4.2. Development of basic model

Most travel time experiments involve collecting a group of fish, marking them with a tag and then releasing them as a group from a single release point. The fish then migrate downstream and are collected at a downstream collection site, often a dam (see Figure 4.1). The river is treated as 1-dimensional.

The modeling effort is directed at determining g(t), which is the probability density function for the distribution of arrival times at the downstream collection site. Also of interest is the position of an individual through time. This is denoted by the random variable X(t) with and t > 0. X is usually further restricted by the physical domain of the system being studied. With a release point at X= 0, and a collection site at X = L (as in Figure 4.1), it is assumed that . Also of interest is p(x,t), which is the density function for an individual occurring at position x at time t. If there are N individuals in a cohort, then N · p (x, t) is the population density. Because individuals leave the river reach .

The travel time of fish through a reach can be thought of in two ways, both of which yield equivalent results. In terms of the process X(t), the travel time T is modeled as the first passage of X(t) from the release point to the collection point. In other words,

(4.1)

(Sacerdote, 1988). In terms of the density function p(x,t), the passage of fish through a reach is modeled in terms of the loss of density at an absorbing boundary. In other words, an absorbing boundary is imposed at X = L, and thus

. (4.2)

The theory of boundary crossing has been extensively developed in the mathematical and statistical literature (e.g., Sacerdote, (1988)), where there has been an effort to derive generalizations about a variety of processes crossing different classes of boundaries. A major application in the statistical literature has been the development of sequential analysis (Siegmund, 1985), where rejecting a hypothesis is related to the probability of a process crossing a boundary. In the biomedical literature, there has been a great deal of activity in applying first passage problems to models of neuronal firing (Lanska, 1988). Boundary crossing models have a lot of potential for ecological applications, where many processes are phenological in nature. There have been a few applications in this area, including applications to the timing of instar development in insects (Kemp, et al., 1989) and population extinction (Dennis, et al., 1991).

In the following section I present some general results for first passage problems. I then focus on the specific case where the parameters are constants. The remainder of the chapter is devoted to statistical methods and applications to data.

assumptions

Several assumptions must be made in order to apply the basic travel time model derived in the following section. In later chapters, I expand the model so that these assumptions are not necessary. The first assumption is that the population of fish is independently, identically distributed. Second, the migration process is time homogeneous - there is no diel or seasonal variation in the migratory behavior. Third, each individual has an equal probability of being sampled at the downstream collection site. This means that survival probabilities are identical among the individuals, and the probability of recapture is also identical.

model development

The travel time model begins with the assumption that the spatial distribution of fish through time is described by an advection-diffusion equation. Several people have suggested using this equation to describe the migration of fish (Saila and Flowers, 1969; DeAngelis and Yeh, 1984; Anderson and Schumaker, 1988; Hiramatsu and Ishida, 1989). Since the available data is of the distribution of fish passing through dams (or collected at traps) through time, the advection-diffusion equation is used to derive the distribution of fish passing a fixed point through time. In this temporal form, the model can be compared to data to determine the validity of the model and estimate parameters.

The advection-diffusion equation is expressed as:

. (4.3)

The parameter r determines the rate of downstream movement, and determines the rate of population spreading. As shown in the Chapter 2, with natural boundaries and a point release at x₀ = 0, the unique solution of equation (4.3) is

. (4.4)

It is not realistic, however, to assume unrestricted boundaries in natural systems. In the case of the Columbia River, dams form delineations, and fish populations are sampled as they pass through dams. To account for this, an absorbing boundary is imposed at the site of a dam. As fish in the population pass a dam, they are "absorbed" from the reservoir and passed through the dam. In terms of the model, we assume that fish are released at X = 0 and are collected at X = L. The boundary conditions are now

.

Note that there still is a natural boundary upstream from the collection point. This allows fish to move upstream from their point of release. With these boundary conditions and the same initial conditions as above, the solution to equation (4.3) is now:

(4.5)

(Goel, Richter-Dyn, 1974). Note that this is similar to equation (4.4) but with an added term that accounts for the loss of density at X = L. An example of this distribution is presented in Figure 4.2 with L = 100. Notice that p(x,t) = 0 beyond X = 100, and that as t increases the area under p(x,t) decreases corresponding to the "loss of probability" at X = 100.

Since the loss of density at the absorbing boundary corresponds to fish passage at the dam, we can use equation (4.5) to derive an arrival time distribution. The first step is to determine the probability of remaining in the river, P (L, t), at a given point in time. This is achieved by integrating equation (4.5):

. (4.6)

is the cumulative distribution of the standard normal distribution. To derive a continuous time pdf for the arrival time distribution at X = L for a group of fish released at X = 0, equation (4.6) is differentiated to determine the rate of loss of density:

,

(4.7)

(Cox and Miller, 1965). Plots of this distribution for various values of r and are contained in Figure 4.3.

With and L held constant, as r decreases the mode of the distribution moves to the right, and the distribution flattens out. With r and L held constant, increasing has the effect of moving the mode to the left and flattening the distribution. To determine the probability of arrival at X = L during a discrete time interval one integrates equation (4.7):

. (4.8)

Further complexity can be added to the model by allowing the parameters r and to vary with time in response to such factors as flow conditions and fish maturity.

It is common to reparameterize equation (4.7) with µ = L/r and = L²/². This parameterization eliminates reach length, L, from the equation. Equation (4.7) then becomes:

(4.9)

in the continuous form, and

(4.10)

in the discrete form. With this parameterization, equation (4.9) has been called the "inverse Gaussian" distribution (Tweedie, 1957a, 1957b; Folks and Chhikara, 1978).

In the appendix to this chapter, I present some useful derivations related to first passage models. In appendix 3.a, I show how to derive first passage distributions with the "method of images," an intuitive approach that produces useful results. In appendix 3.b, I show how to derive the passage pdf (equation (4.7)) using a Laplace transform method. In appendix 3.c, I develop a numerical approximation for the discrete version of the passage pdf (equation (4.8)) that overcomes the "exponential overflow" problem involved in computing the equation. In appendix 3.d, I demonstrate a method for generating inverse Gaussian variates, a method I use in the simulations in the following section.