5.3. Delay in migration

In the previous chapter, I assumed that fish migrate at a constant rate during the entire migration period. In some cases, however, fish may delay their migration. In this section, I examine two types of delay - delay in front of a dam before passage and delay before the fish initiate downstream migration. In both these cases, the delay may be substantial, and incorporating a delay term in the travel time model may be worthwhile.

There is some evidence that fish delay their passage as they encounter a dam. Dams produce turbulence and a significant amount of noise that may deter fish from passing. Also dam passing often involves extreme changes in pressure, which the fish resist. First, I examine the dam delay process by analyzing some chinook radio-tracking data at John Day and Lower Granite Dams. I then incorporate the delay model into the basic travel time model and apply this to PIT tag data.

Sometimes fish are tagged and released before they are ready to initiate migration. This may be the case when hatchery fish are released before they are fully smolted or when wild fish are collected in their rearing habitat, tagged, and then released back in the river. The mid-Columbia fall chinook examined in the previous chapter may be an example of the latter case. These fish were beach seined, and most of the fish were less than 75 mm in length, probably too small to initiate migration. For these fish I incorporate a migratory delay term into the travel time model.

formulation of the model

If the delay probability density function is d(t), and d(t) is independent of the arrival distribution, then we can express the passage distribution incorporating delay as a convolution integral (Mood, et al. 1974):

. (5.7)

g(t) is the arrival distribution without delay (equation (4.7)) and g_D(t) is the arrival distribution with delay included. Note that it does not matter whether the delay occurs before or after the migratory period; the general equation remains the same.

Although the waiting time process is likely to be complex, a reasonable simplification is a waiting time process with instantaneous passage rate (t). This yields a delay pdf of

. (5.8)

Assuming a constant equation (5.7) becomes

, (5.9)

and the average delay is 1/. Figure 5.2 represents the components of equation (5.9) graphically. The delay term (top plot) and reservoir travel time term (middle plot) are both incorporated into the arrival time with delay equation (bottom plot). Assuming the basic travel time distribution (equation (4.7)) for g(t), plots of equation (5.9) are presented in Figure 5.3 for several values of constant . As average delay increases (i.e., as decreases), the mode of the distribution shifts to the right, and the curve flattens out.

application of the delay model to radio-tracking data

It is clear from Figure 5.3 that delay at a dam can produce substantial effects on fish arrival distributions. With most arrival time data, where fish are sampled as they pass a dam, separating river travel time from dam delay is difficult. Fortunately there is some data available where dam delay can be observed directly. These data are from radio tag studies where groups of fish are released upstream from a dam. The time when an individual first reaches the forebay in front of the dam is recorded as well as when the fish passes through the dam. The difference between these two times is dam delay. A distribution of these times is obtained from a group of individuals released at the same time.

Applying delay models to independent data sets has several advantages. Since delay is being observed directly, more accurate parameter estimates can be obtained. These parameter estimates can then either be applied directly to the travel time model with delay (equation (5.7)), or the parameter estimates can be compared to those obtained by applying equation (5.7) to travel time data. Also, these data will allow for a more direct assessment of model performance and for comparison among alternative models.

I will examine three alternative models for delay. The first is a simple model where delay is determined by a constant passage rate. The second model introduces diel behavior with separate passage rates for daytime and nighttime periods. The third model separates the fish into two types: those who pass quickly and those who pass more slowly.

Two radio-tag studies have been performed on juvenile salmonids in the Columbia River system: one at John Day Dam in 1984 (Giorgi et al., 1985) and the other at Lower Granite Dam in 1985 (Stuehrenberg et al., 1986). In these studies fish were fitted with miniature radio-tags and released upstream from the dam. Several receivers were situated at the dam and were able to detect when they first arrived at the front of the dam and when they passed the dam. The difference between these two times is the delay.

In the John Day study, fish were released on 4 days. On the first three days (May 1, May 10, and May 14), 28 fish were released; half were released in the morning and half were released in the afternoon. On the fourth release day, only 11 fish were released, and I did not include these fish in the analysis. The fish were collected from the John Day Dam and released 6.3 km. upstream from the dam. In the Lower Granite study, 4 groups of approximately 100 fish were released 4.8 km. upstream from the dam. These fish were collected at the bypass facilities of Lower Granite and McNary Dams. The first group was not analyzed because of technical difficulties encountered at the dam. The last three groups were released on April 17, April 24 and May 1.

Stuehrenberg et al., (1986) also performed behavioral test to determine the impact of the tags on the fish. They determined that the radio-tags did not significantly affect either swimming velocity or mortality but that the tags may affect the buoyancy of the fish. Because of this problem, these data are not ideal. They are, however, the only data where dam delay is directly observed. For this reason I have chosen to analyze these data to obtain rough parameter estimates and some qualitative results. In addition, the methodology I present will be applicable in the future if better data become available.

