I. Introduction *

I.A. Report Objectives *

I.B. Acknowledgements *

I.C. Fundamental Assumptions *

I.D. Study Site *

I.E. Conventions *

II.A. Habitat Initialization *

II.B. Fish Initialization *

III. Habitat Model *

III.A. Cell boundaries and dimensions *

III.B. Daily Flow and Temperature *

III.C. Depth and velocity *

III.D. Velocity shelter availability *

III.E. Distance to hiding cover *

III.F. Food Production and Availability *

IV. Fish Model *

IV.A. Habitat Selection (Movement) *

IV.C. Survival *

V. Model Schedules *

V.A. Habitat Actions *

V.B. Fish Actions *

V.C. Observer Actions *

V.D. Complete Schedule *

VI. Calibration *

VII. Method Details *

VII.A. Fish Feeding and Energetics *

VII.D. Computer Implementation *

VIII. References *

IX. Input Parameter Index *

X. Input Files *

X.A. Cell Data *

X.B. Cell Hydraulic Data *
 
 

Fig. 1. Sensitivity of EM to the time horizon. EM is shown as a function of depth and velocity for adult trout (15 cm FL) with distance to hiding cover of 5 m and using a velocity shelter. *

Fig. 2. Simulated capture width and observed territory diameters. *

Fig. 3. Observed relation between mean column and focal velocities. *

Fig. 4. Variation in food intake with velocity for each feeding mode. *

Fig. 5. Variation in growth with velocity for each feeding mode. *

Fig. 6. Observed critical swim speeds as a function of fish length. *

Fig. 7. High velocity mortality function. *

Fig. 8. Stranding mortality function. *

Fig. 9. Poor condition mortality function. *

Fig. 10. Terrestrial predation- depth factor. *

Fig. 11. Terrestrial predation- fish length factor. *

Fig. 12. Terrestrial predation- feeding time factor. *

Fig. 13. Terrestrial predation- velocity factor. *

Fig. 14. Terrestrial predation- hiding cover factor. *

Fig. 15. Aquatic predation- depth factor. *

Fig.16. Aquatic predation- fish length factor. *

Fig.17. Aquatic predation- temperature factor. *

Fig.18. Overall survival vs. depth and velocity, 3 cm trout. *

Fig. 19. Overall survival vs. depth and velocity, 5 cm trout. *

Fig. 20. Overall survival vs. depth and velocity, 10 cm trout. *

Fig. 21. Overall survival vs. depth and velocity, 20 cm trout. *
 
  .
 
 
 
 

Table 1. Parameter values for initial fish weight. *

Table 2. Parameter values for identifying destination cells. *

Table 3. Initial population characteristics for calibration. *

Table 4. Ending population characteristics used as calibration targets. *

Table 5. Calibration results. *

Table 6. Parameter values for drift feeding. *

Table 7. Parameter values for maximum consumption, allometric function. *

Table 8. Parameter values for maximum consumption, temperature function. *

Table 9. Parameter values for respiration. *

Table 10. Parameter values for high velocity mortality. *

Table 11. Parameter values for stranding mortality. *

Table 12. Parameter values for poor condition mortality. *

Table 13. Parameter values for terrestrial predation mortality. *

Table 14. Parameter values for aquatic predation mortality. *
 
 

1. Introduction
1. Report Objectives



This report documents the formulation of an individual-based model (IBM) of coastal cutthroat trout in Little Jones Creek, California. The predictions of IBMs are potentially dependent on all of the detailed assumptions, equations, parameter values, and schedules used in the model, and on how these are implemented in computer code. The primary objective of this report is to fully document the methods and parameters that were implemented in the model software. This document describes how the model is initialized by defining the habitat and starting fish populations, then describes the methods used to simulate habitat and fish. The model schedule, which has an important effect on simulation results, is specified explicitly.

The report is organized to allow readers to develop a general understanding of the model before encountering all its details. Sections II through V provide specific information on the major components of the model, but ignore some details for clarity. Full descriptions of the more complicated details of the model are provided in Sect. VII. The input files documenting the habitat data driving the model are provided at Sect. X.

The computer software that implements this formulation (including parameter and input file formats) is described in a separate User Guide (Railsback et al. 1999a). This formulation document uses the same input parameter names as the computer code and compiles them into a master list of model parameters (Sect. IX), but does not otherwise link the formulation to its software implementation.

2. Acknowledgments



The Little Jones Creek cutthroat trout model is being developed and tested under Research Joint Venture Agreement PSW-99-007-RJVA between the Redwood Sciences Laboratory, U. S. Forest Service, and Lang, Railsback & Assoc. (LRA). Collaborative funding for software development and model testing has been provided to LRA by EPRI, the Electric Power Research Institute. Initial funding for the the program was provided by Pacific Gas and Electric Company (PG&E) and Southern California Edison Company (SCE) via a research grant to the Humboldt State University Foundation; the principal investigator for this grant is Dr. Roland Lamberson, Department of Mathematics, Humboldt State University.

A number of concepts used in this model were adapted from approaches and experience developed under the CompMech research program funded by EPRI, PG&E, and SCE and conducted by Oak Ridge National Laboratory (e.g., Van Winkle et al. 1996, 1998). Our approach to fish movement and overall modeling philosophy has been heavily influenced by the Swarm program (www.swarm.org), the Swarm user community, and Glen Ropella and Chris Langton of The Swarm Corporation.

The trout model software is developed and maintained by Steve Jackson, Jackson Scientific Computing, McKinleyville, CA.

3. Fundamental Assumptions

• A one-day time step is used for all model processes: stream flow and temperature, and fish movement, foraging and growth, and mortality.
• The spatial resolution is several square meters. Habitat is modeled as rectangular cells with dimensions typically one to several meters across the stream and several meters in the longitudinal direction. Cell borders are chosen to minimize the habitat variation within cells while capturing the stream’s natural variation among cells.
• The external variables driving the model over time are stream flow, water temperature, and food availability.
• Flow (but not depth and velocity) and temperature are constant over space. We assume that food availability parameters (the mass of drift food per volume of stream water and mass of stationary food per stream area) are constant over both space and time.
• Fish compete for food, not habitat space. We assume that the food intake of a fish is a function of (a) the total food available in a habitat cell (a function of cell size, depth, and velocity) and (b) how much food is consumed by more dominant fish in the cell. This allows fish densities to vary realistically with food availability and the need to avoid mortality risks.
• The fish activity driving trout habitat selection is feeding. Only habitat characteristics related to foraging are simulated. We do not, for example, simulate habitat for hiding and resting when not feeding; we assume all fish can find such habitat.
• Trout move in response to spatial variation in food intake and mortality risks. We assume fish are "intelligent" enough to (1) correctly predict their food intake and survival probabilities at alternative movement destinations and (2) determine which destination offers the highest fitness (Sect. IV.A.2.c). We do not assume or simulate any learning by fish.

1. Study Site



Little Jones Creek is a third-order tributary to the Middle Fork of the Smith River, in Del Norte County, in the northwest corner of California. Resident cutthroat trout are the only fish present; a barrier at the mouth of the creek prevents upstream migration of anadromous fish. Harvey (1998) provides a more complete description of the watershed. The trout model was applied to a reach of Little Jones Creek approximately 1000 m upstream of the mouth.

2. Conventions



This section documents important conventions used in formulating, documenting, and coding the model. These conventions are established as a way to avoid errors resulting from confusion in units, variable names, etc.

1. Units



This formulation uses length units of centimeters (cm), weight units of grams (g), and temperature in degrees centigrade (°C). Stream flow is in units of cubic meters per second (m³/s).

Because the model uses a daily time step, most time-based parameters use day as the time unit. However, there are several exceptions to this convention; for example, flow and velocity variables are per second. Most food intake calculations use hourly rates because the number of hours per day that fish feed is variable.

Fish lengths are evaluated as fork lengths (from nose to the fork in the tail fin) throughout this model formulation. All weight variables for fish and prey (food) use wet weight.

2. Parameter and variable names



So the parameter names in this report match those in the model code’s input files, we follow the naming conventions used in the Swarm simulation software used to code the model (Railsback et al. 1999a). Variable and parameter names typically are made by joining several words. The first word starts with a lower-case letter, and capital letters are used at the start of each subsequent word (e.g., "fishWeightParamA"; see the list of parameter names at Sect. IX).

We use the convention of starting input parameter names with the kind of object that uses the parameter. These objects include fish, habitat cells, and fish mortality sources. Consequently, most parameters start with the words "fish", "cell", "hab", or "mortFish". This convention is not strictly followed for variables calculated by the model instead of read as input.

