8. Vertical distribution models
8.1. Introduction

Several types of models have been formulated to describe distributions of populations in response to environmental stimuli. Clark and Levy (1988) use dynamic programming to model the vertical distribution of sockeye salmon in Lake Babine, British Columbia. In their model the vertical position of an individual is determined by a trade-off between feeding and predator avoidance. Another approach is to model dispersal as a diffusion process with the diffusion parameter a function of some environmental stimulus (Skellam, 1973; Okubo, 1986). Dobzhansky, et al. (1979) used this approach to model the dispersal of fruit flies in a heterogeneous habitat. The chemotaxis model originally developed by Keller and Segel (1971) has received many applications to cellular systems. In this model, a component of organism movement is based on random dispersal, and a component is based on movements dictated by some environmental gradient. There have been few applications of this model to "higher" organisms, possibly because of the difficulty in modeling the organism's response to the gradient. Kareiva and Odell (1987) present one of the few examples, with the distribution of predators (lady bugs) influenced by a gradient of prey (aphids) density.

In this chapter, I apply a chemotaxis type model to the vertical distribution of juvenile salmonids entering the forebay of a dam. The distribution of fish entering the forebay has direct consequences on their passage route through the dam. The main downstream passage routes through dams are the spillway, the turbines, and the fish bypass system; each of these routes has a different mortality rate. The vertical position of a fish is particularly important in determining whether it will pass through the bypass system (higher in the water column) or the turbines (lower in the water column); obviously the bypass system is a more favorable route.

The vertical distribution of fish in the water column can be observed with hydroacoustics (Dawson, et al., 1984a, 1984b). Figure 8.1 shows data for both daytime and nighttime distributions of juvenile salmonids entering the forebay of Lower Monumental Dam in April and May, 1985 (Johnson, et al., 1985). Each plot represents composite distributions over a 5 day period. Some observations from these data are: 1) clear differences exist between daytime and nighttime distributions, indicating that environmental cues may be important; 2) there appears to be consistency in the distributions through time, indicating that there are potential trends to be modeled; and 3) the distributions have quite a bit of spread, indicating that a random dispersal element may be important. One drawback of this type of data is that different stocks or species cannot be distinguished. There appear to be two types of fish in the daytime data - one residing lower in the water column and one residing higher in the column that becomes more prevalent later in the season. The two main groups of juvenile salmonids passing Lower Monumental Dam during this time of year are steelhead and spring chinook. A study by Smith (1974) in the forebay of Lower Monumental Dam showed that during the daytime, steelhead tend to be surface oriented, and chinook tend to migrate lower in the water column. In some cases, it will be possible to compare hydroacoustic data to dam passage counts that distinguish among species.

Two gradients that may be affecting the vertical distributions are light and pressure. Both of these gradients are measurable and are somewhat smooth, making the system amenable to modeling.