Lower Columbia River Fish Population Indexing Analysis 2016

Condition

The expected weight of fish of a given length were estimated from the data using a mass-length model (He et al. 2008). Key assumptions of the condition model include:

• The expected weight is allowed to vary with length and date.
• The expected weight is allowed to vary randomly with year.
• The relationship between weight and length is allowed to vary with date.
• The relationship between weight and length is allowed to vary randomly with year.
• The residual variation in weight is log-normally distributed.

Only previously untagged fish were included in models to avoid potential effects of tagging on body condition. Preliminary analyses indicated that the annual variation in weight was not correlated with the annual variation in the relationship between weight and length.

Growth

Annual growth of fish were estimated from the inter-annual recaptures using the Fabens method (Fabens 1965) for estimating the von Bertalanffy growth curve (von Bertalanffy 1938). This curve is based on the premise that:

\[ \frac{\text{d}L}{\text{d}t} = k (L_{\infty} - L)\]

where \(L\) is the length of the individual, \(k\) is the growth coefficient and \(L_{\infty}\) is the maximum length.

Integrating the above equation gives:

\[ L_t = L_{\infty} (1 - e^{-k(t - t_0)})\]

where \(L_t\) is the length at time \(t\) and \(t_0\) is the time at which the individual would have had zero length.

The Fabens form allows

\[ L_r = L_c + (L_{\infty} - L_c) (1 - e^{-kT})\]

where \(L_r\) is the length at recapture, \(L_c\) is the length at capture and \(T\) is the time between capture and recapture.

Key assumptions of the growth model include:

• The mean maximum length \(L_{\infty}\) is constant.
• The growth coefficient \(k\) is allowed to vary randomly with year.
• The residual variation in growth is normally distributed.

Length-At-Age

The expected length-at-age of Mountain Whitefish and Rainbow Trout were estimated from annual length-frequency distributions using a finite mixture distribution model (Macdonald and Pitcher 1979). Key assumptions of the length-at-age model include:

• There are three distinguishable age-classes for each species: Age-0, Age-1 and Age-2+.
• The proportion of fish in each age-class is allowed to vary with year.
• The expected length increases with age-class.
• The expected length is allowed to vary with year within age-class.
• The expected length is allowed to vary as a linear function of date.
• The relationship between length and date is allowed to vary with age-class.
• The residual variation in log-tranformed length is normally distributed.
• The standard deviation of this normal distribution varies with age-class.

The model was used to estimate the cutoffs between age-classes by year and date within year to account for growth. For the purposes of estimating other population parameters by age-class, Age-0 individuals were classified as fry, Age-1 individuals were classified as subadult, and Age-2+ individuals were classified as adult. Walleye could not be separated by life stage due to a lack of discrete modes in the length-frequency distributions for this species. Consequently, all captured Walleye were considered to be adults.

The results include plots of the age-class density for each year by length as predicted by the length-at-age model. Density is a measure of relative frequency for continuous values.

Observer Length Correction

The bias (accuracy) and error (precision) in observer’s fish length estimates were quantified using a model with a categorical distribution which compared the proportions of fish in different length-classes for each observer to the equivalent proportions for the measured fish. Key assumptions of the observer length correction model include:

• The proportion of fish in each length-class is allowed to vary with year.
• The expected length bias is allowed to vary with observer.
• The expected length error is allowed to vary with observer.
• The expected length bias and error for a given observer does not vary by year.
• The residual variation in length is normally distributed.

The observed fish lengths were corrected for the estimated length biases before being classified as fry, subadult and adult based on the length-at-age cutoffs.

Survival

The annual adult survival rate was estimated by fitting a Cormack-Jolly-Seber model (Kery and Schaub 2011, 220–31) to inter-annual recaptures of adults.

Key assumptions of the survival model include:

• Survival varies randomly with year.
• The encounter probability for adults is allowed to vary with the total bank length sampled.

