CRiSP1.6 Theory & Calibration Manual: II.3 - Fish Migration

II.3 - Fish Migration

II.3.1 - Theoretical Framework

The movement of fish through river segments is described in terms of an average migration velocity and a stochastic velocity that varies from moment to moment. The migration velocity equation for a group of fish is defined by the Wiener stochastic differential equation:

(41)

where

• X = position of a fish down the axis of the river
• dX /dt = velocity of fish in migration
• r = average velocity of fish in the segment; this is a combination of water movement and fish behavior
• = spread parameter setting variability in the fish velocity
• W(t) = Gaussian white noise process to represent variation in velocity.

Numerical simulation of time vs. distance traveled according to eq (41) is illustrated in Fig. 22.

Fig. 22 Movement along axis of segment vs. time. Shown are mean path, three paths, and 95% confidence intervals. For these simulations, r is set at 10 and is set at 20.

Probability Density Function

The stochastic equation describing fish positions is random. As a result, we must define the probability distribution of fish position over time instead of the actual position, which changes from one fish to another. The probability density function (pdf) of the stochastic differential equation (41) can be defined with a Fokker-Planck (Gardiner 1985) equation:

(42)

where p = p (x, t) is the pdf describing the probability density of the fish being at position x at time t given it was at position x = 0 at time t = 0.

Boundary Conditions

To solve the pdf from eq (42), boundary conditions must be identified. We assume that upon release into a segment a fish can move upstream or downstream in the segment; however, once the fish has reached the downstream end of the segment, at x = L, it will move into the next segment. The next downstream segment may be a confluence or the forebay of a dam. The boundary conditions are:

. (43)

Solution

The solution to the partial differential equation (eq (42)) describing the probability distribution of fish in a river segment is a probability density function for the fish. This is:

. (44)

An example of the distribution of p with respect to x for different times is illustrated in Fig. 23. The pdf in the figure can be interpreted as probability of where a fish is in the river at any time. It can also be interpreted as the distribution of a group of fish in a river segment if they have experienced no predation. Notice that the group moves down the segment and spreads over time. At the absorbing boundary representing a dam, the fish enter the boundary regions and pass through to the next segment. Note that the equation cannot define the deterministic path of fish with time.

Fig. 23 Plot of eq (44) for various values of t where r = 5, = 8, and L = 100.

Passage Probability

The probability that a fish that entered the river segment at time t_i is still in the river segment at time t_j is obtained by integrating eq (44) over reservoir length. This is expressed:

(45)

where

• = cumulative distribution of the standard normal distribution
• L = segment length
• r = average migration velocity through the segment (developed in Active Migration Equation section below).

The probability of a fish leaving a segment between time t and t + t is:

. (46)

This is the arrival time distribution at the point L, which is generally a dam or river confluence. The number of fish exiting each river segment is defined by eq (46).

Fig. 24 Fish distribution, p (x, t), at t_j and t_j-1. Size of the shaded area represents probability of fish leaving the segment over the interval t_j - t_{j - 1}

II.3.2 - Migration Models

Active Migration Equation

The goal of the active migration equation is to be flexible enough to capture a variety of migratory behaviors without requiring an excessive number of parameters to fit. The equation has a term that relates migration rate to river velocity and a term that is independent of river velocity. Both terms have temporal components, with migration rate increasing with time of year.

The flow independent migration rate is driven by two parameters, _min and _max. _min is the flow independent migration rate at the time of release (T_RLS), and _max is the maximum flow independent migration rate. In eq (47) below, it is easier to express the equation in terms of the regression coefficients ₀ and ₁, with the following relations:

. (47)

With _max > _min, the fish have a tendency to migrate faster the longer they have been in the river. This tendency can be "turned off" by setting _max = _min (that is, ₁ = 0). Also, flow independent migration can be turned off entirely by setting _max = _min = 0 (that is, ₀ = ₁ = 0).

The magnitude of the flow dependent term is determined by _FLOW. This term determines the percentage of the average river velocity that is used by the fish in downstream migration. This term has a seasonal component determined by the T_SEASN term, which is expressed in terms of julian date. This has the effect of the fish using less of the flow early in the season and more of the flow later in the season. Values of T_SEASN that are relatively early in the season mean that the fish mature relatively early. The parameters determine how quickly the fish mature from early season behavior to later season behavior. Setting ₂ equal to 0 has the effect of "turning off" the flow/season interaction, resulting in a linear relationship between migration rate and river flow.

The full migration rate model (Zabel, Anderson and Shaw 1998) is:

(48)

where

• r(t) = migration rate (miles/day)
• t = julian date
• 's = regression coefficients, described above
• = average river velocity during the average migration period
• ₁ , ₂ = slope parameters
• T_SEASN = seasonal inflection point (in julian days)
• T_RLS = release date (in julian days).

