeakFishArrivalTiming;
real<lower=0,upper=21> sPeakFishArrivalTimingYear;
vector[nYear] bPeakFishArrivalTimingYear;
real<lower=log(10), upper=log(50)> bFishArrivalDuration;
real<lower=0> sFishArrivalDurationSegmentYear;
vector[nSegment * nYear] bFishArrivalDurationSegmentYear;
real<lower=0, upper=2> bDispersionRedds;
real<lower=0, upper=2> bDispersionFish;
model {
vector[nObs] eFishAbundance;
vector[nObs] ePeakFishArrivalTiming;
vector[nObs] eFishArrivalDuration;
vector[nObs] eRedds;
vector[nObs] eFish;
sFishAbundanceYear ~ normal(0, 1);
bFishAbundanceYear ~ normal(0, sFishAbundanceYear);
sFishAbundanceSectionYear ~ normal(0, 1);
bFishAbundanceSectionYear ~ normal(0, sFishAbundanceSectionYear);
bPeakFishArrivalTimingYear ~ normal(0, sPeakFishArrivalTimingYear);
bFishArrivalDurationSegmentYear ~ normal(0, 1);
bFishArrivalDurationSegmentYear ~ normal(0, sFishArrivalDurationSegmentYear);
for (i in 1:nObs) {
eFishAbundance[i] = exp(bFishAbundanceSection[Section[i]] + bFishAbundanceYear[Year[i]] + bFishAbundanceSectionYear[((Section[i] - 1) * Year[i]) + Year[i]]);
ePeakFishArrivalTiming[i] = bPeakFishArrivalTiming + bPeakFishArrivalTimingYear[Year[i]];
eFishArrivalDuration[i] = exp(bFishArrivalDuration + bFishArrivalDurationSegmentYear[((Segment[i] - 1) * Year[i]) + Year[i]]);
eRedds[i] = eFishAbundance[i] * bReddPerFish * bReddObserverEfficiency * (normal_cdf(Doy[i], ePeakFishArrivalTiming[i], eFishArrivalDuration[i]) - normal_cdf(Doy[i], ePeakFishArrivalTiming[i] + bReddResidenceTime, eFishArrivalDuration[i]));
eFish[i] = eFishAbundance[i] * bFishObserverEfficiency * (normal_cdf(Doy[i], ePeakFishArrivalTiming[i], eFishArrivalDuration[i]) - normal_cdf(Doy[i], ePeakFishArrivalTiming[i] + bFishResidenceTime, eFishArrivalDuration[i]));
}
Redds ~ neg_binomial_2(eRedds, 1/bDispersionRedds);
Fish ~ neg_binomial_2(eFish, 1/bDispersionFish);
..Block 1. Model description.
.model {
bAlpha ~ dnorm(90, 50^-2) T(1,)
bAlphaDewatered ~ dnorm(0, 2^-2)
bBeta ~ dnorm(-5, 5^-2)
sRecruits ~ dunif(0, 2)
for(i in 1:length(Stock)) {
log(eAlpha[i]) <- log(bAlpha) + bAlphaDewatered * Dewatered[i]
log(eBeta[i]) <- bBeta
eRecruits[i] <- eAlpha[i] * Stock[i] / (1 + eBeta[i] * Stock[i])
Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
}
..Block 2. Model description.
The suggested citation for this analytic report is:
Thorley, J.L. and Dalgarno, S. (2018) Lower Columbia River Rainbow Trout Spawning Analysis 2018. A Poisson Consulting Analysis Report. URL: https://www.poissonconsulting.ca/f/934446658.
Each spring in the Lower Columbia River (LCR) below the Hugh L. Keenleyside Dam (HLK) and in the Lower Kootenay River (LKR) below the Brilliant Dam, thousands of Rainbow Trout spawn. Since 1992, BC Hydro has stabilized the spring discharge releases from HLK to protect Rainbow Trout redds from dewatering.
The primary goal of the current analysis to estimate the abundance of spawners and the stock-recruitment relationship.
The aerial spawner counts were conducted by Mountain Water Research. The age-1 recruitment estimates were provided by the Fish Population Indexing Program.
The study area was divided into three sections: the LCR above the LKR, the LKR and the LCR below the LKR. Redd and spawner counts upstream of Norns Creek Fan and downstream of Genelle were excluded from the section totals because they constitute less than 0.1% of the total count and were not surveyed in all years. The redd and spawner counts for the Right Upstream Bank above Robson Bridge were also excluded as they appear to be primarily driven by viewing conditions (and constitute less than 2.5% of the total). Viewing conditions were classified as Good or Poor. If information on the viewing conditions was not available a decline in the redd count of more than one third of the cumulative maximum count for a particular section was assumed to be caused by poor viewing conditions.
The data were prepared for analysis using R version 3.5.1 (R Core Team 2017).
Model parameters were estimated using Bayesian methods. The Bayesian estimates were produced using JAGS (Plummer 2015) and Stan (Carpenter et al. 2017). For additional information on Bayesian modelling in the BUGS language, of which JAGS uses a dialect, the reader is referred to Kery and Schaub (2011). For additional information on Bayesian modelling in the Stan language the reader is referred to Stan Development Team (2017).
Unless indicated otherwise, the Bayesian analyses used uninformative uniform or normal prior distributions (Kery and Schaub 2011, 36). The posterior distributions were estimated from 1,500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of three chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that \(\hat{R} < 1.1\) (Kery and Schaub 2011, 40) for each of the monitored parameters (Kery and Schaub 2011, 61).