3. Survival probabilities and mortality sources



A number of factors can cause fish or fish eggs to die in our model. We refer to these factors as "mortality sources". However, we model mortality using survival probabilities, the daily probability of remaining alive. (The term "mortality rate" typically is used to mean the daily probability of dying, equal to one minus the survival probability.) Although we use the word "mortality" in parameter names and our text, the model formulation bases all mortality-related calculations on survival probabilities. This convention simplifies computations and reduces the chances of error: the cumulative survival of several mortality sources is calculated simply by multiplying the individual survival probabilities together.

4. Dates, days, and fish ages



This model uses input in the "MM/DD/YYYY" format (e.g.: 12/7/1999) for dates. The software converts such input to an internal date format that automatically accounts for leap years.

We follow the convention that fish are age 0 when born and the age of all fish is incremented on each January 1.

5. Habitat cell conventions



We developed our own conventions for describing the rectangular cells used to model habitat. These conventions were designed to correspond with computer graphics conventions (so habitat is mapped correctly by our software) and to correspond with terms used in popular instream flow models.

1. "Transects" and "Cells"



Habitat is modeled as rectangular, two-dimensional, depth-averaged "cells"; depth and velocity are modeled for each cell and assumed uniform within the cell. Cells fall along a "transect", a straight row of cells across the stream and floodplain perpendicular to the direction of flow. (The word "transect" also commonly refers to a line across the channel along which depth and velocity is measured. We do not necessarily use this data collection approach so we use "transect" to refer to a row of cells.) Cells on the stream margins will be dry at low flows; the number of habitat cells that have water in them and are usable by fish varies with flow.

2. X and Y dimensions



The hydraulic model we use is one-dimensional so our model assumes the river is straight with all velocities in one direction. The X and Y values referred to here are coordinates (in cm) of cell boundaries.

The X dimension is defined to be in the downstream-upstream direction. The origin (X = 0) is at the downstream end of a reach, so water flows from right to left on an X-Y plot. The Y dimension is across the channel, or along a transect. To correspond with computer graphics, which place the origin (X,Y = 0,0) at the top left of the screen, we define Y to be zero on the left bank facing upstream.

3. Distances between cells

Some calculations in the model require values for the distance between two cells (e.g., for finding the cells that are within a fish’s maximum movement distance). Because cells are two-dimensional, there is no single distance between two cells; as a convention, we evaluate the distance between two cells as the straight-line distance between the centers of the cell.

1. Model Initialization



This section describes the methods used to initialize the habitat and fish populations when each new model run is started. Although this section mentions some of the input types and files, complete documentation of file and input types is provided only in the separate User Guide.

A. Habitat Initialization

A model run starts by reading in the habitat characteristics that do not change during the simulation. These characteristics are the location and dimensions of habitat cells, the values of habitat cell variables that do not change with time (the fractions of cell area with velocity shelters, distance to hiding cover), and the lookup tables used to calculate daily depth and velocity in each cell. (Habitat cells and these variables are described in Sect. III.)

B. Fish Initialization

We build the initial fish population from input data giving the number, mean length, and variance in length for each age of each species. Lengths (cm) for each fish are drawn from a normal distribution defined by the input mean and variance. If the length drawn for a fish is less than half the specified mean, a new length is drawn. The weight (g) of each fish is calculated from its length using a length-weight relation whose parameters are input. (This relation is also used to calculate growth in length from a change in weight; see Sect. IV.B.)

…………………………...….1

Note that this is not simply an observed length-weight regression relationship. It is intended to be a site-specific length-weight relation for fish in good condition.

Parameter values for this relation were calculated from fish observations made at the Little Jones Creek study sites. They were based on observations of fish in relatively good condition, with weight per length that is slightly higher than average. We determined these parameter values using fish electroshocked for diet studies at a number of dates throughout 1998 and 1999. Standard condition factors (100,000 times weight over length cubed) were calculated for each observed fish. We used log-log regression to estimate the parameters from the length and weight of fish that had standard condition factors between 1.1 and 1.3.
 

Table 1. Parameter values for initial fish weight.

Parameter	Definition	Units	Value
fishWeightParamA	Length-weight relation multiplier	g/cm3*	0.0124
fishWeightParamB	Length-weight relation exponent	(none)	2.98

^*Approximately: this is an empirical parameter whose units vary with fishWeightParamB.

By initializing fish using the length-weight relation used in the growth routine, we assume that fish are initially in good condition.

Each fish’s location is assigned randomly to one of the habitat cells. The first day’s movement simulation puts the fish in reasonable starting locations. However, we do not assign fish to cells where the depth is zero because some small fish may have a maximum movement distance (Sect. IV.A.2) too small to allow them to find reasonable habitat on the first move.

II. Habitat Model

The habitat component of our model simulates hydraulic conditions (depth and velocity), temperature, and food availability (a function of food production and the number of fish competing for it).

We follow the lead of preceding habitat models (Bovee 1982) and IBMs (Van Winkle et al. 1998) by representing stream habitat as a collection of rectangular, two-dimensional, depth-averaged cells. However, we have made the important improvement of carefully designing the habitat cell sizes and placement to avoid resolution errors common in other models. In modeling habitat at our Little Jones Creek site, we considered these factors in representing habitat as cells.

• Cell sizes should be appropriate for the scale over which fish use and select habitat. There is little conclusive literature on the most appropriate scale for fish–habitat interactions but it seems clear that fish select habitat over longer distances than the spatial resolutions typically used in PHABSIM habitat suitability studies.
• The very small or narrow cells typically used in PHABSIM models likely would induce major edge-effect errors in an IBM where we model competition among multiple fish for the resources available within each cell.
• Cells should be placed to capture the full range of hydraulic variation and complexity of the stream being modeled. Capturing this variation is much more important to the model’s accuracy than any difficulties that may be encountered in calibrating the hydraulic model to such habitat (Railsback 1999).
• Cells should be placed to minimize habitat variation within each cell, because the model assumes habitat is homogeneous within cells.

1. Cell boundaries and dimensions



All the cells on one transect have the same length in the X (upstream-downstream) dimension, but vary in width- the Y (across channel) dimension. For each transect, the user provides the X coordinate for the upstream end of the cells. For each cell, the user provides the Y coordinate of the cell’s right boundary. These coordinates are measured in the field.

2. Daily Flow and Temperature



At the start of each daily time step, the model reads in the day’s river flow (m³/s) and temperature (° C). Flow is used only to calculate the depth and velocity in each cell.

We assume flow and temperature are constant over space when modeling relatively short reaches that do not contain tributaries, including our sites at Little Jones Creek. We also neglect diurnal variation in them. At Little Jones Creek diurnal variation in temperature is low; measurements made between May, 1998, and May, 1999 show an average diurnal variation of only 1.1° with a maximum of 2.6°.

3. Depth and velocity



The depth and velocity of each habitat cell (and the number of cells that have water) vary with the daily river flow rate. The hydraulic models we use neglect changes in channel direction and cross-channel flows, so the velocity has only one component, in the X direction.

To take advantage of existing stream hydraulic modeling software and avoid having to include hydraulic simulations in our model, we import lookup tables of depth and velocity, as a function of stream flow, for each cell. These depth and velocity lookup tables (Sect. X.B) are generated by external hydraulic simulation software, the RHABSIM package (TRPA 1998).

Daily velocities and depths for each cell are interpolated from the values in the lookup table. Both depth and velocity are assumed by the hydraulic models to be logarithmically related to flow, so we use log interpolation. For any flows below the lowest in the lookup table, we extrapolate the depth and velocity downward from the lowest two flows. Likewise, for flows above the highest in the lookup table, we logarithmically extrapolate depth and velocity upward from the highest two flows.

4. Velocity shelter availability



We assume that a constant (over time) fraction of each habitat cell has velocity shelters available for use by drift feeding fish (Sect. VII.A.6); shelters also affect the high velocity mortality risk (Sect. VII.B.1). The fractions of each cell with velocity shelter are provided by the user as input (variable "cellFracShelter"). These fractions should include any part of the cell with complex hydraulics that could be used by trout to reduce their swimming energy. Such shelters could be provided by boulders, cobbles or other substrates that induce roughness in the bottom, woody debris, roughness in the banks or bedrock channel, or by adjacent cells with near-zero velocities.

5. Distance to hiding cover



The model includes a habitat input variable that is an estimate of how far a fish in the cell would have to move to find hiding cover. This variable (cellDistToHide, m) is used in the terrestrial predation mortality model (Sect. VII.B.4).

