9.2. Summary by chapter

Chapter 2 discusses models of dispersing animals. Models based on an advection-diffusion equation are applicable to migrating populations. The advection term determines the directed movement of the population, and the diffusion term describes the spreading of the population. The diffusion term can be modified to reflect features such as spatial heterogeneity and density dependence. Waiting time models, which determine the time until an event, also capture certain features of dispersing populations such as survival and migrational delay.

The third chapter contains the statistical methods used in comparing models to data. The primary parameter estimation method I use is maximum likelihood, which can be employed analytically or numerically. Goodness-of-fit methods differ depending on whether the data are continuous or discrete and whether parameters are being estimated. I use goodness-of-fit tests based on the chi-square distribution and on the empirical density function. It is often useful to discriminate among alternative models of varying complexity. I present several methods to do this, all based on comparing likelihoods.

The fourth chapter develops a two parameter model of the travel time of fish through a reservoir based on an advection-diffusion equation. One parameter determines the downstream migration rate and one determines the rate of population spread. The model accommodates discrete or continuous time data, and I apply it to several data sets of both types. The model successfully describes travel time distributions of run-of-the-river spring chinook, but describing steelhead and fall chinook is more problematic.

The fifth chapter expands the travel time model to incorporate more complex behavior. Travel time dependent mortality is modelled with a constant hazard rate. This type of mortality does not have much effect on the shape of the travel time distribution, and the data analysis bears this out. Next, a delay term based on a Poisson process is incorporated into the travel time model. Migrational delay can occur as fish hold up before passing a dam or before migration is initiated. Several radio-tracking studies confirm that dam delay occurs for chinook, but this delay is not detectable for Snake River run-of-the-river spring chinook travel time data. The delay term improves the model for Snake River steelhead (based on likelihood ratios), but the results are inconsistent and probably not biologically relevant. For mid-Columbia fall chinook, a delay term, interpreted as a delay before the initiation of migration, substantially improves the travel time model. Finally, I present a hierarchical sequence of models to describe the variation in migration rates for similar groups of fish migrating in a river reach. These regression models are based on date of release and average river flow. A four parameter model, with linear flow relationship and a nonlinear time relationship, worked best with several groups of run-of-the-river spring chinook. The results from the regressions were used to predict travel times for an independent data set.

In chapter 6, I allow for population heterogeneity, with migration rates of individuals related to the factors fish length, date of release, river flow, and river temperature. For the run-of-the-river spring chinook and steelhead, fish length is not an important factor, but it is important for mid-Columbia fall chinook. When several factors are applied sequentially for Snake River fall chinook and mid-Columbia sockeye, date of release, fish length, and average river flow are all important in determining migration rate, while river temperature is not.

In chapter 7, downstream migration is considered in terms of individual movements. I examine two models, one based on the Wiener process that has independent increments and one based on the Ornstein-Uhlenbeck process that incorporates correlation among movements. The models are compared to radio-tracking data, and correlation is determined to be important at the observed time scale (approximately 20 minutes) for the chinook but not for the steelhead.

The vertical distribution of fish in a water column is described in terms of a chemotaxis-type model in chapter 8. In this model, an individual's position is determined by random movement and reaction to an environmental gradient. I apply the model with a light intensity gradient to hydroacoustic data from the forebay of Lower Monumental Dam on the Snake River. The correspondence between the model and data is excellent.