7.4. Application to radio-tracking data

The study site is the John Day reservoir on the Columbia River in front of the John Day Dam (Giorgi, et al. 1986). At this site, the river is relatively straight and is approximately a kilometer wide. The study was conducted during the Summer of 1983. Individual fish were collected at the John Day Dam, radio tagged, and released 6.3 kilometers upstream from the dam. Two boats followed the individuals with the fish's position being noted by hand held receivers at approximately 20 minute intervals. The individuals were followed for up to eight hours with radio tracks up to six kilometers long. 17 chinook and 8 steelhead were released and followed. Many of the individuals had tracks that were too short for adequate analysis. I chose to analyze the tracks of the three chinook and two steelhead that had at least 19 "fixes" and track durations of at least six hours. Since the primary interest in these data is the downstream movement of the individuals, I ignored the horizontal movements of the fish and converted the data to downstream displacements. Figure 7.2 contains plots of downstream displacement versus time for the five individuals.

results

The results of the data analysis are contained in Table 7.1. For two out of three of the chinook and one out of two steelhead, normality is not rejected based on Liliefor's test. For both of these chinook, though, zero or negative correlation is rejected at the = 0.05 level. For the steelhead (steelhead 170), zero or negative correlation is not rejected, and thus the two properties of the Wiener process are not rejected for this individual.

Table 7.1 Results from the radio tracking data analysis. The test on the correlation coefficient is only conducted for the individuals where normality is not rejected based on Liliefor's test. For Liliefor's test, normality is rejected for low p-values (typically p < .05). Based on the BIC value, the null model (the Wiener drift model) is rejected for positive values. Other details of the analysis are contained in the text.
track information			Wiener process					O-U based model		likelihoods
Individual	# of fixes	length (min.)	parameters		Liliefor's	correlation				ratio	BIC
					T		p
chinook 627	24	445	9.59	68.07	0.145b	0.40	0.032	19.37	0.080	4.847	1.78
chinook 633	19	487	7.57	35.00	0.208e	0.61	-	7.64	0.064	7.630	4.80
chinook 876	21	399	13.58	58.91	0.175c	0.60	0.003	15.94	0.058	5.794	2.85
steelhead 170	24	453	9.87	34.55	0.136a	-0.026	0.547	266.74	119.06	0.000	-3.09
steelhead 667	20	529	7.52	57.19	0.281f	0.34	-	18.20	0.157	0.650	-2.24

For the three chinook, the O-U displacement model is supported over the Wiener drift model based on BIC values. For the two steelhead, the opposite is true, and the simple Wiener drift model is supported.

With only 5 fish analyzed, it is not possible to determine whether either of the models is "appropriate". For the chinook, the Wiener drift model appears to be inadequate, with the results of the correlation test and the likelihood ratio comparisons indicating that some type of correlation structure is required to accurately model the data. More analysis is required to determine if the O-U displacement model is consistent with the chinook's behavior, though. For the steelhead, one of the fish's behavior is consistent with the Wiener process, as the normality and independence properties are not rejected. Again, more fish will need to be analyzed to make conclusive statements.

It should be emphasized that the results are dependent on time scale. In this case, the average time increment is approximately 20 minutes. At a shorter time scale, correlation may be important for the steelhead, and at a longer time scale, the correlation may cease to be important for the chinook.

discussion

Although I have not encountered any studies that have applied the O-U process to the movements of individuals, it appears to have promise. The conditional distribution of the displacement of an individual given the last time period's displacement is easily formulated. Also, the theory can accommodate unequal time intervals.

There are two features of the O-U process that are consistent with the behavior of migrating juvenile salmon. The first feature of the O-U process is that if a particle is moving with a certain velocity, there is a tendency to remain at that velocity in the short run. This feature is very appropriate for dispersing organisms. Another feature of the O-U process is that there is a tendency to bring particles back to their mean velocity - the further a particle's velocity is from the mean velocity, the greater the tendency. This is also a desirable property. Migrating juvenile salmon appear to undergo a relatively passive migration process (Smith, 1982), expending little energy as they are carried downstream with the current. There are reasons, however, for individuals to move out of this "low energy" state (e.g, predator avoidance, feeding behavior) and actively move in either the upstream or downstream direction. Because of the swimming energetics of juvenile salmon, the fish cannot maintain this energy expenditure for an extended period of time before they must return to the "low energy" state and replenish their oxygen debt (Brett, 1965). This is reminiscent of the O-U process.

While it is improbable that migrating salmon are strict adherents to the O-U process, there does seem to be some value in applying the model. On the time scale of days and kilometers, the Wiener process with drift is a useful model of migrating juveniles and is being used to predict their arrival times at dams (chapter 4). Looking at migratory process on the time scale of hours and meters is a valuable exercise because it can lend validity to the migration model at the longer time scales.