6. Food Production and Availability



The amount of food available to fish is a very important habitat variable, probably more important than flow or temperature in determining fish population abundance and production except under extreme conditions. Unfortunately food availability is poorly understood: although such studies as Morin and Dumont (1994) and Railsback and Rose (1999) indicate that food availability or consumption can vary with factors including flow, temperature, trout abundance, and physical habitat characteristics, there is little information available as a basis for modeling how it varies over time and space at scales relevant to individual-based models. Modeling food production is also complicated by the multiple sources of food available to fish; we assume all food is either (a) "drift" food, moving with the current; or (b) "search" food that is relatively stationary and must be searched out by the fish. Both drift and search food may originate with benthic (stream bottom) production or with drop-in from above the stream.

Our model assumes fish compete for the food available in each habitat cell. Therefore, the habitat model includes methods to determine (a) how much food is produced each day, and (b) how much of the food production has been consumed by the fish in the cells.

1. Production



In the absence of established models of trout food production, we developed models that are simple yet mechanistic and easily calibrated using observed trout growth and survival (Sect. VII.A.9). We make the simple assumption that the concentration of food items in the drift ("habDriftConc", grams of prey food per cm³ of stream water) and the production of stationary food items in the stream benthos or overhead drop-in available via the search feeding strategy ("habSearchProd", grams of prey food produced per cm² of stream bottom per hour) are constant over time and space.

Our feeding formulation allocates the amount of drift and search food available per hour in each cell among fish (Sect. VII.A.8). The total amount of search food available ("searchHourlyCellTotal", g/h) is simply the cell area times habSearchProd. The total drift food available in a cell ("driftHourlyCellTotal", g/h) is a function of the cell’s cross-sectional area (perpendicular to the flow) and velocity, the drift food concentration parameter habDriftConc, and a drift regeneration parameter "habDriftRegenDist":

driftHourlyCellTotal (g/h) = 3600 s/h ´ cell width (cm) ´ depth(cm) ´ velocity (cm/s) ´ habDriftConc (g/cm³)

´ [cell length (cm) / habDriftRegenDist (cm)]…………………………………...2

The last term in this equation has two purposes. First, it simulates the regeneration of prey consumed by drift-feeding fish. Second, it makes the amount of drift food available per cell area independent of the cell’s length; without this term, five transects with cells 2 m long would have five times the food availability of one 10 m-long transect.

The parameter habDriftRegenDist should theoretically approximate the distance over which drift depleted by foraging fish is regenerated. This parameter is actually used to calibrate habitat selection and survival of starvation (Sect. VII.A.9); smaller values of habDriftRegenDist provide higher amounts of food in a cell. The parameter habDriftConc also affects the amount of food in a cell but (unlike habDriftRegenDist) affects the rate at which drift-feeding fish capture food (Sect. VII.A.2). Estimation of values for these food parameters is discussed in Sect. VII.A.9.

2. Availability

When a fish is conducting its daily evaluation of potential movement destinations (Sect. IV.A), it considers how much food is available to it in each cell. We model food availability to a fish as the total food production minus the food consumed by any larger fish in the cell. Availability is tracked separately for drift and search food. At the start of each day, the total daily food availability is calculated for each cell, for both drift and search food. Total daily availability is equal to the hourly rates described above times the number of hours per day we assume fish feed (Sect. VII.A.1). The daily amount of food remaining available to additional fish is set equal to this total availability before fish movement simulations begin each day.

Each time a fish moves into a cell, its food consumption is subtracted from that remaining available for additional fish. When a fish’s consumption is limited by the amount of food available in the cell (Sect. VII.A), its consumption will equal the remaining availability and no food will be available for additional fish. Any fish moving into a cell where all the food is consumed by preceding fish will consequently have zero food consumption.



7. Day Length

…………………………3

where:

……………………………………………………..4

For the Little Jones Creek site, habLatitude is 42 degrees.

1. Fish Model

 

This section describes the methods used by the model’s fish. We use the same methods for all fish regardless of age or size. Although many of the model’s functions depend on a fish’s size, we assume no ontogenetic changes in equations or parameter values.

The fish conduct three daily actions. (An "action" is part of the model representing one major biological process; actions are specifically ordered in the model’s schedule, described in Sect. V). These actions are to: move in response to flow and temperature, grow, then survive or die according to survival probabilities that vary with habitat cell and fish characteristics. The order in which these actions are scheduled is discussed in Sect. V.B.

The coastal cutthroat trout (Oncorhynchus clarki clarki) is closely related to the rainbow trout (O. mykiss) and they have similar life history characteristics (Stearley 1992). Less laboratory and field information is available for cutthroat than for rainbow trout, so in many parts of the formulation we use equations and parameter values originally developed for rainbows.

1. Habitat Selection (Movement)



Fish movement simulation involves two related steps: a fish determines whether to move each day (using the "departure rules"), and determines the destination it moves to (using the "destination rules"). Our formulation does not explicitly assume fish are territorial. Instead, it assumes the number of fish in a cell is an emergent property of fish movement to maximize fitness, where fitness is defined as expected probability of surviving and growing to reproductive maturity over a specified time horizon. This formulation was selected following a detailed review of alternatives by Railsback et al. (1999b).

1. Departure rules



We use a simple departure rule: fish examine potential destinations every day and move to a location offering higher fitness if one is available. This method assumes fish are aware of their surroundings and know when better habitat is available nearby (e.g., Hughes 1992). This method allows fish to escape habitat that becomes detrimental (cells that go dry when flow decreases; excessive velocities during floods); such habitat provides low fitness and fish immediately move away from it.

2. Destination rules



We assume fish move to the habitat cell that (a) is known and accessible, as determined by a maximum movement distance and exclusion of cells where depth is zero; and (b) provides the highest value of the fitness measure used to evaluate destination cells. This approach is implemented using the following steps.

1. Move in order of dominance



The destination rules are dependent on fish moving in order of decreasing dominance, implementing the assumption (tested successfully by Hughes 1992) that stream salmonids rank feeding positions by desirability and the most dominant fish obtain the most desirable sites. The most dominant fish move first, and cannot be displaced by smaller fish. Dominance in the model is determined by fish length; Hughes (1992) showed that dominance is usually, but not always, proportional to length for arctic grayling.

2. Identify potential destination cells



Potential destination cells are limited by a distance limitation and depth. Note that the number of fish already in a cell does not limit its availability as a destination. Following our approach of having fish compare conditions between its current cell and destination cells, our term "destination cells" does not include the cell a fish is currently in.

A habitat cell is excluded as a potential destination if it is beyond a certain distance. This maximum movement distance should be considered the distance over which a fish is likely to know its habitat well enough to be aware when desirable destinations are available, over a daily time step. The maximum movement distance is not necessarily a function of the fish’s swimming ability.

We assume the maximum movement distance is a function of length:

……………………....5

It should be noted that our model lets fish follow a gradient toward better habitat if the gradient is detectable within the maxMoveDistance, but it does not give fish the ability to find and move toward some specific target if that target is beyond maxMoveDistance. For example, if habitat generally improves in an upstream direction, fish will have an incentive to gradually move upstream. However, if an isolated good location for some fish exists farther away than its maxMoveDistance, the fish will not be aware of it.

Our formulation for maxMoveDistance is a conservative representation of observed distances over which trout select habitat. Literature observations indicate that adult trout commonly select habitat over large distances. Harvey et al. (1999) showed fall and winter movements of adult (18-24 cm length) cutthroat trout of up to about 55 m in one day at the Little Jones Creek study site, with fish often moving long distances back and forth among specific locations. Bunnell et al. (1998) showing that large brown trout routinely use different habitats that are > 80 m apart (sometimes over 1000 m) within a day, usually returning to the same spots.

June (1981) observed little movement in newly emerged cutthroat trout <3 cm; dispersal started after they exceeded 3 cm in length.

We selected parameter values to estimate maxMoveDistance as 6 m for newly emerged trout with length of 3 cm, 10 m for juveniles 5 cm long, 20 m for trout 10 cm long, and 40 m for trout 20 cm long (approximately the largest in our simulations).

Table 2. Parameter values for identifying destination cells.
 

Parameter	Definition	Units	Value
fishMoveDistParam	Multiplier for maximum movement distance	None*	200

*fishLength and maxMoveDistance are both in cm.

The maximum movement distance could preclude very small fish from having any potential destination cells if cells are large. This artificial barrier to movement (an artifact of the model’s spatial scale) could be important, for example by preventing newly emerged fish from moving from the cell where their natal redd was to habitat where survival probabilities are higher. In such a situation, competition among new fry for food would largely be an artifact of the cell’s size, which controls how much food is in it (III.F). To address this problem we always include as potential destinations four cells bordering the sides of a fish’s current cell.

Note that this formulation provides another advantage to rapidly growing fish: they are able to look further for good habitat.

