II.6.1 - Predation Mortality Theory

• Simple predation relationship where reservoir volume has no effect on predation rate.
• Predator density/reservoir volume interaction where changing the reservoir elevation changes effective predator densities. The effect is that lowering reservoir elevation concentrates predators and increases the predation rate.

The general predator mortality rate coefficient M_p is defined

(64) where

• a = Predator Activity Coefficient
• P(E) = Predator Density/Reservoir Volume Coefficient (a function of the elevation of the river segment below full pool)
• = temperature in °C
• u = Predator Activity Temperature Exponent.

Predation Activity Coefficient

The predation activity coefficient is fixed for a given species but may have additive deterministic (a₀) and stochastic parts (a'₀) so

(65)

Predator Density / Reservoir Volume

The predator enhancement equation expressing how the effective predator density may change with reservoir elevation change is

(66)

where

• V(0) = river segment volume at full pool conditions and is fixed for each river segment
• V(E) = river segment volume when elevation is lowered by E as defined in the Flow-Velocity-Elevation section II.3.4; may change with Julian Day
• P(0) = predator density in a river segment per unit area at full pool (predators km^-2).

The term a is specific to species and can have deterministic and stochastic parts. Individual terms are identified for the reservoir, tailrace and forebay.

For reservoirs with variable pool level the predator density can be defined by eq(66) where

• E = elevation of the river segment below full pool
• V(0) = river segment volume at full pool conditions
• V (E) = river segment volume when the elevation is lowered by an amount E
• P = predator density at full pool.

(67)

• h (0) = forebay or tailrace depth at full pool conditions
• h (E) = forebay or tailrace depth at elevation E

Predator density is required for calibrating the activity coefficient and for estimating reservoir mortality rate. Calibration of predator density uses data from the predation studies in John Day Reservoir conducted between 1984 and 1986 and from predator indexing between 1990 and 1993. The estimation of predator density has been complicated by an ongoing predator removal program that has been implemented since 1992. These alterations are adjusted for in the model calibration using information on predator exploitation rates.

Predator Density in John Day Reservoir

Total predator density in each region of John Day Reservoir is required to calculate predator activity coefficient from predation consumption data in the reservoir. Since there are three major predators, squawfish, smallmouth bass and walleye, but the model only considers a generic predator, the effect of all predators was expressed in equivalent densities that reflect each species' rate of smolt consumption. The equivalent measure was adjusted to squawfish over 250 mm in length, this being the minimum fish length that consumes smolts. The adjustment was based on the ratio of consumption rates for smallmouth bass and walleye to squawfish. In addition, the prdator population was partitioned into forebay, pool and tailrace regions.

Rieman et al. (1991) indicates that squawfish are distributed in each zone of the reservoir according to the zone's dimension and the catch per unit effort (CPUE) in each zone, while walleye and bass are mostly confined to the reservoir proper.

Expressing densities in squawfish equivalents and applying a CPUE correction, predator densities in each zone are as follows.

The equivalent squawfish predator density in the reservoir is

(68)

where

• P_re = equivalent density of predators in the reservoir zone including smallmouth bass, walleye and squawfish
• c_name = average salmonid consumption rate of walleye (w), smallmouth bass (b) or squawfish (s) in zones (From Table 23)
• P_name = total density of walleye (w), smallmouth bass (b) or squawfish (s) (From Table 13)
• A_name = area of each zone [tailrace (tr) reservoir (re) and forebay (fb)] (From Table 22)
• w = squawfish distribution factor based on CPUE according to the equation (From Table 14). This is the fraction of total predators in the entire reservoir that is in the forebay and reservoir pool

(69)

where

• CPUE_name = catch per unit effort in the tailrace (tr) or the combined regions of the forebay and reservoir (pool).

(70)

The equivalent squawfish predator density in the tailrace, which contains only squawfish due to the high flow speeds, is defined by

(71)

where

• w = squawfish distribution factor expressed by eq(69).

CPUE per month of northern squawfish in John Day Reservoir is given in Table 14 (Beamesderfer and Rieman 1991).

The resulting densities using eq(69), eq(70) and eq(71) are given in Table 15.

The mean predator density in each region is obtained by averaging each region over all months. The result is shown in Table 16.

