# Curtis and Qua Creeks Productivity Analysis 2017

The suggested citation for this analytic report is:

*Thorley, J.L. and Dalgarno, S. (2018) Curtis and Qua Creeks Productivity Analysis 2017. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/611560309.*

## Background

Water chemistry, invertebrate and fish data were collected from Curtis and Qua Creeks from 2015 to 2017 and periphyton data were collected in 2015 and 2016 to assess their productivity. The fish data were collected by depletion electrofishing.

## Methods

### Data Collection

The data were collected by the Salmo Streamkeepers and Poisson Consulting in 2015 and by the Salmo Streamkeepers in 2016 and 2017. The invertebrate data were processed by Westcott Environmental Services and the water chemistry and periphyton data by CARO Analytic. The data were provided to Poisson in the form of an Excel database.

### Data Preparation

The data were manipulated using R version 3.4.3 (R Core Team 2015) and imported into an SQLite database.

### Statistical Analysis

Model parameters were estimated using Bayesian methods. The Bayesian estimates were produced using JAGS (Plummer 2003). For additional information on Bayesian estimation the reader is referred to McElreath (2016).

Unless indicated otherwise, the Bayesian analyses used uninformative normal prior distributions (Kery and Schaub 2011, 36). 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 \(\hat{R} \leq 1.1\) (Kery and Schaub 2011, 40) and \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61). Where \(\hat{R}\) is the potential scale reduction factor and \(\textrm{ESS}\) is the effective sample size.

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 \(\mathrm{sd}/\mathrm{mean}\) 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.

Where informative, model adequacy was confirmed by examination of residual plots for the full model(s).

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 3.4.3 (R Core Team 2015) and the `mbr`

family of packages.

### Model Descriptions

#### Periphyton Growth

Periphyton growth can be modelled using sigmoidal curves (Rodriguez 1987). Here we use the flexible unified-Richards model (Tjørve and Tjørve 2010)

\[P_t = A(1 + (d - 1)\exp(-K(t - T_i)/d^{d/(1-d)}))^{1/(1-d)}\] to model the periphyton density at time \(t\) (\(P_t\)) where \(A\) is the maximum density, \(K\) is the maximum relative growth rate, \(T_i\) is the time at maximum growth, and \(d\) is the relative value at maximum growth (Tjørve and Tjørve 2010).

Key assumptions of the growth model include:

- \(A\) can vary randomly by series within location within creek within year
- \(K\) can vary randomly by series within location within creek within year
- \(T_i\) is 0
- \(d\) is 0
- The residual variation in \(P_t\) is log-normally distributed.

#### Rainbow Trout Density

The density of Rainbow Trout each site was estimated using a removal model (Wyatt 2002).

Key assumptions of the removal model include:

- Lineal density varies by creek and randomly by site.
- Capture efficiency varies randomly by pass within site within year.
- The number of fish caught on each pass is binomially distributed.

Preliminary analyses indicated that year was not a significant predictor of density.

### Model Templates

### Periphyton Growth

```
model{
bA ~ dnorm(0, 5^-2)
bK ~ dnorm(-5, 5^-2)
sAYearCreekLocationSeries ~ dunif(0, 1)
sKYearCreekLocationSeries ~ dunif(0, 1)
for(i in 1:nYear) {
for(j in 1:nCreek) {
for(k in 1:nLocation) {
for(l in 1:nSeries) {
bAYearCreekLocationSeries[i, j, k, l] ~ dnorm(0, sAYearCreekLocationSeries^-2)
bKYearCreekLocationSeries[i, j, k, l] ~ dnorm(0, sKYearCreekLocationSeries^-2)
}
}
}
}
sPeriphyton ~ dunif(0, 1)
for (i in 1:length(Periphyton)) {
log(eA[i]) <- bA + bAYearCreekLocationSeries[Year[i],Creek[i],Location[i],Series[i]]
log(eK[i]) <- bK + bKYearCreekLocationSeries[Year[i],Creek[i],Location[i],Series[i]]
ePeriphyton[i] <- eA[i] * (1 - exp(-eK[i] * Days[i]))
Periphyton[i] ~ dlnorm(log(ePeriphyton[i]), sPeriphyton^-2)
}
..
```

Template 1. The model description.