We exclude cells as destinations if they have depth ? 0. This rule is imposed only to reduce computer execution times. The movement fitness measure and stranding survival probability also give fish a very strong incentive to avoid moving to dry cells: movement to maximize fitness would keep fish from using cells where stranding mortality risk is very high. However, specifically excluding movement to dry cells significantly reduces the computations needed to select a destination cell, with very little anticipated effect on model results. We do not, however, require a fish to move if the depth in its current cell drops to zero; in this case it must either chose to move or suffer stranding mortality. If the flow decreases so that the nearest cell with non-zero depth is farther away than a fish’s maximum movement distance, then the fish suffers stranding mortality (Sect. VII.B.2).

3. Evaluate potential destination cells

The destination cell is the one with the highest value of "expectedMaturity" where:

……………………….….6

The variable "nonstarvSurvival" is the calculated probability of survival for all mortality sources except poor condition (which is closely related to starvation; Sect. VII.B.3), over a specified time horizon given by the variable "fishFitnessHorizon". This probability of survival is calculated assuming that the current day’s survival probabilities will persist for the number of days specified by "fishFitnessHorizon". The value of "nonstarvSurvival" is calculated as:

…………………………7

where S_i, S_ii, S_iii, etc. are the daily survival probabilities (d^-1) for various mortality sources, evaluated for the current day, fish, and cell. Details of survival probability methods are presented in Sect. VII.B. The value of nonstarvSurvival is determined for the fish’s size before the daily growth that would occur at the potential destination cell; this assumption is made to simplify the model’s software. (It would be more consistent with the rest of our model to calculate nonstarvSurvival using the fish size after the growth it would obtain at the destination cell; however, we expect the effect of this assumption to be negligible in almost all cases.)

The value of "starvSurvival" is the probability of surviving the risk of poor condition over the number of days specified by "fishFitnessHorizon". This term introduces the effects of food intake to the fitness measure. The value of starvSurvival is determined by these steps (Railsback et al. 1999b).

• Determining the foraging strategy, food intake, and growth (g/d) for the fish and habitat cell in question, for the current day. The potential growth at a destination cell is determined by simulating how much food would be obtained and how much energy it takes to swim in the cell. These fish energetics methods are described in Sect. VII.A.
• Projecting the fish’s weight, length, and condition factor K that would result if the current conditions persisted for the number of days specified by "fishFitnessHorizon". The daily growth is multiplied by fishFitnessHorizon to determine the change in weight; the corresponding change in length and K are determined using the methods described in Sect. IV.B.
• Approximating the probability of survival over the fitness horizon using the equation:

The parameter "fishSpawnMinLength" is given a value of 15 cm for the small cutthroat of Little Jones Creek.

The time horizon variable "fishFitnessHorizon" is the number of days over which the terms of the expected maturity fitness measure equation are evaluated. The biological meaning of this variable is the time horizon over which fish evaluate the tradeoffs between food intake and mortality risks to maximize their probability of surviving and reproducing. It is discussed in the "unified foraging theory" literature (e.g., Mangel and Clark 1986). The value of the time horizon affects how EM varies among habitats (). At short time horizons (e.g., 10 d) starvation over the time horizon is unlikely even if growth is strongly negative. Consequently, with a short time horizon EM depends mainly on non-starvation survival and the variation in EM with depth and velocity is similar in pattern to that of daily survival probability. Starvation over the time horizon becomes more likely as the horizon increases; at a time horizon of 40 d, EM is very low in habitats providing highly negative growth. A time horizon of 90 d restricts high levels of EM to regions with both high daily survival probabilities and growth rates near or above zero. Longer time horizons result in little additional change in the variation of EM with habitat (Figure 1).

There is little literature addressing the issue of fitness time horizons but two studies are relevant. Bull et al. (1996) used a model similar to ours and assumed overwintering juvenile salmon used the remaining winter period as a time horizon. Thorpe et al (1998) proposed using the duration of various salmonid life stages as time horizons. If we follow the lead of this literature and assume fish anticipate seasonal changes in habitat conditions and their life stage, it makes sense to assume they use a habitat selection time horizon of several months. We use fishFitnessHorizon equal to 90 d.

Fig. 1. Sensitivity of EM to the time horizon. EM is shown as a function of depth and velocity for adult trout (15 cm FL) with distance to hiding cover of 5 m and using a velocity shelter.

1. Move to best destination

1. Feeding and Growth

Unlike previous individual-based stream trout models, we assume that fish compete with each other for the available food instead of for feeding space. We assume each cell has a certain amount of food produced in it each day and the food available to a fish in the cell is limited by the cell’s food production and the amount of food eaten by other fish in the cell (Sect. III.F).

Our model includes two alternative feeding strategies. Drift feeding, in which the fish remains stationary and captures food as it is carried past by the current, is the most studied and often the most profitable strategy (e.g., Fausch 1984, Hill and Grossman 1993, Hughes and Dill 1990). We model drift food intake (Sect. VII.A.2) as a function of stream depth and velocity and fish length; intake peaks at an optimal velocity that is higher for larger fish. Metabolic costs for drift feeding increase with water velocity, but use of velocity shelters reduces this cost. Actively searching for benthic food and food dropped into the stream from overhead is an alternate strategy that can be important when food competition is high or when conditions for drift feeding are poor (e.g., Nielsen 1992; Nislow et al. 1998). We assume the energetic benefits of search feeding are mainly a function of food availability, with energetic cost depending on water velocity (Sect. VII.A.3).

For both of these strategies, we model the potential food intake and metabolic costs a fish would experience in a cell; the fish then selects the strategy that provides the highest net energy (which often can be negative). Following standard bioenergetics approaches, we assume growth is proportional to net energy intake.

From the daily growth, the model determines the changes in length and condition factor. We adopt the simplistic approach developed by Van Winkle et al. (1996), and also use their nonstandard definition of a condition factor, K, a unitless index of how much weight a fish has relative to its length. This condition factor can be considered as the fraction of "normal" weight a fish is, given its length. The value of K is 1.0 when a fish has a "normal" weight for its length, according to a length-weight relation input to the model (Sect. II.B). Fish grow in length whenever they gain weight while their value of K is 1.0. Condition factors less than 1.0 indicate that the fish has lost weight, and condition factors greater than 1.0 do not occur.

Weight, length, and K are calculated in this way.

• The fish’s weight is changed by the value of dailyGrowth (calculated using methods in Sect. VII.A; note that daily growth can be negative).
• The length of a healthy fish with the fish’s new weight ("fishWannabeLength") is calculated from the inverted length-weight relation (with the same parameters used to initialize new fish, Sect. II.B):

………………………………9

• If the fish’s actual length is less than fishWannabeLength (indicating that the fish is not underweight), then its length is set to fishWannabeLength.
• If the fish’s actual length is greater than fishWannabeLength (indicating the fish is underweight for its length), its length is not changed.
• The new condition factor (unitless) is equal to the fish’s new weight divided by the "normal" weight for a fish its length:

It is important to note several limitations of this formulation.

• It does not allow fish to store a high-energy-reserve condition. Fish will have a condition of 1.0 only on those days when daily growth is positive. Even if a fish has eaten well for many days in succession, its condition factor can only be as high as 1.0 and one day of negative net energy intake causes condition to fall below 1.0. This could be a serious problem where short periods of high food intake are common.

• This formulation locks in a length-weight relationship for growing fish. Calibration of growth to situations where this relationship is valid will be automatic, but calibration to situations where the relationship is not valid will be impossible. For example, our model cannot predict the existence of unusually fat fish.
• The energetics of reproduction are not considered. We do not model how the need to store energy for gonad development affects length and weight, nor how the loss of gonads in spawning affects weight.

1. Survival

• High velocity (representing exhaustion and inability to maintain position),
• Stranding (including predation risk associated with extremely shallow habitat),
• Poor condition and starvation,
• Predation by terrestrial animals, and
• Predation by fish.

Understanding and calibrating mortality in the model requires that we evaluate each potential mortality source separately; if we calculated one daily overall survival probability for each fish each day, we would not be able to attribute mortality to any particular source. Therefore, we treat each mortality source as independent from the others. For each mortality source, we:

• Obtain the survival probability from the fish’s mortality methods (Sect. VII.B).
• Obtain a random number from a uniform distribution between zero and one.
• If the random number is greater than the survival probability, then the fish dies as a result of the mortality source.

1. Model Schedules

 

The schedule in which events occur is a key factor determining the outcome of individual-based models. This section defines the order in which trout model actions occur.

We followed several guidelines in defining the schedule. In general, an action is placed in the schedule before any others that might be affected by it. The schedules for imposing mortality sources consider the fact that placing a mortality source earlier in the schedule makes it slightly more likely to cause mortality (a mortality source cannot kill a given fish on a given day if a preceding mortality source does so first). Therefore, we schedule widespread, less random mortality sources (e.g., high velocities) first; survival probabilities for these sources tend to be either negligible (1.0) or very low when some mortality event occurs.