The parameters are summarised in terms of the point estimate, standard deviation (sd), the z-score, lower and upper 95% credible limits (CLs) and the p-value (Kery and Schaub 2011, 37, 42). A p-value of 0.05 indicates that the lower or upper 95% CL is 0. The estimate is the median (50th percentile) of the MCMC samples, the z-score is \(\mathrm{mean}/\mathrm{sd}\) and the 95% CLs are the 2.5th and 97.5th percentiles.
The results are displayed graphically by plotting the modeled relationships between particular variables and the response(s) with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their mean and first level values, respectively, while random variables are held constant at their typical values (expected values of the underlying hyperdistributions) (Kery and Schaub 2011, 77–82). When informative the influence of particular variables is expressed in terms of the effect size (i.e., percent change in the response variable) with 95% confidence/credible intervals (CIs, Bradford, Korman, and Higgins 2005).
The analyses were implemented using R version 3.5.1
(R Core Team 2017) and the jmbr and smbr packages.
The spawner abundance and spawn timings were estimated from the aerial fish and redd counts for the three sections using an Area-Under-The-Curve (AUC) model.
Key assumptions of the AUC model include:
The relationship between the number of spawners and the resultant number of age-1 fish was estimated using a Beverton-Holt stock-recruitment model (Walters and Martell 2004):
\[ R = \frac{\alpha \cdot S}{1 + \beta \cdot S} \quad,\]
where \(S\) is the spawners (stock), \(R\) is the recruits, \(\alpha\) is the recruits per spawner at low density and \(\beta\) determines the density-dependence.
Key assumptions of the stock-recruitment model include:
The mean of 90 for \(\alpha\) was based on an average of 2,900 eggs per female spawner, a 50:50 sex ratio, 50% egg survival, 50% post-emergence fall survival, 50% overwintering survival and 50% summer survival.
The carrying capacity is \(\alpha / \beta\).
data {
int<lower=0> nObs;
int<lower=0> nSection;
int<lower=0> nYear;
int Year[nObs];
int Section[nObs];
real Doy[nObs];
int Spawners[nObs];
int Redds[nObs];
parameters {
vector<lower=5,upper=10>[nSection] bSpawnerAbundanceSection;
real<lower=0,upper=2> sSpawnerAbundanceYear;
vector[nYear] bSpawnerAbundanceYear;
real<lower=0,upper=2> sSpawnerAbundanceSectionYear;
vector[nSection * nYear] bSpawnerAbundanceSectionYear;
real<lower=0.8,upper=1.0> bSpawnerObserverEfficiency;
real<lower=0.0,upper=2.0> bReddObserverEfficiency;
real<lower=1, upper=2> bReddPerSpawner;
real<lower=14, upper=21> bSpawnerResidenceTime;
real<lower=30, upper=40> bReddResidenceTime;
real<lower=100, upper=150> bPeakSpawnerArrivalTiming;
real<lower=0,upper=21> sPeakSpawnerArrivalTimingYear;
vector[nYear] bPeakSpawnerArrivalTimingYear;
real<lower=log(10), upper=log(50)> bSpawnerArrivalDuration;
real<lower=0,upper=2> sSpawnerArrivalDurationSectionYear;
vector[nSection * nYear] bSpawnerArrivalDurationSectionYear;
real<lower=0, upper=2> bDispersionRedds;
real<lower=0, upper=2> bDispersionSpawners;
model {
vector[nObs] eSpawnerAbundance;
vector[nObs] ePeakSpawnerArrivalTiming;
vector[nObs] eSpawnerArrivalDuration;
vector[nObs] eRedds;
vector[nObs] eSpawners;
bSpawnerAbundanceYear ~ normal(0, sSpawnerAbundanceYear);
bSpawnerAbundanceSectionYear ~ normal(0, sSpawnerAbundanceSectionYear);
bPeakSpawnerArrivalTimingYear ~ normal(0, sPeakSpawnerArrivalTimingYear);
bSpawnerArrivalDurationSectionYear ~ normal(0, sSpawnerArrivalDurationSectionYear);
for (i in 1:nObs) {
eSpawnerAbundance[i] = exp(bSpawnerAbundanceSection[Section[i]] + bSpawnerAbundanceYear[Year[i]] + bSpawnerAbundanceSectionYear[((Section[i] - 1) * Year[i]) + Year[i]]);
ePeakSpawnerArrivalTiming[i] = bPeakSpawnerArrivalTiming + bPeakSpawnerArrivalTimingYear[Year[i]];
eSpawnerArrivalDuration[i] = exp(bSpawnerArrivalDuration + bSpawnerArrivalDurationSectionYear[((Section[i] - 1) * Year[i]) + Year[i]]);
eRedds[i] = eSpawnerAbundance[i] * bReddPerSpawner * bReddObserverEfficiency * (normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i], eSpawnerArrivalDuration[i]) - normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i] + bReddResidenceTime, eSpawnerArrivalDuration[i]));
eSpawners[i] = eSpawnerAbundance[i] * bSpawnerObserverEfficiency * (normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i], eSpawnerArrivalDuration[i]) - normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i] + bSpawnerResidenceTime, eSpawnerArrivalDuration[i]));
}
Redds ~ neg_binomial_2(eRedds, 1/bDispersionRedds);
Spawners ~ neg_binomial_2(eSpawners, 1/bDispersionSpawners);
..Template 1. AUC model.
model {
bAlpha ~ dnorm(90, 50^-2) T(1,)
bAlphaDewatered ~ dnorm(0, 2^-2)
bBeta ~ dnorm(-5, 5^-2)
sRecruits ~ dunif(0, 2)
for(i in 1:length(Stock)) {
log(eAlpha[i]) <- log(bAlpha) + bAlphaDewatered * Dewatered[i]
log(eBeta[i]) <- bBeta
eRecruits[i] <- eAlpha[i] * Stock[i] / (1 + eBeta[i] * Stock[i])
Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
}
..Template 2.