Site Fidelity

The extent to which sites are closed, i.e., fish remain at the same site between sessions, was evaluated with a logistic ANCOVA (Kery 2010). The model estimates the probability that intra-annual recaptures were caught at the same site versus a different one. Key assumptions of the site fidelity model include:

• The expected site fidelity is allowed to vary with fish length.
• Observed site fidelity is Bernoulli distributed.

Length as a second-order polynomial was not found to be a significant predictor for site fidelity.

Capture Efficiency

The probability of capture was estimated using a recapture-based binomial model (Kery and Schaub 2011, 134–36, 384–88).

Key assumptions of the capture efficiency model include:

• The capture probability varies randomly by session within year.
• The probability of a marked fish remaining at a site is the estimated site fidelity.
• The number of recaptures is described by a binomial distribution.

Abundance

The abundance was estimated from the catch and bias-corrected observer count data using an overdispersed Poisson model (Kery and Schaub 2011, 55–56).

Key assumptions of the abundance model include:

• The capture efficiency is the point estimate from the capture efficiency model.
• The efficiency varies by visit type (catch or count).
• The lineal fish density varies randomly with site, year and site within year.
• The overdispersion varies by visit type (count or catch).
• The catches and counts are described by a gamma-Poisson distribution.

Stock-Recruitment

The relationship between the adults and the resultant number of age-1 subadults 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 adults (stock), \(R\) is the subadults (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 prior for \(\log(\alpha)\) is a truncated normal distribution from \(\log(1)\) to \(\log(5)\).
• The recruits per spawner varies with the proportional egg loss.
• The residual variation in the number of recruits is log-normally distributed.

The carrying capacity \(K\) is given by the relationship:

\[ K = \frac{\alpha}{\beta} \quad.\]

Scale Age

The age of Mountain Whitefish was estimated from scales using an automated algorithm. The algorithm estimates the age of individual fish by finding annuli among the growth circuli in the scale. Specifically, annuli represent blocks of slow growth – short inter-circuli distance – during the winter. The algorithm takes scale growth between circuli, calculates a moving average growth rate, and then calculates a moving average of the rate of change in growth. For brevity, we refer to the “rate of change in growth” as “dGrowth”.Periods of rapid deceleration (dGrowth < 0) correspond to winter and mark an annulus. Specifically, we defined an annulus as a run of negative dGrowth values. Because of noise in the measurements, these runs had to be of a minimum length. We estimated the window size for calculating moving averages and this minimum length by tuning the algorithm to most accurately predict scale ages from recaptured fish of known age. During tuning, we found that a window size of seven circuli was best for calculating the moving average growth rate; a window size of 17 circuli was best for calculating dGrowth. Note that a larger window size leads to greater smoothing in the series. We estimated the minimum length for a run to be four. Once the algorithm was tuned, we applied to fish scales of unknown age.

Key assumptions of the scale age algorithm include:

• Growth decelerates in the fall/winter and accelerates in spring/summer
• The actual age is the year of capture minus the hatch year
• The scale age varies linearly with age.

Age-Ratios

The proportion of Age-1 Mountain Whitefish \(r^1_t\) from a given spawn year \(t\) is calculated from the number of Age-1 & Age-2 fish \(N^1_t\) & \(N^2_t\) respectively, which were lead or lagged so that all values were with respect to the spawn year:

\[r^1_t = \frac{N^1_{t+2}}{N^1_{t+2} + N^2_{t+2}}\]

As the number of Age-2 fish might be expected to be influenced by the percentage egg loss \(Q_t\) three years prior, the predictor variable \(\Pi_t\) used is:

\[\Pi_t = \textrm{log}(Q_t/Q_{t-1})\]

The ratio was logged to ensure it was symmetrical about zero (Tornqvist, Vartia, and Vartia 1985).

The relationship between \(r^1_t\) and \(\Pi_t\) was estimated using a hierarchical Bayesian logistic regression (Kery 2010) loss model.

Key assumptions of the final model include:

• The log odds of the proportion of Age-1 fish varies linearly with the log of the ratio of the percent egg losses.
• The numbers of Age-1 fish are extra-Binomially distributed.