This section is one of the few places in this document where we consider the computer implementation of the model, by scheduling the "observer" actions that allow us to see the model’s state as it executes. This is because the model user must understand how the observer schedule affects the model’s output.

1. Habitat Actions



Habitat actions include updating the daily habitat variables (flow, depth, velocity, temperature, food availability) in habitat cells. These are conducted first because subsequent fish actions depend on the day’s habitat variables.

2. Fish Actions



There are three fish actions that are scheduled with respect to each other: movement, growth, and survival. Each of the three fish actions are conducted one fish at a time, from longest to shortest. Each action is conducted for all fish before the next action is begun.

Movement is scheduled as the first fish action each day to allow fish to move in response to each day’s new habitat conditions before feeding and survival simulations. Movement strongly affects both growth and survival. Note that movement decisions are based in part on survival probabilities, which vary with fish size. We base movement decisions on survival probabilities evaluated for the fish’s current size (before the current day’s growth, which depends on its movement destination).

Growth is scheduled before survival because changes in a fish’s length or condition factor affect its probability of survival.

Fish survival includes evaluation of a number of mortality sources, and the number of fish killed by each mortality source depends on the order in which survival probabilities for each source are evaluated. We schedule the least random sources (high velocity) first and the most random (predation) last.

This fish schedule has important but subtle implications for testing the model software. Methods used to determine fish movement and growth (e.g., food intake, activity respiration) can produce different results when executed during the movement method than they do when executed during growth. This is because these methods depend on how many other fish are in the fish’s cell, which changes as movement is executed; the availability of food and velocity shelters decreases as more fish are added to a cell during the movement action. For this reason, re-calculating net energy intake for a fish after the movement action is completed would produce a result that was not valid at the time the fish made its movement decision. Likewise, mortality risks evaluated during movement will differ slightly from those used in the survival action because the fish has acquired its daily growth after movement and before survival actions, and survival probabilities depend on fish size.

3. Observer Actions



We schedule observer actions as the last of the daily model actions. This means that the model’s graphical and file outputs represent the state of the model after all the habitat and fish actions have been completed for a time step. Intermediate states of the model (e.g., the size of a fish at the time it made its movement calculations but before its simulated growth; the food availability in a cell before all its fish moved into it) can only be observed by specifically telling the software to save the desired values.

4. Complete Schedule

1. Habitat Actions: Update flow, depth, velocity, temperature of cells
1. Fish Actions

A. Movement (move fish in order of decreasing dominance)
B. Growth
C. Survival
ii. High velocity
iii. Stranding
v. Poor condition
vi. Terrestrial predation
vii. Aquatic predation
1. Observer Actions: Model outputs

1. Calibration

• Individual components of the model formulation (e.g., feeding and growth methods) are tested and calibrated as they are designed. In this document, discussion of such calibration is included in the sections describing individual formulation components.
• The full model is calibrated after all model components have been designed and calibrated individually, the entire model has been implemented in computer code, and the code checked thoroughly.

To calibrate food availability and predation risks at our Little Jones Creek study site, we developed the following criteria.

Because our simulations are short- to mid-term and mostly for summer periods, calibration was conducted using dates between day 200 (July 19) through 275 (October 2). We used the year 1998 as it is the only year for which we currently have flow and temperature data. The initial population had these characteristics, estimated as a typical population from field data collected in July, 1999. (The abundance of cutthroat trout at the site was much lower in 1999 than in previous years, as indicated by pre-1999 informal sampling. Therefore, we doubled the abundance observed in 1999.)

The number of age 0 fish could not be estimated meaningfully in July because not all juveniles were emerged or large enough to see. Consequently, we calibrated survival for age 1 and older fish, then used the model to estimate the day 200 abundance of age 0 fish.
 
 

Table 3. Initial population characteristics for calibration.

Age, yr	Number	Mean Length, cm	Variance in Length, cm
0	unknown, > 450	5.0	0.67
1	50	11.4	1.69
2 and above	20	16.9	7.56

 

The following target ending population characteristics were estimated from field observations collected in at the Little Jones study site at the end of September, 1999. These characteristics were also compared to data from a field site on the Tule River, Sierra Nevada, California, from which extensive field data are available (Studley et al. 1995; Railsback and Rose 1999). This site (upstream of the Tule River diversion dam) is roughly similar to Little Jones Creek in habitat type and structure, but has higher summer temperatures and brown trout that compete with the rainbow trout we use data for. The Tule River data for rainbow trout show typical summer growth rates (cm/day) of 0.04 for age 0, 0.01 for age 1, and 0.02 for age 2 and older. For our assumed starting sizes, these growth rates translate into ending mean lengths of 8.0, 12.2, and 18.4 cm.

The Tule River data also do not provide a meaningful mortality estimate for age 0 fish (as at Little Jones Creek, apparently not all juvenile fish had emerged or become observable when the July samples were taken). For both age 1 and 2 and older age classes, the Tule River data show overall summer mortality rates (for a time period roughly equal to our 75-day calibration period) of 40%.

Table 4. Ending population characteristics used as calibration targets.

Age, yr	Number	Mean Length, cm
0	450	6.9
1	40	12.8
2 and above	15	19

 

Calibration was conducting using Expected Reproductive Maturity as the movement objective. We calibrated growth (change in length) for age 1 and older fish by adjusting the food availability parameter that represents the concentration of drift food in the water column (parameter habDriftConc; Sect. III.F). Many age 0 fish used the active searching feeding strategy, so we used the density of search food (parameter habBenthicProd; Sect. III.F) to calibrate age 0 growth. We assumed that aquatic predation is the most important (along with poor condition) and least certain mortality source for age 0 trout. The parameter mortFishAqPredMin (Sect. VII.B.5) was therefore used to calibrate age 0 survival. Similarly, we assumed that terrestrial predation is the most important and least certain mortality source for age 1 and older fish, and used the parameter mortFishTerrPredMin (Sect. VII.B.4) for survival calibration of these age classes.

We did not attempt to calibrate survival and growth precisely, but to ensure that rates and causes of mortality were reasonable and that growth rates are reasonable. The calibration process produced these values of the calibration parameters:

• habDriftConc: 1.50E-10
• habBenthicProd: 7.0E-7
• mortFishAqPredMin: 0.997
• mortFishTerrPredMin: 0.99

We conducted five replicate model runs (initialized with different random number seeds), with the following results. These can be compared to the calibration targets in the previous table.
 
 

Table 5. Calibration results.

Output	Run 1	Run 2	Run 3	Run 4	Run 5	Mean (SD)
Age 0 abundance	472	446	451	429	444	448 (16)
Age 1 abundance	39	35	38	39	32	37 (3)
Age 2+ abundance	18	15	17	19	17	17 (1)
Age 0 length	6.4	6.6	6.6	6.6	6.7	6.6 (0.1)
Age 1 length	12.9	13.0	13.5	12.4	13.1	13.0 (0.4)
Age 2+ length	18.6	18.2	18.8	18.5	18.2	18.5 (0.3)

 

1. Method Details

 

This section provides the formulation details for parts of the model that are too complicated to be explained in full in the preceding sections. The goal of this section is to unambiguously describe the model in sufficient detail for it to be implemented in computer code or otherwise reproduced.

1. Fish Feeding and Energetics



The feeding and energetics methods determine the net energy benefits (joules of food energy per day available for growth) and growth (grams of fish mass per day) a fish would obtain if it were in a specific habitat cell. These methods are used both to evaluate potential destinations during movement (Sect. IV.A) and to simulate growth (Sect. IV.B).

The energetics methods are a simplified version of widely used fish bioenergetic models (Hanson et al. 1997). Net food energy benefits of a site are equal to the fish’s food intake (a function of food availability, depth and velocity, and fish size), minus respiration costs (a function of fish size, temperature, and swimming speed). Intake and costs differ between two foraging strategies: stationary drift feeding and active searching for food. Food availability can be limited by the consumption of other fish in a habitat cell, implementing competition for food. We assume fish select the most profitable of these two strategies. This approach is modified from that of Van Winkle et al. (1996).

Sections VII.A.1 through VII.A.7 describe input to the net energy intake and growth calculations, which are detailed in Sect. VII.A.8. Parameter estimation and calibration for food intake and growth are discussed in Sect. VII.A.9.

1. Activity budget



Energy intake and costs differ between feeding vs. resting fish. To avoid the need for sub-daily simulations, we parameterize the fish’s daily activity schedule. Energetic calculations are based on hourly energy rates (j/h), and the daily energy totals depend on how many hours are spent feeding vs. resting.

