# Kootenay Lake Piscivorous Trout Inlake Abundance Estimation 2015

The main intepretive report by Redfish Consulting, which was prepared for the Fish and Wildlife Compensation Program, is available upon request.

The suggested citation for this online appendix is:

Hogan, P.M. and Thorley, J.L. (2015) Kootenay Lake Piscivorous Trout Inlake Abundance Estimation 2015. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/533988777.

## Background

With the recent decline in the number of Kokanee, there is an imperative to estimate the abundance of piscivorous Gerrard Rainbow Trout and Bull Trout in Kootenay Lake.

In this report we estimate the abundance of both species of trout by age- and size-class for 2011, because in this year:

1. Gerrard Rainbow Trout and Bull Trout numbers were at a peak;
2. Tag-based survival and exploitation probability estimates are available for this period (Thorley 2014);
3. A creel survey was conducted in this year;
4. Juvenile Gerrard Rainbow Trout snorkel surveys were conducted.

## Methods

We derive the number of adults ($$N_A$$) from the following abundance parameters, where adults are considered to be individuals with a fork length $$\geq$$ 500 mm:

Parameter Description
$$N_S$$ Number of spawners
$$\sigma$$ Probability of spawning
$$N_C$$ Number of adults caught
$$\gamma$$ Annual exploitation probability
$$\omega_A$$ Annual adult survival probability
$$N_1$$ Number of age-1 juveniles
$$\lambda$$ Maximum adult age

The number of adults are estimated assuming the following relationships:

$N_{A} = \frac{N_S}{\sigma} \qquad\text{or}\qquad N_{A} = \frac{N_C}{\gamma}$

The number of adults are partitioned into age-classes ($$N_i$$) based on the adult survival and the assumption that all adults are between 4 and $$\lambda$$ years of age, where for Gerrard Rainbow Trout $$\lambda^{RT} = 9$$ and for Bull Trout $$\lambda^{BT} = 14$$:

$N_i = N_{4} \omega_A^{i-4} \qquad\text{where}\qquad N_{4} = \frac{N_A}{1 + \sum_{i = 5}^{\lambda} \omega_A^{i-4}} = N_A \cdot \frac{(\omega_A - 1) \, \omega_A^3}{\omega_A^\lambda - \omega_A^3}$

In addition when the number of age-1 juveniles are known, the abundance of subadults are estimated under the assumption that the annual survival probability from age-1 juveniles to age-4 adults ($$\omega_J$$) is constant:

$N_i = N_{1} \omega_J^{i-1} \qquad\text{where}\qquad \omega_J = \sqrt[3]{\frac{N_4}{N_1}}$

Finally, the abundance by age-class estimates are converted into abundance by length-class estimates assuming a von Bertalanffy growth curve. The growth parameters for Gerrard Rainbow Trout will be taken from Thorley (Thorley 2015); the growth parameters for Bull Trout will be assumed to be the same.

### Data Preparation

The parameters used for the analysis must be provided as two files located in the folder .\input\. The abdunace parameters used in this analysis, which must be in the file .\input\parameters.csv, are:

species parameter estimate lower upper sd
RB N_S NA NA NA NA
RB sigma NA NA NA NA
RB N_C 8637.00 7.0e+03 1.00e+04 1000.00
RB gamma 0.13 8.0e-02 1.90e-01 0.03
RB omega_A 0.48 3.6e-01 6.30e-01 0.06
RB N_1 126000.00 9.7e+04 1.63e+05 16900.00
RB lambda 9.00 NA NA NA
BT N_S NA NA NA NA
BT sigma NA NA NA NA
BT N_C 4845.00 4.0e+03 6.00e+03 500.00
BT gamma 0.17 1.1e-01 2.40e-01 0.03
BT omega_A 0.48 3.5e-01 6.30e-01 0.07
BT N_1 NA NA NA NA
BT lambda 14.00 NA NA NA

The von Bertalanffy growth parameters used in this analysis, which must be in the file .\input\vB.csv, are:

species parameter estimate lower upper sd
RB bK 0.17204 0.15224 0.19128 0.01034
RB bLInf 924.18000 904.69000 951.64000 11.83000
RB sLength 20.59600 17.63400 23.95700 1.54000
RB t0 -0.85200 -1.64400 -0.28300 0.34900
BT bK 0.17204 0.15224 0.19128 0.01034
BT bLInf 924.18000 904.69000 951.64000 11.83000
BT sLength 20.59600 17.63400 23.95700 1.54000
BT t0 -0.85200 -1.64400 -0.28300 0.34900

These files may be modified at will, with parameters added or removed as appropriate; the code will check that the structure is correct for each species and for both sets of spawner and catch parameters, and will run the analysis and produce plots for each valid combination. Additional fish species may also be added to the end of the input files, provided the structure is identical to the above.