To analyze the data I use the following procedure. First I estimate the parameters using maximum likelihood. If numerical solutions are required, I use the downhill simplex method (Nelder and Mead, 1965; Press, et al., 1988). Also, log likelihoods are computed for comparisons among models within a data set. In addition, I perform an X² goodness-of-fit test, following the procedure for continuous data outlined in chapter 3.

The first model is the waiting time model with constant . The pdf for delay is

. (5.10)

The maximum likelihood estimate for is

, (5.11)

where N is the number of fish in the cohort, t_i is the waiting time of the ith individual and is the average waiting time for the group.

The second model includes a different passage rate for daytime and nighttime, _d and _n respectively. The delay pdf for passage during the day is

. (5.12)

In this notation t_d is the time spent waiting during the day, and t_n is time spent waiting during the night period. The pdf for passage occurring during the night period is the same as equation (5.12) but with _n substituted for _d in front of the exponential term on the right side. Note that since individual fish arrive at the dam at different times of the day, each fish will have a different waiting time pdf. The mle's of the two parameters _n and _d are determined numerically.

A third model is a double exponential model. The model essentially divides the population into two groups: those that pass the dam quickly and those that pass more slowly. The model is expressed as

. (5.13)

_f corresponds to the fast passage rate, _s is the slow passage rate, and wt assigns a weight to the two types of passage, with . Again, the mle's of the parameters are determined numerically, and a log likelihood is computed.

results

The parameter estimates, likelihoods and goodness-of-fit results for the Lower Granite data are contained in table Table 5.3, and for the John Day data in Table 5.4. In these tables, ₁= for the simple model, ₁ = _n and ₂ = _d for the diel-delay model, and ₁ = _f and ₂ = _s for the double-exponential model. The BIC values reported are those for the individual models (not comparisons between models as in the last application). According to this criterion, the most desirable model is the one with the largest BIC value.

Table 5.3 Delay model results from the Lower Granite data. For the simple model, 1=. For the diel delay model, 1=n, and 2=d. For the double exponential model, 1=fand 2=s. Based on BIC values, the "best" model has the largest value.
model	1	2	wt	lik	ratio	BICi	X2	p
Release data: April 17; n=61
simple	1.13	-	-	-247.19	-	-498.50	105.64	< 0.001
diel	1.42	0.91	-	-245.70	3.00	-499.61	106.26	< 0.001
2 exp	66.87	0.85	0.26	-224.14	46.10	-460.62	32.13	0.010
Release date: April 24; n=65
simple	3.48	-	-	-190.45	-	-385.08	123.54	< 0.001
diel	6.29	1.33	-	-173.83	33.24	-356.02	100.15	< 0.001
2 exp	113.70	2.07	0.41	-151.30	78.31	-315.12	14.22	0.58
Release date: May 1; n=70
simple	2.91	-	-	-217.68	-	-439.61	272.86	< 0.001
diel	7.01	0.70	-	-181.43	72.51	-371.35	152.86	< 0.001
2 exp	70.65	1.21	0.59	-148.06	139.24	-308.87	27.71	0.048

Table 5.4 Delay model results from the John Day data. For the simple model, 1=. For the diel delay model, 1=n, and 2=d. For the double exponential model, 1=f and 2=s. Based on BIC values, the "best" model has the largest value.
model	1	2	wt	lik	ratio	BICi	X2	p
Release data: May 1; n=19
simple	6.53	-	-	-43.74	-	-90.42	30.89	< 0.001
diel	8.78	5.07	-	-43.03	1.42	-91.94	30.89	< 0.001
2 exp	71.67	2.59	0.63	-30.93	25.62	-70.69	4.37	0.89
Release date: May 10; n=25
simple	13.38	-	-	-39.60	-	-82.42	25.44	0.008
diel	29.86	4.18	-	-29.39	20.43	-65.21	24.40	0.007
2 exp	101.59	7.87	0.45	-33.98	11.24	-77.62	12.96	0.23
Release date: May 14; n=23
simple	13.51	-	-	-36.22	-	-75.58	27.30	0.004
diel	38.61	4.04	-	-23.34	25.76	-52.95	19.39	0.036
2 exp	473.63	9.88	0.27	-29.25	13.93	-67.91	17.13	0.072

Figures 5.4 and 5.5 contain plots of the fitted models versus the data. In these plots, the percentiles of the data are plotted against the percentiles predicted by the model. A straight line through the origin and the point (1.0, 1.0) would signify an exact correspondence between the two. The columns of plots represent the three models, and the rows represent the three data sets.