We simply assume that the modeled fish spend all daylight hours feeding. Hill and Grossman (1993) observed that rainbow trout in a small stream spent 98% of daylight (including dusk and dawn) feeding, except that no feeding occurred at temperatures below 2° C. We include one hour before sunrise and after sunset in the feeding period. Consequently, we calculate the time spent feeding as:

feedTime (h/d) = daylength + 2;

if temperature ? fishMinFeedTemp, then feedTime = 0……………………………..…..11

This assumption is not completely accurate for Little Jones Creek, where nocturnal activity and diurnal changes in activity have been documented (Harvey et al. 1999). However, a more detailed depiction of activity budget would significantly complicate the model and is not necessary for our analysis of habitat selection objectives.

The input parameter fishMinFeedTemp is a temperature threshold below which the trout do not feed. Using the observations of Hill and Grossman (1993), we use a value of 2° C for this parameter. Temperatures this low have not been observed in Little Jones Creek.

2. Food intake: Drift feeding strategy



Drift feeding fish wait and capture food items as they are carried within range by the current. Our drift feeding energy intake formulation is modified and simplified from that of Van Winkle et al. (1996), which was based largely on the work of Hughes and Dill (1990), Hughes (1998), and Hill and Grossman (1993). This literature shows clearly that prey items at greater distance from a fish are less likely to be captured, and the distance over which fish can capture food increases with trout size and decreases with water velocity. (Because of its potential to add complexity to the model, we neglect variation in prey size and its effect on capture probability.)

We assume drift feeding fish capture all food items that pass within a "capture area"- a rectangular area perpendicular to the current, the dimensions of which depend on depth, velocity, fish size, and temperature. A fish’s intake per hour is calculated as the mass of prey passing through the capture area:

driftIntake (g/h) = habDriftConc (g/cm³) ´ velocity (cm/s) ´ captureArea (cm²) ´ 3600 s/h….12

In this equation, habDriftConc is a habitat cell variable (Sect. III.F.1) and captureArea is calculated using a reactive distance approach.

We use the approach and parameter values of Van Winkle et al. (1996) to identify a "reactive distance" within which all prey items are captured. This equation and its parameters were derived from data reported by Hill and Grossman (1993) relating velocity to capture success of rainbow trout.


............13
 

At high velocities (e.g., >50 cm/s for fish 5 cm long; >155 cm/s for 15 cm fish), this equation produces negative values for reactDistance. We set such values to zero.

This formulation has the advantage of being developed from the data of Hill and Grossman (1993), who used a wide range of fish lengths (5-12 cm) and velocities (between 0 and 40 cm/s) for rainbow trout, which are relatively similar to the cutthroat trout we are modeling; it also produces very reasonable results (Sect. VII.A.9). The temperature dependence was determined from data collected at 5 and 15°C, approximately the range observed in Little Jones Creek. However, the approach has several potential limitations. Data were collected at reactive distances only up to 2.5 times the fish length were tested, whereas the model predicts reactive distances of over 4 times the fish length. The data of Hill and Grossman were collected using only one size of prey (chironomids, which are small), so the model may underestimate the distance over which fish capture larger prey. The model of Hughes and Dill (1990) has been used in several other models (e.g., Gowan 1995), but has the disadvantage of having parameter values available only for arctic grayling; cutthroat trout may have significantly different swimming ability than grayling.

We define the width of the rectangular capture area as twice the reactive distance; this implements the assumption that fish are able to capture all drift that comes within the reactive distance to their left and right (as they face into the current). We compared the width of the capture area calculated by our drift-feeding method to the diameter of territories estimated by the field observations assembled by Grant and Kramer (1990). The capture area varies with velocity as well as fish size (hence the three lines for velocities of 10, 20, and 40 cm/s), but in general our model and the territory model of Grant and Kramer (1990) are very similar in how much stream width they allocate per fish, especially in the range of fish lengths (5-12 cm) used by Hill and Grossman (1993) to evaluate their parameters. A similar capture area approach was developed independently by Gowan (1995).



Fig. 2. Simulated capture width and observed territory diameters.

We define the height of the capture area to be the minimum of the reactive distance and the depth, as we assume fish are more likely to be near the stream bottom than at mid-depth when feeding. The reactive distance is typically 3-5 body lengths in velocities of less than 50 cm/s. Especially in shallow habitat and for large fish, the depth will be less than reactive distance, so we must limit the height of the capture area by the depth.

captureArea = [2 ´ reactDistance] ´ [min(reactDistance, depth)]……………..14

Increasing velocities increase the rate at which prey items are carried past a fish but decrease the reactive distance. As a consequence, the value of driftIntake peaks at intermediate velocities and reaches zero at high velocities (see the graphs in Sect. VII.A.9).
 
 

Table 6. Parameter values for drift feeding.

Parameter	Definition	Units	Value
fishReactParamA	Reactive distance constant	none	-5.91
fishReactParamB	Reactive distance velocity parameter	none	0.847
fishReactParamC	Reactive distance temperature parameter	1/C°	-0.0473
fishReactParamD	Reactive distance fish length parameter	none	1.74

1. Food intake: Active searching strategy



Actively searching for benthic or drop-in food is an alternative to the drift-feeding strategy. We simulate the food intake from this active searching strategy simply as:

…….15

The value of searchIntake is set to zero if this equation produces a negative value. This equation assumes food intake rate for searching fish (searchIntake, g/h) varies linearly with the rate at which search food becomes available to fish (habSearchProd, g wet weight/h-cm²; Sect. III.F.1). The proportionality constant (fishSearchArea, cm²) can be loosely interpreted as the area over which the production of stationary (non-drifting) food is consumed by one fish. This search area however may not be a contiguous piece of stream area: a small fish searching a small area closely may obtain the same food intake as a big fish spot-searching over a much larger area. Because fishSearchArea would be very difficult to measure, it is a good parameter to use for calibration. We do not make searchIntake a function of fish size, except for the effect of size on maxSwimSpeed, so active searching is more likely to be the desirable strategy for smaller fish.

This equation also includes a term causing searchIntake to decrease linearly with the habitat cell’s mean velocity and reach zero when velocity equals the fish’s maximum sustainable swim speed (maxSwimSpeed). The equation for maxSwimSpeed is presented in the formulation for high velocity mortality (Sect. VII.B.1). This term is included to decrease the ability of a fish to see and search for food as velocity increases. (It does not represent the energetic cost of swimming at high velocities, which is considered in the respiration formulation in Sect. VII.A.6.)

Our estimation of habSearchProd and other growth parameters is discussed in Sect. VII.A.9.

2. Food intake: Maximum consumption



As part of the net energy intake calculations, we check to make sure calculated intake does not exceed the physiological maximum daily intake (Cmax, g/d). Field bioenergetics studies (e.g., Preall and Ringler 1989; Railsback and Rose 1999) indicate that actual food intake does not approach Cmax under typical conditions. However, Cmax serves the purpose of restricting intake and growth during low temperatures, a function otherwise lacking in the model (except that the time spent feeding becomes zero at temperatures below a threshold; Sect. VII.A.1). Evidence that low food assimilation efficiencies and gut evacuation rates, which can be represented by Cmax, limit energy intake in cold temperature is cited by Cunjak et al. (1998).

There are a number of published equations for Cmax that include (a) an allometric function, relating Cmax to fish size; and (b) a temperature function. We use the equation:

Cmax (g/d) = fishCmaxParamA ´ [fishWeight (g)]^{(1+fishCmaxParamB)} ´ cmaxTempFunction……..16

with the allometric parameters developed by Rand et al. (1993) for rainbow trout.
 
 

Table 7. Parameter values for maximum consumption, allometric function.

Parameter	Definition	Units	Value
fishCmaxParamA	Allometric constant in maximum intake equation	d-1	0.628
fishCmaxParamB	Allometric exponent in maximum intake equation	none	-0.3


 

For the Cmax temperature function, our experience with lab studies indicates that only a simple approach is appropriate because Cmax is poorly defined and highly variable with fish condition, activity, food type, etc. (PG&E 1997; Myrick 1998). We use a simplified temperature function, interpolating values between the points estimated from laboratory studies on rainbow trout (Myrick 1998).
 
 

Table 8. Parameter values for maximum consumption, temperature function.

