Unlike steelhead and salmon, sea-run cutthroat continue to feed during their freshwater residence and, thus, are able to retain their physical condition for extended periods in the rivers. While they will certainly eat salmon eggs when they are available, the common explanation that they are "following" the salmon doesn't begin to explain the timing of their migration. In the Stillaguamish system, for instance, there are usually fishable numbers of cutthroat in the river by the middle of August, long before there are significant numbers of salmon and none which have yet begun to spawn. Before construction of the dams on the Cowlitz, locals expected the first runs of "harvest trout" by the Fourth of July.

Back in the day when I still killed a cutthroat or two on occasion, I found their stomach contents to be more likely to consist of October caddis pupae and other insect foods, sculpins and other baitfish, even when salmon were present and their eggs were available.
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PS