### Rainbow Trout Density

```
model{
bDensity ~ dnorm(0, 5^-2)
bDensityCreek[1] <- 0
for (i in 2:nCreek) {
bDensityCreek[i] ~ dnorm(0, 5^-2)
}
sDensitySite ~ dunif(0, 5)
for (i in 1:nSite) {
bDensitySite[i] ~ dnorm(0, sDensitySite^-2)
}
bEfficiency ~ dnorm(0, 2^-2)
sEfficiencyYearSitePass ~ dunif(0, 2)
for(i in 1:nYear) {
for(j in 1:nSite) {
for(k in 1:nPass) {
bEfficiencyYearSitePass[i,j,k] ~ dnorm(0, sEfficiencyYearSitePass^-2)
}
}
}
for (i in 1:length(SiteLength)) {
log(eDensity[i]) <- bDensity + bDensityCreek[Creek[i]] + bDensitySite[Site[i]]
eAbundance[i] ~ dpois(eDensity[i] * SiteLength[i])
eRemaining[i,1] <- eAbundance[i]
for(j in 1:nPass) {
logit(eEfficiency[i,j]) <- bEfficiency + bEfficiencyYearSitePass[Year[i], Site[i], j]
Catch[i,j] ~ dbin(eEfficiency[i,j], eRemaining[i,j])
eRemaining[i,j+1] <- eRemaining[i,j] - Catch[i,j]
}
}
..
```

Template 2. The model description.

## Results

### Tables

### Periphyton Growth

Table 1. Parameter descriptions.

Parameter | Description |
---|---|

`bA` |
Intercept for `eA` |

`bAYearCreekLocationSeries[i,j,k,l]` |
Effect of `i` ^{th} year at `j` ^{th} creek at `k` ^{th} location at `l` ^{th} series on `bA` |

`bK` |
Intercept for `eK` |

`bKYearCreekLocationSeries[i,j,k,l]` |
Effect of `i` ^{th} year at `j` ^{th} creek at `k` ^{th} location at `l` ^{th} series on `bK` |

`Days[i]` |
Time since deployment in creek at `i` ^{th} observation (days) |

`eA[i]` |
Expected maximum `ePeriphyton` for `i` ^{th} observation |

`eK[i]` |
Expected maximum relative growth of `ePeriphyton` for `i` ^{th} observation |

`ePeriphyton[i]` |
Expected `Periphyton` for `i` ^{th} observation |

`Periphyton[i]` |
Measured Chlorophyll-A density at `i` ^{th} observation (mg/m2) |

`sAYearCreekLocationSeries` |
SD of `bAYearCreekLocationSeries` |

`sKYearCreekLocationSeries` |
SD of `bKYearCreekLocationSeries` |

`sPeriphyton` |
SD of residual variation in `Periphyton` |

Table 2. Model coefficients.

term | estimate | sd | zscore | lower | upper | pvalue |
---|---|---|---|---|---|---|

bA | 1.7528139 | 0.0899591 | 19.489965 | 1.5829985 | 1.9389502 | 7e-04 |

bK | -2.8275844 | 0.2016678 | -13.981585 | -3.2136251 | -2.4016112 | 7e-04 |

sAYearCreekLocationSeries | 0.2212607 | 0.0907603 | 2.421603 | 0.0352755 | 0.4009219 | 7e-04 |

sKYearCreekLocationSeries | 0.4597242 | 0.1800653 | 2.592648 | 0.0908912 | 0.8500394 | 7e-04 |

sPeriphyton | 0.3643397 | 0.0278003 | 13.140708 | 0.3129817 | 0.4217670 | 7e-04 |

Table 3. Model summary.

n | K | nchains | niters | nthin | ess | rhat | converged |
---|---|---|---|---|---|---|---|

137 | 5 | 3 | 500 | 100 | 1056 | 1.005 | TRUE |

### Rainbow Trout Density

Table 4. Parameter descriptions.