For the Lower Granite data, the average waiting time (1/) is approximately 20 hours for the first group and approximately 8 hours for the second and third groups. In all three cases, the plots (Figure 5.4) show that the simple waiting time model cannot adequately describe the data. The model under predicts early fish passage and overpredicts late fish passage. The results of the goodness-of-fit tests (all p-values below 0.001) confirm this.

Comparing the likelihood ratios and BIC values from second model to the first show that this model is a marked improvement in the last two data sets, but the first model would be selected for the first data set. In all three cases, the fish are more inclined to pass during the nighttime hours, with a tenfold difference between nighttime and daytime passage rates in the last data set. These results are consistent with diel behavior and a tendency toward nighttime passage. However, the plots show that this model still does not adequately describe the data and suffers from the same shortcomings as the first model.

The third model was partly motivated by the shortcomings of the first two. Based on likelihood ratios and BIC values, this model is a substantial improvement over the first two. Also, the plots show that this model does a reasonable job of describing the data. Among the three data sets, the estimates of _f and _s are roughly of the same order of magnitude, with _f fifty to one hundred times larger than _s. For example, in the first group of fish, average waiting time for the fast fish is on the order of 20 minutes, while the slow group waits for more than a day, on average, before passing the dam. There is a noticeable difference among the three data sets in the estimates of the parameter wt (the proportion of fish in the fast group), which increases with release date, ranging from 0.257 in the earliest release to 0.594 in the latest release. This is consistent with the fish being more eager to migrate later in the season.

For the John Day data, the average waiting time is under 4 hours for the first group and under 2 hours for the last two groups. As with the Lower Granite data, the simple model cannot adequately describe the data based on the plots (Figure 5.5). Based on BIC values, the diel delay model would be selected over the simple model in 2 out of 3 cases, but the plots indicate that this model is inadequate for the John Day data. The double exponential model is a clear improvement over the simple model, based on the BIC values. Also, the plots and the goodness-of-fit results indicate that this model represents the data well.

Some general conclusions from this analysis are the following. Fish passed John Day more rapidly than they passed Lower Granite, and fish passed more rapidly later in the season at both dams. The simple waiting time model could not adequately describe the data. The diel passage model is an improvement but still did not adequately describe the data. The double exponential model did an excellent job of describing the data.

application to travel time data

In this section, I apply the travel time/delay model equation (5.9) to pit tag data. In this application, I use treatment groups from the Snake River spring chinook and steelhead and the mid-Columbia fall chinook. The cohorts are identified by year and cohort number, so these results can be directly compared to those found in Table 4.4 - Table 4.6 (basic travel time model results) and release information can be found in Appendix I. For each cohort, I numerically calculate maximum likelihood estimates of r, and . I also report the likelihoods for the travel time/delay model and the null model, which is the basic travel time model (equation (4.7)). I also report the ratios of these likelihoods and the BIC values. The BIC value reported is the difference between the value for time/delay model and the null model. A negative value lends support to the null model.

For the spring chinook, all but one of the cohorts had slightly higher likelihoods for the model with the delay component (Table 5.5). For none of the cohorts, though, would the delay model be selected over the basic travel time model based on BIC criterion. Also, the maximum likelihood estimates of varied substantially ranging from =.202 (corresponding to an average waiting time of ~ 5 days) to = 6.81 (average waiting time under 4 hours). On the other hand, many of the 's were in the 3-4 range, which is consistent with the results from the radio-tracking data.

Table 5.5 Results from the application of the travel time/delay model to Snake River, spring chinook, PIT tag data. Each row is a cohort. A negative BIC value lends support to the null model. See text for further details of the analysis.
cohort	# of fish	parameter estimates			likelihoods
		r			l0	lA	ratio	BIC
1989
3	57	3.70	6.68	0.202	-196.66	-195.56	2.19	-1.86
10	52	4.48	7.18	0.243	-168.49	-168.00	0.96	-2.99
15	55	6.58	9.02	0.202	-171.46	-171.46	0.00	-4.01
17	53	5.32	8.53	3.012	-151.52	-151.48	0.09	-3.88
26	60	8.21	10.12	3.011	-143.13	-142.85	0.55	-3.54
33	41	12.90	9.50	0.963	-79.23	-79.22	0.03	-3.69
34	64	12.39	17.07	3.171	-138.73	-138.15	1.16	-3.00
1990
3	52	8.56	9.86	3.123	-119.82	-119.48	0.70	-3.25
8	62	5.88	10.74	1.921	-177.67	-177.34	0.65	-3.47
10	80	4.79	9.73	1.670	-246.49	-246.30	0.38	-4.00
1991
4	84	3.80	5.19	1.265	-248.59	-248.53	0.11	-4.33
10	62	5.27	8.48	3.303	-177.90	-177.89	0.04	-4.09
16	63	10.32	11.91	4.889	-137.90	-137.63	0.54	-3.60
1992
2	57	3.91	7.03	1.581	-178.31	-178.01	0.61	-3.44
1993
4	59	4.43	6.22	0.311	-182.60	-181.88	1.43	-2.65
9	47	6.45	7.59	4.305	-117.86	-117.83	0.06	-3.79
15	58	10.49	7.32	6.797	-114.68	-113.49	2.38	-1.68
21	69	11.68	10.82	4.158	-135.66	-135.57	0.19	-4.04
26	84	12.96	12.41	4.074	-160.80	-160.35	0.90	-3.53

