Methods

The Model

Figure 1 . A schematic diagram of the travel time problem. The river is considered one-dimensional, with a group of fish released at X = 0 and collected downstream at X = L.

overview of migration models

The migration of individuals is commonly developed in terms of a biased random walk (DeAngelis and Yeh 1984) where the probability of moving downstream is greater than the probability of moving upstream during a time increment. Travel times are determined by recording when an individual first reaches a downstream point. A travel time distribution is generated by conducting many realizations of the process. This procedure is intuitive but cumbersome.

Population level migration models are more complex mathematically but are easier to apply to data because resulting distributions can be stated as an explicit equation instead of simulation results. We have chosen this approach because of its applicability.

modeling travel times in terms of population density

. (1)

This equation describes the rate of change of population density with respect to time in terms of an advection (or drift) term (first term right hand side) and diffusion term (second term right hand side). The parameter r determines the average rate of downstream movement, and determines the rate of population spreading.

The first step is to solve equation (1) for p(x,t). With unrestricted boundaries and initial conditions of a point release at X = 0 and t = 0, p(x,t) is a normally distribution (with respect to X for fixed t) with mean rt and variance ²t. Both the mean and variance increase linearly with time, corresponding to the population moving downstream and spreading with time.

While fish can move upstream or downstream in a reservoir, once they pass a dam, they cannot move upstream from that site. To account for this, an absorbing boundary is imposed at the dam (X = L). Once fish reach a dam, they are "absorbed" and passed to the next reservoir. Movement upstream from the release point (X = 0) is unrestricted, but the advection term moves fish away from this point. With these boundary conditions and the same initial conditions as above, equation (1) can still be solved for p(x,t) (Goel and Richter-Dyn 1974) but is more complex than in the previous case.

Since the absorbing boundary corresponds to fish passage at a dam, p(x,t) can be used to derive an arrival time distribution at a dam. The first step is to determine the probability of remaining above the dam at a given point in time (denoted P(L,t)). This is achieved by integrating the probability density upstream of the dam:

. (2)

,

(3)

Plots of this distribution for various values of r and are shown in Figure 2. Note that the distribution is unimodal and right skewed, which is consistent with most observed fish travel time distributions. Decreasing r moves the mode of the distribution to the right and flattens out the distribution. Increasing has the effect of moving the mode to the left and flattening the distribution.

Figure 2 . Plots of equation (3), the travel time distribution, with various parameter values. In the top figure, is set at 25, and r is varied. In the bottom plot, r is set at 25, and is varied. In both plots, L = 120.

Statistical methods

assumptions

parameter estimation and confidence intervals

The maximum likelihood estimators (mles) for the two parameters r and are:

(4)

, (5)

where t_i is the observed arrival time of the ith individual, is the average travel time of the cohort, and N is the number of recaptured individuals in the cohort. Notice that the mle for r is the average migration rate and the mle for involves the difference between the harmonic mean and reciprocal of the arithmetic mean of the travel time. The maximum likelihood estimators of these two parameters are independent (Chhikara and Folks 1989), and much of the statistical inference involving the inverse Gaussian distribution parallels that of the normal distribution.

Confidence intervals for the parameters r and are available (Tweedie 1957a, 1957b; Folks and Chhikara 1978; Chhikara and Folks 1989). A 100(1-) percent confidence interval for r is

if , and

otherwise. (6)

Here, a is the (/2)th percentile of Student's t distribution with N - 1 degrees of freedom. A 100(1-) percent confidence interval for can be constructed as:

, (7)

where a is the (1-/2)th percentile of the ² distribution with N - 1 degrees of freedom and b is the (/2)th percentile of the same distribution.

Notice that the confidence interval for r is determined by the estimates of r and , but the confidence interval for is determined only by the estimate of . Also, the confidence intervals of r and are dependent on sample size, with the confidence intervals decreasing as N increases.

goodness-of-fit

, , (8)

, (9)

where c() is the (1-)th quantile of the standard normal distribution, with = 0.05. Once the number of cells is set, the limits of integration in equation

Data

Figure 3 . Map showing the release and recapture sites for the Snake River spring chinook salmon.

The chinook salmon in this study were active migrants of unknown origin (hatchery versus wild). Although the run type (spring versus fall) of these fish was not determined at tagging, it is likely that the vast majority of these fish were spring (yearling) chinooks based on their lengths (most fish longer than 110 millimeters) and migration timing (early spring). The migratory season begins in early April, and fish released after May 2 were excluded because after this date average fish length began declining, indicating a possible presence of fall chinook. Treating these fish as spring chinook is consistent with other treatments of this group of fish (e.g., Fish Passage Center 1991).

The PIT tag, 12 mm long, is inserted in the fish's body cavity (Prentice, et al. 1990). At monitoring sites the tag emits a unique signal in response to excitation from an interrogation system. The signal is decoded to yield information about instantaneous passage times of individuals. The tags do not adversely affect the fish's survival or swimming performance (Prentice, et al. 1990). Fish were tagged and released as groups on a daily basis throughout the migration season.

Based on the results of simulations and plots of the confidence intervals and bias (Zabel 1994), we used a target cohort sample size of 100 fish observed at the downstream collection site, with a minimum sample size of 80. Release groups were lumped together (if necessary) from up to three consecutive days of release to construct "cohorts" with these sample sizes. If a minimum sample size of 80 could not be obtained from release groups over a three day period, these groups were excluded from the analysis. Also, if a cohort had a sample size of 100 fish or more, no more release groups were lumped into that cohort. 46 cohorts were analyzed, with releases occurring over the six year period 1989-1994.