Both the flow dependent and flow independent components of eq (48) use the logistic equation (term in brackets). The logistic equation is expressed in general as:

. (49)

This equation has a minimum value of ₀ and a maximum value of ₀ + ₁. T₀ determines the inflection point, and determines the slope. Fig. 25 contains example plots of the equation and demonstrates how varying a parameter affects the shape of the curve.

The logistic equation is used instead of a linear equation because upper and lower bounds can be set. This eliminates the problem of unrealistically high or low migration rates that can occur outside observed ranges with linear equations. Also, for suitable parameter values, the logistic equation effectively mimics a linear relationship.

Fig. 25 Examples of the logistic equation (eq (49)) with various parameter values. In all four plots, the parameter values for the solid curves are: ₀ = 1.0, ₁ = 2.0, = 0.2, and T₀ = 20. In the upper left plot ₀ is varied, and ₁ is varied in the upper right. In the lower left plot, is varied, and T₀ is varied in the lower right.

Other Migration Model Options

As mentioned in the previous section, simpler models are nested within the full migration model. For example, setting ₁ = 0 removes the flow-independent experience term. The resulting model:

(50)

has only the flow-dependent experience factor, which assumes that fish migrate more rapidly later in the season by migrating in high flow regions of the river and/or by spending a greater portion of the day in the river rather than holding up along the shore.

By also setting ₂ = 0, all experience related migration rate increases are removed. The resulting model:

(51)

assumes a linear relation between migration rate and river velocity. Other combinations of assumptions are also available in CRiSP.1.

Velocity Variance

The spread parameter sets the variability in the migration velocity. This term represents variability from all causes including water velocity and fish behavior. In CRiSP.1, ² = V_var which is the variance in the velocity. This can vary on a daily basis.

Variance in Migration Rate

Variance in the migration rate is applied for each release, thus randomly representing differences in the migration characteristics of each release. Although studies suggest differences in migration can partly be attributed to differences in fish condition and perhaps stock to stock variations, these factors have not been sufficiently identified so their contribution to differences in travel time is randomized. The equation is:

(52)

where

• r(t) = determined from eq (48)
• V(i) = variance factor that varies between releases only.

V(i) is drawn from the broken-stick distribution. The mean value is set at 1, representing r(t), and the upper and lower values are set with sliders in Migration Rate Variance window in the Behavior menu.

Pre-smolt Behavior

In some cases, fish are released into the river before they are ready to initiate migration. This may be the case with hatchery releases or fish that are sampled and released in their rearing grounds. The probability of moving from the release site is determined by two dates, smolt_start and smolt_stop:

. (53)

In other words, the probability of initiating migration is 0 before smolt_start, 1 after smolt_stop, and linearly increasing with time between the two values. Fish are subjected to predation prior to the onset of smoltification. The predation activity coefficient for pre-smolt mortality uses the activity coefficient for the first day of smoltification t = 1.

Implementing the Travel Time Algorithm

The basic unit of the travel time algorithm is a reach of river between two nodes, where a node is a dam, confluence of two rivers, or a release point (Fig. 26). The travel time algorithm passes a group of fish from node to node and determines the distribution of travel times from an upstream node to the next downstream node.

Fig. 26 Schematic diagram of a river system. Arrows represent the migration of release groups 1 and 2 through reaches. At the confluence, groups are combined for counting purposes only, i.e they still exhibit their unique migration characteristics.

CRiSP.1 groups fish according to user preference. The user defines species (and stocks, if desired) in the columbia.desc file¹ and associates behavioral characteristics with each species through the user interface or the yearly input data file. For instance, the user may decide that all chinook 1's should be treated identically or that wild and hatchery stocks should be treated separately. All releases that are treated similarly are referred to as a release group, except for the random selection of a migration rate variance.

During one iteration of the travel time submodel, fish from a release group pass through a reach. The input to CRiSP.1 is the number of fish from the release group that are ready to depart a node during the time interval. This input group is passed to the next node downstream with the travel time distributions determined by eq (45) and (46). Fig. 27 demonstrates a single iteration of the travel time algorithm.

Fig. 27 Plots of a single iteration of the travel time algorithm through a single reach. 1000 fish released at the upstream node are distributed through time at the next downstream node. Parameter: r = 10, = 8, L = 100.

¹ As configured, the columbia.desc file defines three species: chinook 1 = spring (yearling) chinook,
chinook 0 = fall (subyearling) chinook, and steelhead.

CRiSP1.6 Theory & Calibration Manual: II.3 - Fish Migration