Parameter | Description |
---|---|

`bDensity` |
Intercept for `log(eDensity)` |

`bDensityCreek[i]` |
Effect of `i` ^{th} `Creek` on `bDensity` |

`bDensitySite[i]` |
Effect of `i` ^{th} `Site` on `bDensity` |

`bEfficiency` |
Intercept for `logit(eEfficiency)` |

`bEfficiencyYearSitePass[i,j,k]` |
Effect of `i` ^{th} `Year` at `j` ^{th} `Site` at `k` ^{th} `Pass` on `bEfficiency` |

`Catch[i, j]` |
Catch on `j` ^{th} pass at `i` ^{th} site visit |

`Creek` |
Creek on `i` ^{th} site visit |

`eAbundance[i]` |
Expected total number of fish at `i` ^{th} site visit |

`eDensity[i]` |
Expected lineal density at `i` ^{th} site visit (fish/m) |

`eEfficiency[i]` |
Expected efficiency during `i` ^{th} site visit |

`eRemaining[i, j]` |
Expected number of fish remaining after `j` ^{th} pass at `i` ^{th} site visit |

`sDensitySite` |
SD of `bDensitySite` |

`sEfficiencyYearSitePass` |
SD of `bEfficiencyYearSitePass` |

`Site` |
Site on `i` ^{th} site visit |

`SiteLength` |
Length of creek surveyed during `i` ^{th} site visit (m) |

`Year` |
Year on `i` ^{th} site visit |

Table 5. Model coefficients.

term | estimate | sd | zscore | lower | upper | pvalue |
---|---|---|---|---|---|---|

bDensity | -0.1524739 | 0.1891862 | -0.7885377 | -0.5147539 | 0.2396398 | 0.3587 |

bDensityCreek[2] | 0.4053819 | 0.2260175 | 1.7787547 | -0.0499179 | 0.8754357 | 0.0720 |

bEfficiency | -0.1455399 | 0.3143788 | -0.5314010 | -0.8252120 | 0.4081983 | 0.6187 |

sDensitySite | 0.1622042 | 0.1655570 | 1.1666716 | 0.0139174 | 0.5560167 | 0.0007 |

sEfficiencyYearSitePass | 0.6789001 | 0.1946906 | 3.5447247 | 0.3461363 | 1.1020117 | 0.0007 |

Table 6. Model summary.

n | K | nchains | niters | nthin | ess | rhat | converged |
---|---|---|---|---|---|---|---|

58 | 5 | 3 | 500 | 10 | 318 | 1.015 | TRUE |

### Bull Trout

Table 7. Bull Trout captured during electrofishing surveys.

Waterbody | Dayte | Year | EFSite | Location | Species | ForkLength | BodyWeight |
---|---|---|---|---|---|---|---|

Curtis Creek | 1972-08-03 | 2016 | CUR1 | 1 | BT | 91 | 10 |

Curtis Creek | 1972-08-03 | 2016 | CUR1 | 1 | BT | 95 | 10 |

Curtis Creek | 1972-08-03 | 2016 | CUR2 | 2 | BT | 92 | 10 |

Curtis Creek | 1972-08-14 | 2017 | CUR2 | 2 | BT | 115 | 18 |

### Figures

### Water Temperature

### Periphyton Growth

### Benthic Invertebrates

### Rainbow Trout Length-Weight

### Rainbow Trout Density

### Chemical

## Recommendations

- Early season periphyton growth should be monitored until late October

## Acknowledgements

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

- Fish and Wildlife Compensation Program
- Crystal Klym

- Poisson Consulting
- Robyn Irvine

- Salmo Streamkeepers
- Gerry Nellestin
- Eleanor Duifhuis
- Tanya Chi Tran

- Westcott Environmental Services
- Lynn Westcott

- CARO Analytic

## References

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

McElreath, Richard. 2016. *Statistical Rethinking: A Bayesian Course with Examples in R and Stan*. Chapman & Hall/CRC Texts in Statistical Science Series 122. Boca Raton: CRC Press/Taylor & Francis Group.

Plummer, Martyn. 2003. “JAGS: A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling.” In *Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003)*, edited by Kurt Hornik, Friedrich Leisch, and Achim Zeileis. Vienna, Austria.

R Core Team. 2015. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org/.

Rodriguez, Marco A. 1987. “Estimating Periphyton Growth Parameters Using Simple Models.” *Limnology and Oceanography* 32 (2): 458–64.

Tjørve, Even, and Kathleen M.C. Tjørve. 2010. “A Unified Approach to the Richards-Model Family for Use in Growth Analyses: Why We Need Only Two Model Forms.” *Journal of Theoretical Biology* 267 (3): 417–25. https://doi.org/10.1016/j.jtbi.2010.09.008.

Wyatt, Robin J. 2002. “Estimating Riverine Fish Population Size from Single- and Multiple-Pass Removal Sampling Using a Hierarchical Model.” *Canadian Journal of Fisheries and Aquatic Sciences* 59 (4): 695–706. https://doi.org/10.1139/f02-041.