### Statistical Analysis

Bayesian models were fitted to the data using R version 3.2.1 (Team 2013) and JAGS 3.4.0 (Plummer 2012) which interfaced with each other via jaggernaut 2.3.0 (Thorley 2013). 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, 41–44).

Unless indicated otherwise, the models use prior distributions in the form of a normal distribution defined by the parameters provided for the analysis, truncated by the lower and upper credible limits. The posterior distributions were estimated from a minimum of 1,000 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 Rhat (Kery and Schaub 2011, 40) was less than 1.1 for each of the parameters in the model (Kery and Schaub 2011, 61).

The posterior distributions of the parameters are summarised in terms of a point estimate (mean), lower and upper 95% credible limits (2.5th and 97.5th percentiles), the standard deviation (SD), percent relative error (half the 95% credible interval as a percent of the point estimate) and significance (Kery and Schaub 2011, 37, 42).

The results are displayed graphically by plotting the modeled relationships between particular variables and the response with 95% credible intervals (CRIs) with the remaining variables held constant.

### Model Code

The JAGS model code, which uses a series of naming conventions, is presented below.

#### Abundance

##### Abundance - Model1
model{

bN ~ dnorm(N.mean, N.sd^-2) T(N.lower, N.upper)
bP ~ dnorm(p.mean, p.sd^-2) T(p.lower, p.upper)

N_A <- bN/bP

bOmega_A ~ dnorm(omega_A.mean, omega_A.sd^-2) T(omega_A.lower, omega_A.upper)

N_4 <- N_A * ((bOmega_A - 1) * bOmega_A^3)/(bOmega_A^iLambda - bOmega_A^3)

for(i in 4:iLambda){
N[i] <- N_4 * bOmega_A^(i-4)
}

N_1 ~ dnorm(N_1.mean, N_1.sd^-2) T(N_1.lower, N_1.upper)

bOmega_J <- ifelse(N_1 == 1, 0, (N_4/N_1)^(1/3))

for(i in 1:3){
N[i] <- ifelse(N_1 == 1, 0, N_1 * bOmega_J^(i-1))
}
}

## Results

### Model Parameters

The posterior distributions for the fixed (Kery and Schaub 2011 p. 75) parameters in each model are summarised below.

#### Abundance - Rainbow Trout - Catch

Parameter Estimate Lower Upper SD Error Significance
N_A 67590 45590 101190 14610 41 7e-04
N[1] 126600 100040 155700 14650 22 7e-04
N[2] 82310 65920 100790 8980 21 7e-04
N[3] 53920 39600 72960 8680 31 7e-04
N[4] 35590 22610 55340 8440 46 7e-04
N[5] 16920 11380 25230 3670 41 7e-04
N[6] 8151 4888 13002 2094 50 7e-04
N[7] 3979 1953 7055 1329 64 7e-04
N[8] 1968 753 4000 843 83 7e-04
N[9] 985 292 2278 523 100 7e-04
Convergence Iterations
1 1000

#### Abundance - Bull Trout - Catch

Parameter Estimate Lower Upper SD Error Significance
N_A 29420.00 20460.00 42620.00 5740.00 38 7e-04
N[10] 219.80 45.20 580.50 144.10 120 7e-04
N[11] 113.90 16.50 349.10 88.40 150 7e-04
N[12] 59.70 6.10 207.00 53.80 170 7e-04
N[13] 31.60 2.20 126.70 32.50 200 7e-04
N[14] 16.96 0.82 75.23 19.63 220 7e-04
N[4] 15250.00 9590.00 23440.00 3520.00 45 7e-04
N[5] 7241.00 5021.00 10490.00 1412.00 38 7e-04
N[6] 3495.00 2138.00 5334.00 820.00 46 7e-04
N[7] 1713.00 849.00 2969.00 549.00 62 7e-04
N[8] 852.00 326.00 1706.00 362.00 81 7e-04
N[9] 429.80 121.90 982.40 231.20 100 7e-04
Convergence Iterations
1.01 1000

## Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

• Redfish Consulting
• Greg Andrusak

## References

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

Plummer, Martyn. 2012. “JAGS Version 3.3.0 User Manual.” http://sourceforge.net/projects/mcmc-jags/files/Manuals/3.x/.

Team, R Core. 2013. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org.

Thorley, J. L. 2013. “Jaggernaut: An R Package to Facilitate Bayesian Analyses Using JAGS (Just Another Gibbs Sampler).” Nelson, B.C.: Poisson Consulting Ltd. https://github.com/poissonconsulting/jaggernaut.

———. 2014. “Kootenay Lake Exploitation Study 2013.” Poisson Consulting Analysis Report. http://www.poissonconsulting.ca/f/1088509848.

———. 2015. “Kootenay Lake Gerrard Rainbow Trout Condition and Growth Analysis 2014.” Poisson Consulting Analysis Report. http://www.poissonconsulting.ca/f/723490014.