Predator Density in System

Predator abundances in Columbia and Snake River reservoirs are estimated from the Predator Index studies conducted by USFWS, ODFW and WDFW between 1990 and 1993 (Table 17). Squawfish density indices are defined as 1 divided by the square root of the proportion of electrofishing runs in which no northern squawfish were caught. Each reservoir is referenced to John Day Reservoir. This involves estimating the reference to the John Day reservoir during the period of the predator studies, 1984-1986. Estimates of the change in density from predator removal can be derived from information on the productivity of the fishery and exploitation rate (Table 19). From this information predator density estimates can be constructed for a number of years.

The predator density in a given reservoir expressed in terms of the John Day reservoir is given by

(72)

where

• P_i = squawfish density (predators km^-2) in a given segment i
• I_i = Northern squawfish density index in segment i. The predator density index is calculated according to Ward and Peterson Table 5 (in press) as the Index value 1 divided by the square root of the proportion of the zero catch electrofishing runs.
• P_ref = reference average squawfish density = 472 (predators km^-2)

Reference density is calculated using information on total population number in the reservoir in a given year and the index for that year from each segment. The procedure is as follows. First the average predator index I_ave for the reservoir is calculated for each area according to the equation

(73)

where I_i is the index for each reservoir segment and is obtained from year 1991 in Table 17. The reference density, which is the density for which the index is 1, can be calculated from the average index, the total population of squawfish, N, and the total area of the reservoir (Area_res). The formula and result for 1991 using a predator population number in Ward et al. (1993 page 390) is

(74)

The squawfish predator densities given in Table 18 are calculated using information from eq(72) and indices given in Table 17.

Predator removal adjustments

The above predator density estimates were indexed in specific reservoirs for specific years. These densities must be adjusted for changes in density resulting from predator removal efforts that were initiated during the index studies. This adjustment is problematic since it requires knowledge of population growth and natural mortality as well as the exploitation rate.

The response of predators to exploitation was studied by Rieman and Beamesderfer (1990) using a generalized population model designed for simulating age-structured populations. The model included age-specific natural mortality, exploitation rates and recruitments described by either a Beverton-Holt or Ricker stock-recruitment function.

Since the population dynamics in CRiSP1 use a combined predator density, age-specific predator population information must be represented as a combined generic predator. The approach was to represent the dynamics of an age-specific predator population model by a continuous logistic curve which is equivalent to the Beverton-Holt stock-recruitment equation. The parameters of the logistic model were then adjusted to provide the same response as the stock recruitment-based model of Rieman and Beamesderfer (1990).

The logistic-based population model incorporates a generalized net growth rate and a death rate term that depends on population size. The equation is

(75)

where

• P = predator population density
• g = intrinsic predator population growth rate: the difference between growth mortality rate elements independent of population size
• f = exploitation rate from the predator removal program
• K = carrying capacity of predators in a reservoir. This is taken as the predator index-based value prior to predator removal.

(76)

where

• P(t) = predator population density at time t
• P(t + t) = predator population density at time t + t

To estimate the carrying capacity assume that prior to exploitation the population was at its carrying capacity. As a first approximation of carrying capacity we take the predator index study which was initiated in the first years of predator removal efforts. The carrying capacities for the reservoirs are thus taken from Table 18.

The predator population growth coefficient was adjusted to fit the population growth suggested in by Reiman and Beamsderfer (1990). In this analysis the suggested range for the Beverton-Holt recruitment factor was 0.5 to 0.98. This gave a population that upon termination of exploitation would return to 90% of carrying capacity in 6 to 30 years. The same response from the logistic equation gave a growth parameter of g = 0.375 yr.^-1 for the 6 year return time and g = 0.075 yr.^-1 for the 30 year return time.

The exploitation rate was estimated in the predator index studies. The exploitation rates for the reservoirs is given in Table 19. Exploitation rates for 1990, 1991, 1992 and 1993 were obtained from Ward et al. 1993, Ward et al. 1994, and Ward et al. 1995.

Using the above estimates of the growth rate in eq(76) the predator densities in the reservoirs can be corrected for exploitation.

The choice of a representative population growth rate is bounded by the estimates conforming to the work of Reiman and Beamsderfer (1990). They discussed the possible range of growth and concluded that other studies supported their estimates. The consequences of the different estimates of growth on predator density and the ability of the model to fit the survival studies is discussed in the validation section of the manual. The growth rate choice has a small effect on estimates of predator density and this in turn has a small effect on smolt survival.

In the model the predator density was taken as the average of the densities estimated from the two growth rate. These densities are given in Table 20.