Parameter Name	Temperature (° C)	Parameter Name	cmaxTempFunction, cutthroat trout
fishCmaxTempT1	0	fishCmaxTempF1	0.05
fishCmaxTempT2	2	fishCmaxTempF2	0.05
fishCmaxTempT3	10	fishCmaxTempF3	0.5
fishCmaxTempT4	22	fishCmaxTempF4	1.0
fishCmaxTempT5	23	fishCmaxTempF5	0.8
fishCmaxTempT6	25	fishCmaxTempF6	0
fishCmaxTempT7	100	fishCmaxTempF7	0

3. Food intake: Daily food availability

 

In determining the daily food intake rate for fish, our feeding formulation uses the total amount of drift (driftDailyCellTotal, g/d) and search (searchDailyCellTotal, g/d) food available each day in each cell. These daily food availability values are a function of the fish’s feeding time because we cannot count food produced during non-feeding hours as available to the fish. We obtain the daily food availability rates from the hourly food availability rates described in Sect. III.F.2.

driftDailyCellTotal (g/d) = driftHourlyCellTotal (g/h) ´ feedTime (h/d)………………..17

searchDailyCellTotal (g/d) = searchHourlyCellTotal (g/h) ´ feedTime (h/d)…………..18

4. Respiration costs and use of velocity shelters

Swim speeds. We assume drift-feeding fish swim at a speed equal to their habitat cell’s water velocity unless they have access to velocity shelter. Fish using the active search feeding strategy are also assumed to swim at a speed equal to their cell’s velocity.

If a drift-feeding fish has access to velocity shelter, then we assume its swimming respiration is a fraction of the cell velocity determined by the input parameter fishShelterSpeedFrac. A number of studies have shown that "focal" water velocities (the velocity measured as closely as possible to the location of a drift-feeding fish) are related to, but less than, the mean column velocity at the same location. For example, we plotted focal vs. mean column velocity for rainbow trout using data from Baltz and Moyle (1984), Baltz et al. (1987) and Moyle and Baltz (1985); this graph shows the focal velocity is consistently about 77% of the mean column velocity.

However, the data plotted above do not relate focal velocity to a mean habitat cell velocity, and we unaware of studies that do evaluate this relation. Accurately measuring the swimming speed of a trout holding in a velocity shelter is not easy and the focal velocity measurements reported in the literature have key uncertainties. The measurements rely on an underwater observer to estimate the fish’s exact location, and on being able to place a velocity meter exactly where the fish was observed, in a highly variable velocity field. These measurements also typically use velocity probes that are larger than fish and cannot replicate a fish’s ability to continuously adjust to velocity variations. The data plotted above also do not distinguish between trout that were and were not clearly using velocity shelters; they probably include some unsheltered fish, reducing the apparent difference between focal and mean column velocity. Finally, the above graph relates focal velocity to the mean column velocity at the spot where the fish was observed, which is likely to be less than the mean velocity of the surrounding habitat cell because the fish’s velocity shelter (e.g., a boulder) may slow the current locally.

In the absence of a reliable way to estimate the daily swimming speed of a trout using velocity shelter, we assume a value of 0.3 for the parameter fishShelterSpeedFrac. We also use this formulation and parameter in the high velocity mortality function (Sect. VII.B.1).

Velocity shelter access. We use the following steps to determine whether each fish has access to shelter in a habitat cell.

• Each cell has a limited area of velocity shelter; this area varies among cells but is constant over time (Sect. III.D).
• Each drift-feeding fish is assumed to use up an area of velocity shelter equal to the square of its length.
• A fish has access to velocity shelter only if the sum of shelter areas occupied by more-dominant fish in a cell is less than the cell’s total shelter area.

Respiration cost model. We adopt the Wisconsin Model equation 1 for respiration (Hanson et al. 1997), as modified by Van Winkle et al. (1996) to apply the activity respiration rate only during active feeding hours (feedTime). We use the parameters that Rand et al. (1993) developed for steelhead trout (converted from calories to joules). This formulation breaks respiration into two parts: standard respiration takes place 24 h/d and assumes no activity; activity respiration is the energy needed to swim during feeding. Total respiration is the sum of these two. Respiration costs are in j/d. The equations are:

....19

and

….20

where swimSpeed is the fish’s swimming speed (cm/s) during feeding.

The data of Myrick (1998) indicate that the above standard respiration formulation overestimates the effect of temperature on respiration rates and does not account for an observed decrease in respiration at temperatures above 22° . However, temperatures this high do not occur at our study site.
 
 
 

Table 9. Parameter values for respiration.

Parameter	Definition	Units	Value
fishRespParamA	Allometric constant in standard respiration equation	(j/d) ´ g-0.78*	30
fishRespParamB	Allometric exponent in standard respiration equation	none	0.784
fishRespParamC	Temperature coefficient in standard respiration equation	1/° C	0.0693
fishRespParamD	Velocity coefficient in activity respiration equation	s/cm	0.03
fishShelterSpeedFrac	Swim speed reduction for fish using velocity shelter	none	0.3

*Empirical parameter with units that depend on fishRespParamB.

1. Other energy losses



Most fish bioenergetic formulations include terms for energy losses due to egestion, excretion, and specific dynamic action. We do not include these terms because their effect is small compared to the large uncertainties in food availability and in the bioenergetics formulation and parameter values. These terms may be important at extremely low or high temperatures when the ability to digest food can limit growth, but we use the Cmax function to limit food consumption at extreme temperatures.

2. Feeding strategy selection, net energy benefits, and growth

"Food" variables are in grams of prey; "Energy" variables are in joules. Prey energy density is used to convert grams of prey eaten to joules of energy; we adopt the value of Van Winkle et al. for habPreyEnergyDensity of 2500 j/g. Daily growth, the change in fish weight (g/d), is equal to the net energy intake (j/d) divided by the energy density of a fish ("fishEnergyDensity", j/g). We use the energy density of Van Winkle et al., 5900 j/g.

1. Determine the daily drift intake that would be obtained in the absence of more dominant fish in the cell. This "dailyPotentialDriftFood" is determined from the hourly intake rates and hours spent feeding:

dailyPotentialDriftFood (g/d) = driftIntake (g/hr) ´ feedTime (h/d)………………….....21
2. Determine how much drift intake is really available after more dominant fish have consumed their intake: this "dailyAvailableDriftFood" is equal to driftDailyCellTotal minus the drift intake of all drift-feeding fish already in the cell (Sect. III.F.2).
3. Calculate the actual drift intake, considering actual food availability and the physiological maximum intake, Cmax:

dailyDriftFoodIntake = min (dailyPotentialDriftFood, dailyAvailableDriftFood, Cmax)…………………………………………………………………………………….22
1. Convert daily drift intake in grams of food to joules of energy:

dailyDriftEnergyIntake (j/d) = dailyDriftFoodIntake ´ habPreyEnergyDensity (j/g)…..23
2. Conduct the bioenergetics energy balance to get net energy intake for drift feeding, with activity respiration depending on whether the fish has velocity shelter.

dailyDriftNetEnergy (j/d) = dailyDriftEnergyIntake - respStandard - respActivity.……24
3. Determine the daily active searching intake that would be obtained in the absence of more dominant fish in the cell, "dailyPotentialSearchFood" is determined from the hourly intake rates and hours spent feeding:

dailyPotentialSearchFood (g/d) = searchIntake (g/hr) ´ feedTime (h/d)………………..25
4. Determine how much search intake is really available after more dominant fish have consumed their intake: this "dailyAvailableSearchFood" is equal to searchDailyCellTotal minus the search intake of all search-feeding fish already in the cell (Sect. III.F.2).
5. Calculate the actual search intake:

dailySearchFoodIntake = minimum of (dailyPotentialSearchFood, dailyAvailableSearchFood, Cmax)………………………………………………………26
1. Convert daily search intake in grams of food to joules of energy:

dailySearchEnergyIntake (j/d) = dailySearchFoodIntake ´ habPreyEnergyDensity (j/g)…..27
2. Conduct the bioenergetics energy balance to get net energy intake for search feeding (no search-feeding fish have velocity shelter):

dailySearchNetEnergy (j/d) = dailySearchEnergyIntake - respStandard - respActivity…….28
1. Select the most profitable feeding strategy:

bestNetEnergy (j/d) = maximum of (dailyDriftNetEnergy, dailySearchNetEnergy)…….29
1. Convert net energy intake to grams of growth:

dailyGrowth (g/d) = bestNetEnergy / fishEnergyDensity (j/g)…………………………..30

1. Feeding and growth calibration

There are many variables affecting growth so it must be calibrated incrementally, making sure that reasonable results are likely before trying to calibrate growth in the full model, where flow and movement are additional major factors affecting growth. We started the calibration process by estimating initial parameter values for the food intake and growth formulation after coding it in a spreadsheet. Parameter estimation concentrated on adjusting the search and drift food availability parameters so that feeding strategies and growth rates met criteria that we developed from field observations of habitat use and laboratory growth data. For this process, we ignored depletion of food by competing fish.

We used the following criteria for initial estimates of food intake and growth parameters.