The relationship between percent dewatering and subsequent recruitment is expected to depend on stock abundance (Subbey et al. 2014) which might be changing over the course of the study. Consequently, preliminary analyses allowed the slope of the regression line to change by year. However, year was not a significant predictor and was therefore removed from the final model. The effect of dewatering on Mountain Whitefish abundance was expressed in terms of the predicted percent change in Age-1 Mountain Whitefish abundance by egg loss in the spawn year relative to 10% egg loss in the spawn year. The egg loss in the previous year was fixed at 10%. The percent change could not be calculated relative to 0% in the spawn or previous year as \(\Pi_t\) is undefined in either case.

Table 1. Parameter descriptions.

Mountain Whitefish

Table 2. Model coefficients.

Table 3. Model summary.

Rainbow Trout

Table 4. Model coefficients.

Table 5. Model summary.

Walleye

Table 6. Model coefficients.

Table 7. Model summary.

Table 8. Parameter descriptions.

Mountain Whitefish

Table 9. Model coefficients.

Table 10. Model summary.

Rainbow Trout

Table 11. Model coefficients.

Table 12. Model summary.

Walleye

Table 13. Model coefficients.

Table 14. Model summary.

Table 15. Parameter descriptions.

Mountain Whitefish

Table 16. Parameter estimates from length-at-age model. Averages are taken across yearly coefficients.

Table 17. Estimated length cutoffs (mm) by year from length-at-age model. Age0-1 is the cuttoff between fry and subadult fish; Age1-2+ is the cuttoff between subadult and adult fish. Reported cutoffs are for the median sampling date.

Rainbow Trout

Table 18. Parameter estimates from length-at-age model. Averages are taken across yearly coefficients.

Table 19. Estimated length cutoffs (mm) by year from length-at-age model. Age0-1 is the cuttoff between fry and subadult fish; Age1-2+ is the cuttoff between subadult and adult fish. Reported cutoffs are for the median sampling date.

Table 20. Parameter descriptions.

Mountain Whitefish

Table 21. Model coefficients.

Table 22. Model summary.

Rainbow Trout

Table 23. Model coefficients.

Table 24. Model summary.

Walleye

Table 25. Model coefficients.

Table 26. Model summary.

Table 27. Parameter descriptions.

Mountain Whitefish

Table 28. Model coefficients.

Table 29. Model summary.

Rainbow Trout

Table 30. Model coefficients.

Table 31. Model summary.

Walleye

Table 32. Model coefficients.

Table 33. Model summary.

Table 34. Parameter descriptions.

Mountain Whitefish

Table 35. Model coefficients.

Table 36. Model summary.

Rainbow Trout

Table 37. Model coefficients.

Table 38. Model summary.

Walleye

Table 39. Model coefficients.

Table 40. Model summary.

Table 41. Parameter descriptions.

Mountain Whitefish

Rainbow Trout

Walleye

Table 52. Parameter descriptions.

Mountain Whitefish

Rainbow Trout

Walleye

Table 63. Parameter descriptions.

Mountain Whitefish

Table 64. Model coefficients.

Table 65. Model summary.

Rainbow Trout

Table 66. Model coefficients.

Table 67. Model summary.

Table 68. Parameter descriptions.

Mountain Whitefish

Table 69. Model coefficients.

Table 70. Model summary.

Subadult

Mountain Whitefish

Rainbow Trout

Adult

Mountain Whitefish

Rainbow Trout

Walleye

Mountain Whitefish

Rainbow Trout

Walleye

Mountain Whitefish

Rainbow Trout

Adult

Mountain Whitefish

Rainbow Trout

Walleye

Mountain Whitefish

Subadult

Adult

Rainbow Trout

Subadult

Adult

Walleye

Adult

Mountain Whitefish

Subadult

Adult

Rainbow Trout

Subadult

Adult

Walleye

Adult

Mountain Whitefish

Rainbow Trout

Mountain Whitefish

Mountain Whitefish