Activity Coefficient Estimation: Theory

The predation activity coefficient in reservoirs is estimated from data on the rate of consumption of smolts by predators. The general strategy is to construct a simple equation describing the flux of smolts through a reservoir in terms of the smolt migration rate, the predator density, and predation rate. Assuming a steady state smolt distribution on a monthly basis, an equation can be developed describing the consumption rate coefficient in terms of the flux of smolts, the consumption rate of each predator, and the number of predators in the reservoir. The calculation is applied sequentially to the tailrace, reservoir proper, and forebay of John Day reservoir, and yields the activity coefficient for each region. The coefficients are then adjusted for temperature using information on the effect of temperature on squawfish feeding behavior. Since CRiSP.1 only tracks one predator type, the contribution to predation by fish other than squawfish is accounted for by expressing other predator densities in terms of their equivalence in squawfish where the equivalence is related by the rate of smolt consumption for the different predators.

The above calibration approach estimates predator activity coefficients for each region of John Day Reservoir. These coefficients are then applied to other reservoirs throughout the system. The coefficient describing the rate of predation on smolts was estimated using data derived from both experiments and field studies. The analysis uses data from John Day Reservoir because this is the only reservoir in which extensive predation studies were conducted. In order to define an equation describing the rate of change of smolts in a single reservoir, three regions are defined as in Fig. 35.

Fig. 35 Regions with fish fluxes F, smolt number S, and rates R

Smolt equation

The equation balances the rate of change in the number of smolts in each region with the flux and the consumption of smolts by predators in the region. The equation is

(77)

where

• dS_i / dt = rate of change of smolts in region i
• F_i = flux of smolts into the region
• F_i+1 = flux of smolts out of the region
• R_i = consumption rate of smolts by predators in the region.

(78)

where

• S and P = average numbers of smolts and predators in the region which could also be expressed on a unit area basis
• r_i = consumption rate coefficient.

(79)

The calibration strategy is to work with eq(77) to estimate the rate term R, which itself is defined in eq(78). The important parameter is the activity coefficient r_i describing the chance of predation given that predator and prey encounter each other. The densities of prey and predator and the residence time of prey in the river segment determine the number of encounters. The following equations develop this algorithm.

Smolt exit flux

The smolt flux out of a region is assumed to be

(80)

where

• b_i = rate at which smolts exit the region.

(81)

Taking the average travel time TT_ave, which is the time for 50% of the fish to exit a segment, the coefficient b is

(82)

Travel times are measured as medians which differ from the average travel time since the distribution of fish is skewed with increasing time. The relationship between mean and median is dependent on the travel time parameters. This difference can be estimated from the model by following a single release through John Day reservoir.

The average travel time was estimated from the median travel time using a ratio of the two from CRiSP output through John Day Reservoir. The conversion is expressed

(83)

A sensitivity analysis indicated that the transformation of travel time from median to average was a sensitive parameter and could change the results by tens of percentage points.

Steady state

Assuming that local smolt density is at steady state, such that the rate of change of smolts is zero, the calculations are greatly simplified. This assumption does not produce a significant bias in the estimates since the rate parameter is calculated over a period of time in which the smolt concentration first increases and then decreases in the reservoir. The result is that rate estimates with positive and negative biases are averaged so that the net bias is small.

The resulting population balance at steady state using eq(77), eq(78) and eq(80) is,

(84) where

• = steady state average smolt density in region i

(85)

where c_i is the consumption rate smolts predator^-1day^-1. Assuming that predator consumption rate coefficient, r_i, does not vary during the migration period of a species it follows that

(86)

Rearranging eq(85) the equilibrium smolt density is

(87)

Combining eq(86) and eq(87), the consumption rate coefficient is

(88)

The flux into the next adjacent region downstream is expressed

(89)

Temperature factor

The model applies a temperature correction to the rate term in order to express the rate coefficient independent of temperature. Based upon the work of Vigg et al. (1990) the equation is

(90)

where

• a = predation activity coefficient defined in eq(64).
• u = temperature coefficient for the predation rate. This rate changes with temperature. CRiSP.1 uses u equal to 0.207 °C^-1. This value was inferred from a relationship of how squawfish satiation changes with temperature (Vigg & Burley 1991).
• = water temperature in °C.