• Daily food intake should be in the range of 20% to 50% of Cmax. Food intake should rarely if ever be limited by Cmax. This criterion is based on field research in which food intake was estimated from observed growth and bioenergetics models (Preall and Ringler 1989; Railsback and Rose 1999). (This criterion may not be valid in unusual situations where food is extremely abundant and trout growth rates very high, or at very low temperatures where Cmax is very low.)
• Drift feeding should be more profitable than active search feeding, except at low velocities. Trout are rarely observed feeding only with the search strategy, and where both strategies are available drift feeding has been observed to be preferred (Nielsen 1992; Nislow et al. 1998).
• Growth under good conditions (high food intake, low swimming velocity) should not exceed growth rates observed in lab studies where fish were fed as much as they could eat (Myrick and Cech 1996; Myrick 1998). These lab growth rates are in the range of 2-6% of body weight per day, varying with temperature.

The value of habDriftRegenDist was estimated by assuming a cell that contains 15 cm trout, each having a square territory 150 cm on each side. This assumption is based on the observations collected by Grant and Kramer (1990), which indicate that 15 cm trout have an average territory diameter of 150 cm. Further, we assume that the fish get an intake of 30% of Cmax, or 0.13 g/h, and that under these conditions drift food production equals consumption by the trout. To provide this level of drift food production, the value of habDriftRegenDist must be approximately 500 cm.

The assumptions used to estimate search intake parameters are (a) a search-feeding fish consumes the production of two square meters, so the value of fishSearchArea is 20,000 cm²; and (b) a 5 cm trout can maintain zero growth by search feeding for 16 h/d at 15° (an intake of 0.006 g/h). These conditions give a value of 3´ 10^-7 g/cm²/h for habSearchProd.

These parameter estimates provide the food intake levels (evaluated as the fraction of Cmax) and growth (evaluated as percent of body weight per day) illustrated by the following graphs. For comparison to these parameter estimates, we did not locate any published estimates of the rate at which food becomes available, or is eaten by trout, at the stream bottom; published estimates of invertebrate production do not separate drift from any invertebrates eaten at the benthic surface. The rate at which food drops in from overhead (part of our search food production) is also rarely measured. Poff and Huryn (1998) report overall food production rates (in Atlantic salmon streams) in the range of 4-24 g dry weight per m² per year, which converts to 10 – 60´ 10^-7 g/cm²/h (assuming a typical ratio of 20 for dry:wet weight; Hanson et al. 1997). Our estimate of habSearchProd appears reasonable compared to this value; we would expected habSearchProd to be a relatively small but not negligible fraction of the total production rate.

The following two graphs show food intake and growth rates for 5 and 15 cm trout, as a function of cell velocity. For these graphs, the temperature was 15° , depth was 50 cm, and feeding time was 16 h/d. Food intake is reported as the percent of Cmax, growth as percent body weight per day.

Fig. 4. Variation in food intake with velocity for each feeding mode.

Fig. 5. Variation in growth with velocity for each feeding mode.

Several observations from these graphs are interesting and realistic.

• Conditions providing high intake do not always provide high growth, due to the metabolic costs of swimming (especially for fish drift feeding without velocity shelters).
• The use of velocity shelters for drift feeding is very beneficial.
• Search feeding is a profitable strategy only for small fish in low velocities.
• The relative benefits of drift feeding increase with fish size.
• Larger fish can drift feed profitably over a wider range of velocities, and at higher velocities, than can smaller fish.

1. Fish Survival Probabilities



The survival probability methods determine the daily probability of a fish surviving individual mortality sources, typically as a function of habitat and fish size and condition. These methods are used for two purposes: they are used in the movement destination rules to evaluate each potential destination site (Sect. IV.A.2), and in survival simulations to determine whether and why each fish dies each day (Sect. IV.C). We use the same survival probabilities for each of these purposes, assuming that fish are completely aware of actual mortality risks.

We do not provide a calibration parameter for any risks except the two predation risk functions. A calibration parameter lets the user adjust the magnitude of the risk without changing how it is related to fish and habitat variables. To reduce the number of processes adjusted in calibration, we assume that survival probabilities for high velocity, stranding, and poor condition are less uncertain than predation and should not be used for calibration.

Users should be aware that seemingly high survival probabilities can result in low survival over time. For example, a survival probability of 0.99 d^-1 results in mortality of 26% of fish within 30 days (0.99³⁰ = 0.74). Survival probabilities should be well above 0.99 if they are not to cause substantial mortality.

The survival probability formulations make extensive use of logistic functions, which are useful for depicting how many survival factors vary between 0 and 1 in a nonlinear way. We define these logistic curves using parameters that specify the point at which the logistic function equals 0.1 and 0.9. The logistic functions are defined as:

…………………………………………………………………..31

where









.

These equations evaluate the daily survival probability S given the input "habitatVariable", the value of the habitat variable driving the logistic function. The parameters "habVarAtS0.1" and "habVarAtS0.9" are the values of the habitat variable at which daily survival is defined to be 0.1 and 0.9, respectively.

We simulate five mortality sources. Parameter estimation and calibration of survival is discussed at Sect. VII.B.6.

1. High velocity



The high velocity survival function represents the potential for trout to suffer fatigue or lose their ability to hold position in a cell with high velocity. This function is included not because we expect it to kill fish, but because it is a real mortality risk that trout must move in response to. Mortality due to high velocities are not observed in nature because fish can avoid them via movement; we must include this risk to cause such movement to occur in the model. Velocities posing mortality risk are spatially extensive at high flows. Our feeding model (Sect. VII.A.9) predicts that growth rates become negative at water velocities just below the onset of high velocity risk.

The survival probability is based on the ratio of the swimming speed required to remain in a cell to the fish’s maximum sustainable swim speed, a function of fish size; larger fish are better able to resist higher velocities than smaller fish are. We determine the fish’s swimming speed the same way we do for calculating respiration energy costs (Sect. VII.A.6). Fish are assumed to swim at the cell’s water velocity unless they are drift-feeding with access to velocity shelters. Fish using velocity shelters are assumed to swim at a speed equal to the cell’s velocity times the parameter fishShelterSpeedFrac.

First we estimate the fish’s maximum sustainable swim speed, maxSwimSpeed (cm/s). As used in our model with its daily time step, this variable should be a speed that fish can swim for hours, not a burst or short-term maximum speed. Myrick (1998) measured "critical swimming speed", a high estimate of sustainable speed; Myrick cites references indicating that trout may start to use white (fast-twitch) muscle fibers at 90-95% of the standard critical swimming speed. A better estimate of the speed fish can sustain for long periods is 90% of the critical speed (C. Myrick, Department of Fish, Wildlife, and Conservation Ecology, University of California, Davis, pers. comm. with S. Railsback, 10 May 1999). Myrick (1998) measured critical swim speed at temperatures between 10 and 19°C for four strains of O. mykiss.; he also cites (Table 9 of his Chapter II) other studies in which critical swimming speed has been measured for O. mykiss and cutthroat trout at similar temperatures. These measurements are subject to a number of uncertainties, including that laboratory fish may not be in good exercise condition. As the following figure shows, these data show a lack of temperature dependence, and good correspondence among studies.



Fig. 6. Observed critical swim speeds as a function of fish length.

[Griffiths and Alderdice (1972) made extensive measurements of swimming speed over temperatures between 2 and 26° C, for juvenile coho salmon. These were the basis of the swimming speed model of Stewart (1980), which was also adopted in the individual-based model of Van Winkle et al (1996). These measurements showed gradual increase in sustainable swimming speed as temperature increased from 2 to about 20° , approximately doubling over this range. Swimming performance dropped off sharply at temperatures above about 20° . However, the data presented by Griffiths and Alderdice do not allow fish length effects to be separated from temperature effects. It seems reasonable to conclude from all the evidence that temperature has a relatively small effect on sustainable swimming speed except at temperatures approaching a fish’s physiological tolerance limits. The model used by Van Winkle et al. (1996) estimates swim speeds well above the critical swim speed measurements presented above.]

We use a maximum sustainable swim speed equation that is a simple linear function of fish length. It was developed by assuming sustainable swim speeds are 0.9 times the measured critical swim speeds, and using regression on the data in the above figure. We did not include the apparent outlier measurement with fish length of 30.8 cm and critical speed of 54 cm/s. The predicted sustainable swim speeds for trout of 4, 10, and 30 cm length are 40, 51, and 88 cm/s.

maxSwimSpeed = [fishSwimParamA ´ fishLength] + fishSwimParamB………………32

A decreasing logistic function relates survival probability to the habitat cell velocity divided by the fish’s maximum swim speed. The parameters for this function are chosen so that high velocity mortality is negligible at cell velocities less than maxSwimSpeed, reflecting that (a) the laboratory apparatus for measuring swim speeds does not have the kinds of turbulence and fine-scale velocity breaks that trout can use to swim at speeds less than the cell mean, and (b) stream fish are likely to be