5.2. Time dependent mortality

If the reservoir mortality rate is (t), then as shown in equation (2.24), the probability of surviving through time t is

. (5.1)

This mortality can be incorporated into the migration model as follows:

. (5.2)

Solutions of equation (5.2) have the form

. (5.3)

Referring to this density as p_m(x, t) and using equation (4.5) for p(x, t), we have:

. (5.4)

Carrying out the same procedure as before, that is, integrating from to A with respect to x and differentiating with respect to t, we end up with:

, (5.5)

where g_L(t) represents the loss from the reservoir (due to both dam passage and mortality) and g(t) is equation (4.7), the basic arrival time distribution. The first term in the right side of equation (5.5) is loss due to fish leaving the reservoir, and the second term is loss due to mortality. This makes intuitive sense because g(t) is the pdf for dam passage in the absence of mortality, and represents the probability of surviving through time t. In the second part of equation (5.5), the term in brackets represents the fish remaining in the reservoir, and is the survival probability density function. To obtain a probability density function, g_m(t), for the arrival time given time dependent in-river mortality, the passage portion of equation (5.5) must be normalized:

. (5.6)

The simplest case is when is constant, and Figure 5.1 contains plots of equation (5.6) for various values of constant . Note that as increases, the mode of the distribution shifts to the left, and the right tail becomes thinner. An of 0.02 corresponds roughly to 18 per cent mortality after 10 days. At this and higher levels of mortality, including mortality has little effect on the shape of the arrival time distribution.

The model discrimination methods described in chapter 3 can determine the ability of the model to detect travel time dependent mortality. In this case, the null model is the basic arrival time distribution, equation (4.7). The alternative model is the arrival distribution described by equation (5.6). I should emphasize that this will test for the ability of the model to detect travel time dependent mortality in the river. Accepting the null hypothesis does not necessarily mean the effect does not exist.

results

The results of the data analysis for the Snake River trap chinook and steelhead are contained in Table 5.1 and Table 5.2. Each line in the table represents the results from a single cohort. The cohorts are identified by year and cohort number, so these results can be directly compared to those found in Table 4.4 and Table 4.6 (basic travel time model results) and release information can be found in Appendix I. These tables also contain parameter estimates for the travel time model with mortality, likelihoods for the null and alternative models, and the likelihood ratios and BIC values. The BIC value in these tables is the difference between the BIC values for the alternative and null models. A negative value lends support to the null model, and positive one lends support to the alternative model.

Little support exists for including travel time dependent mortality in the model for the Snake River chinook (Table 5.1). 16 out of the 19 cohorts had likelihood ratios less than 0.01, and for none of the cohorts would the null model be rejected based on a likelihood ratio test or based on the BIC values. This is not to say that travel time dependent mortality is not occurring for these groups, but this model cannot detect it with these data. Other types of data are necessary to observe this effect. On a positive note, the fact that this type of mortality seems to have little effect on the arrival time distribution makes modeling arrival times less complex.

The results from the steelhead cohorts are a bit perplexing. As with the chinook, the majority of cohorts (14 out of 19) had likelihood ratios less than 0.01. But the remaining 5 cohorts all had fairly large ratios, and for all these cohorts the null hypothesis would be rejected based on a likelihood ratio test or BIC values. The estimates of the mortality term, , for these groups is quite variable, ranging from less than 0.0001 to 0.058. This leads me to believe that these results are spurious - the added term allows for a better fit of the model to the data but not in a biologically meaningful way.

Table 5.1 Results from the application of the travel time dependent mortality model to Snake River chinook PIT tag data. Each row is a cohort. A negative value for BIC 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
1	48	2.55	5.46	0.012	-166.88	-166.88	0.00	-3.87
7	54	2.96	6.48	0.022	-180.04	-180.04	0.00	-3.99
13	54	3.09	6.02	0.018	-176.02	-176.02	0.00	-3.99
18	66	5.70	7.49	0.011	-174.07	-174.07	-0.00	-4.19
24	59	8.85	11.94	0.008	-139.46	-139.46	-0.00	-4.08
32	71	8.38	7.30	0.015	-159.62	-157.80	3.64	-0.62
37	61	6.70	10.25	0.016	-159.96	-159.96	-0.00	-4.11
1990
1	59	5.10	7.55	0.019	-162.63	-162.63	-0.00	-4.08
9	70	4.93	9.78	0.012	-207.84	-207.84	0.00	-4.25
12	41	5.64	7.06	0.013	-106.59	-106.59	0.00	-3.71
1991
5	69	2.91	5.33	0.014	-225.63	-225.63	0.00	-4.23
8	55	3.49	6.08	0.012	-172.85	-172.85	0.00	-4.01
17	53	10.32	6.59	0.003	-91.08	-91.08	0.00	-3.97
1992
6	46	5.08	9.96	0.015	-133.54	-133.54	0.00	-3.83
1993
1	47	3.51	6.57	0.011	-149.96	-149.96	0.00	-3.85
8	43	5.42	6.40	0.007	-111.69	-111.69	0.00	-3.76
16	60	10.25	8.67	0.013	-116.58	-116.58	0.00	-4.09
17	53	11.11	4.93	0.031	-82.08	-81.05	2.07	-1.90
24	67	9.10	8.36	0.003	-139.99	-139.65	0.68	-3.53

Table 5.2 Results from the application of the travel time dependent mortality model to Snake River steelhead PIT tag data. Each row is a cohort. A negative value for BIC 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	66	16.83	18.61	0.0052	-117.72	-117.72	-0.00	-4.19
6	63	17.75	12.71	0.0000	-108.89	-104.11	9.57	5.42
13	80	26.33	15.85	0.0034	-87.89	-87.89	0.00	-4.38
1990
5	52	14.77	10.91	0.0000	-85.12	-85.12	-0.00	-3.95
10	55	11.35	5.61	0.0581	-91.19	-87.94	6.49	2.49
14	53	13.27	8.32	0.0187	-100.79	-91.91	17.75	13.78
21	50	7.64	9.05	0.0483	-115.68	-115.68	-0.00	-3.91
25	57	20.04	13.01	0.0000	-75.97	-75.97	0.00	-4.04
1991
1	57	8.27	10.51	0.0370	-131.50	-131.50	-0.00	-4.04
9	113	13.87	9.47	0.0145	-179.38	-179.38	0.00	-4.73
12	84	14.17	9.16	0.0167	-128.96	-128.96	0.00	-4.43
18	58	17.14	16.01	0.0000	-97.93	-97.93	-0.00	-4.06
1992
3	64	8.73	6.08	0.0505	-118.79	-118.79	-0.00	-4.16
7	180	13.23	6.12	0.0050	-278.24	-262.08	32.32	27.12
18	42	7.23	15.34	0.0050	-112.78	-112.78	0.00	-3.74
1993
5	57	14.65	11.08	0.0000	-93.72	-93.72	0.00	-4.04
10	217	19.58	12.67	0.0000	-305.57	-301.26	8.64	3.26
15	93	15.67	15.07	0.0000	-164.82	-164.82	-0.00	-4.53
19	84	24.75	18.39	0.0032	-108.48	-108.48	-0.00	-4.43