9.3. Recommendations for salmon population management

The two parameter travel time model (equations (4.7) and (4.8)) is particularly effective for describing arrival distributions of run-of-the-river, yearling chinook, for which abundant data exists. The model accommodates both discrete and continuous data and is easily applied. In continuous form, g(t), the probability density function for the arrival times of fish at the downstream collection site, is expressed as

, (9.1)

where L is the length of the river reach. The parameters are intuitive and biologically meaningful: r is the downstream migration rate, and describes the rate of spreading of the population. The model, in its simplest form, does not work as well for steelhead and fall chinook. Although the model captures the important features of steelhead arrival time distributions, more modeling efforts are needed to understand the departure of observed steelhead travel time distributions from model-predicted distributions.

The travel time model is improved for fall chinook by incorporating a delay term, which corresponds to a delay in the initiation of migration. In its simplest form, this is modeled as an exponential waiting time process. More complexity can be added by relating the instantaneous departure rate, , to time (for example, the fish are more likely to initiate migration as the season progresses) and covariates, X, particularly fish length. The delay model is then expressed as

. (9.2)

This equation is easy to evaluate if the form of (t, X) is not complex.

The delay in front of a dam before fish passage is an important component of downstream migration. I developed three alternative models to describe this delay process and applied the models to radio tag data, where exact times of arrival to the forebay and dam passage are observed. These data show that dam delay can be substantial; one group of chinook delayed for an average of 20 hours at Lower Granite Dam. The model that works best to describe these data splits the fish in two groups: those that pass quickly with rate _f, and those that pass slowly with rate _s. This model works substantially better than one with daytime and nighttime passage rates. Unfortunately, dam delay is difficult to detect with travel time data and is difficult to observe directly. More work is necessary to determine the extent of dam delay and how it varies from dam to dam.

Utilizing the travel time model in a predictive manner involves selecting model parameters a priori. I related the observed variation in parameter estimates to the factors date of release and average river flow in regression equations. I tested several alternative equations and determined that the following set worked the best to predict values of r and :

(9.3)

and

. (9.4)

In the first equation, migration rate is linearly related to flow, F_i, and the term in the brackets represents a nonlinear relationship with date of release, where migration rate begins at a lower rate early in the season and increases to an upper level as the season progresses. The second regression equation linearly relates , the rate of population spread, to migration rate. These two regression equations were applied to four groups of run-of-the-river chinook (composed primarily of yearling chinook of both wild and hatchery origins). The regression equation for r had R² values ranging from .855 to .945, and the regression equation for had R² values ranging .589 to .845. These regression equations can be used to determine model parameters based on date of release and river flow. The travel time model can then be implemented to predict the downstream arrival distributions.

When information on the variability of individuals within a cohort was included in the travel time model, fish length was determined to be an important factor for mid-Columbia subyearling chinook but not for Snake River yearling chinook and steelhead. Also, for sequential releases of Snake River subyearling chinook and Columbia River sockeye, I determined that fish length, date of release, and average river flow are important factors at the level of the individual, but river temperature is not. River temperature may be important, though, in determining the timing of runs on a year to year basis. The importance of fish length in the fall chinook may be partly due to its relation to the onset of migration, and incorporating fish length into the delay term can account for this.

The vertical distribution model can benefit future modeling applications. The position of fish in the water column as they approach the dam is related to their passage route through the dam - spillway, fish bypass system, or turbines. Since each pathway has a different associated mortality, utilizing a vertical distribution model to predict passage routes will be useful in ascribing total passage mortality. The modeling demonstrated that observed vertical distributions are predictable and that different species have different distributions. Future experimental work in this are will help to identify underlying mechanisms of the vertical distribution process.

Overall conclusions are as follows. First, simple models based on diffusion equations are quite tractable mathematically and capture many of the features of the distributions of migrating juvenile salmonids. Statistical techniques, primarily based on likelihood functions, are readily applied to these models to estimate parameters, assess model goodness-of-fit, and to compare among alternative models. This combination of modeling and statistics is a powerful method in establishing models as predictive tools for management purposes.

Spatial and Temporal Models of Migrating Juvenile Salmon with Applications.