I was not able to successfully apply the more complex models (diel delay and double exponential delay) to these data. There are probably too many parameters to be fit.

The results from the steelhead are somewhat perplexing. On the one hand, for 13 out of 19 cohorts, we would select the model with the delay component based on the BIC criterion (Table 5.6). The parameter estimates, however, have a great deal of variability with some unrealistically high values for r and unrealistically low values for . I would be very hesitant to use these results. The added component seems to make up for some of the deficiency of the null model for this data but in a biologically unrealistic and inconsistent manner.

Table 5.6 Results from the application of the travel time/delay model to Snake River steelhead PIT tag data. Each row is a cohort. A negative BIC value lends support to the null model. See text for further details of the analysis.
cohort	# of fish	parameter estimates			likelihoods
		r			l0	lA	ratio	BIC
1989
4	45	44.49	3.93	0.869	-83.59	-75.15	16.88	13.08
6	63	29.93	4.48	0.968	-81.60	-75.04	13.11	8.97
11	79	41.33	0.10	0.885	-101.52	-88.74	25.57	21.20
1990
2	51	20.43	5.25	0.771	-79.07	-77.15	3.84	-0.09
7	86	17.14	3.01	0.779	-134.72	-124.82	19.80	15.34
15	80	23.72	1.94	0.712	-152.68	-133.65	38.06	33.67
18	55	14.58	18.35	2.822	-109.54	-108.75	1.60	-2.41
24	60	23.45	0.20	0.761	-86.53	-80.06	12.93	8.83
1991
3	49	9.61	10.63	3.297	-107.54	-107.10	0.88	-3.01
6	68	21.22	6.62	0.758	-111.19	-106.90	8.58	4.36
14	85	24.91	4.13	1.318	-112.02	-94.14	35.76	31.31
16	339	23.70	14.98	8.373	-445.62	-440.32	10.60	4.77
1992
6	72	26.24	7.99	0.911	-131.02	-110.83	40.39	36.11
9	69	22.26	0.69	0.600	-123.11	-105.97	34.28	30.04
13	40	12.85	10.38	5.954	-71.46	-71.28	0.35	-3.34
1993
2	51	11.80	15.63	2.954	-110.26	-109.50	1.52	-2.41
9	72	18.92	15.95	6.944	-113.12	-112.68	0.89	-3.39
11	97	36.90	0.82	0.711	-156.05	-131.73	48.63	44.06
20	61	41.29	2.40	0.917	-81.87	-69.57	24.61	20.50

The results for the fall chinook are contained in Table 5.7. The results appear to be positive - 5 out of the 6 cohorts had positive BIC values, some of which were quite high. Also there is a fair degree of consistency among parameter estimates, which is desirable. Most of the values of are close to 0.1, resulting in an average waiting time of 10 days. The 1992 results are not as positive as the 1991 and 1993 results. 1992 was an extremely low flow year, and the behavior of the fish may have been affected by this.

Table 5.7 Results from the application of the travel time dependent mortality model to mid-Columbia fall chinook PIT tag data. Each row is a cohort. A negative BIC value lends support to the null model. See text for further details of the analysis.
cohort	# of fish	parameter estimates			likelihoods
		r			l0	lA	ratio	BIC
1991
1	154	4.60	11.95	0.098	-655.93	-642.39	27.07	22.03
1992
2	73	4.76	6.23	0.113	-272.21	-271.75	0.91	-3.38
3	68	4.14	5.74	0.306	-250.37	-244.51	11.72	7.50
1993
2	81	5.65	3.83	0.081	-316.33	-309.19	14.27	9.88
4	75	5.74	1.90	0.090	-272.00	-264.44	15.13	10.82
5	118	5.36	3.39	0.100	-446.97	-426.93	40.08	35.31

Figure 5.6 contains plots of the cumulative of the best fit model compared to the cumulative travel times for the six cohorts. The plots of the cohorts from 1991 and 1992 (the first three plots) indicate some inconsistency between the model and data. The plots of the 1993 cohorts, on the other hand, show a great deal of consistency between the model and data. Clearly more years of data will help to elucidate these differences.