The data for estimates of the predator activity coefficient are obtained from the following sources:

Flux estimate

The flux of salmonids into John Day Reservoir is given in Table 21 from data presented in Rieman et al. (1991). The flux estimates were computed from passage indices at McNary Dam and number of hatchery fish released into John Day reservoir on a monthly basis and using the fish guidance efficiency through the counting facilities at McNary Dam. The guidance efficiency estimated for salmon was 40% and for steelhead was 75% (Giorgi and Sims 1987). Fish removed at McNary Dam and transported through the reservoir in barges were accounted for in the estimates.

Travel time

Travel times (Table 21) in April and May were estimated from studies by Stevenson and Olson (1990). Travel times during June, July and August were obtained from reports by Giorgi et al. (1990). Temperatures in Table 21 represent monthly average temperatures for John Day reservoir as generated from CRiSP data files. The model was calibrated to these temperatures since model runs use temperatures generated from headwaters.

Table 21 John Day reservoir properties. F1 = smolts month-1 entering reservoir. Median TT is travel time in days.
Month	F1 salmon	F1 steelhead	TT (days)	cal1 (°C)
April	1567000	129000	5.8	8.09
May	5896000	955000	5.8	10.47
June	4465000	210000	12.9	14.26
July	5246000	3000	18.2	18.03
August	801000	1000	24.6	19.78

Consumption rate estimate

The consumption rates of smolts by squawfish were obtained from Vigg et al. (1991). In that study, stomach contents of adult northern squawfish were analyzed and identified as either salmon or steelhead. The rate of consumption was determined from the number and weight of digested prey and the literature on gut evacuation rates of squawfish. Further detail on how the data were distributed by sampling station and month, were provided by J. Peterson (personal communication) (Table 22). The John Day study also measured consumption rates of secondary predators: walleye and smallmouth bass. The results of these studies on a whole-reservoir basis are given in Table 23 along with a reservoir averaged consumption rate for squawfish.

Calculation of activity coefficients from consumption rate data in John Day Reservoir required combining data into three groups representing the regions defined in CRiSP.1.

To obtain a consumption rate for salmon, excluding steelhead, the consumption rates (Table 22) are scaled by the ratio of salmon to all salmonids found in squawfish stomachs (Table 24).

Average salmon consumption rates in the three regions--tailrace, reservoir, and forebay--can be generated from the equation

(91)

where

• G_m = fraction of salmon in the salmonids samples for month m as given in Table 24
• C _{i, m} = salmonid consumption rate for zone i in month m as given in Table 22
• A_m = area of zone in as given in Table 22.

The indices for the model regions for eq(91) are as follows:

McNary Dam tailrace: j = 1, k = 1

John Day Reservoir: j = 2, k = 4

John Day Dam forebay: j = 5, k = 5.

Consumption rates of salmon by squawfish as developed in eq(91) are given in Table 25.

The consumption rate for steelhead can be estimated from the same data since the remaining salmonids were steelhead. The steelhead consumption rate can be expressed by the equation

(92)

where all terms are analogous to those in eq(91). Steelhead consumption rates are given in Table 26.

The estimation of fish average travel time from the mean travel time is given in Table 27. The difference was calculated by using 1986 flow levels and the movement of spring and fall chinook through John Day reservoir.

Activity Coefficient Calibration

Using data from the tables in the section on Data for Activity Coefficients estimates the predator activity coefficients on a per month basis in each zone can be calculated for steelhead and salmon. The calculations use eq(88), eq(89) and eq(90) and the coefficient is designated a.

The majority of fish in the system in April and May are spring chinook while fall chinook are dominant June through August. Steelhead predominantly pass in May. Thus as a first order approximation, activity coefficients for each species can be estimated by averaging coefficients according to the weights of smolt passage numbers. Weighting coefficients as fractions of the total flux are shown in Table 28.

The average activity coefficient for each species in each region is now obtained by averaging the month specific activities using the weighting functions in Table 28. The equation for average predator activity coefficient for a species is

(93)

where the factor is used to convert the measure from an activity coefficient based on predator equivalents to a predator activity coefficient based on squawfish only.

The predator adjustment (factor) is based on the following: From Table 16 the equivalent predator density in John Day reservoir for the period 1984-1986 is 456 predators km^-2. The average squawfish density in the reservoir is 326 squawfish km^-2. The factor converting the activity coefficient from one based on predator equivalents to one based on squawfish is 456/326 = 1.39. This coefficient is applied to the reservoir and forebay, these being the two regions where walleye and smallmouth bass are found. Since only squawfish are found in the tailrace the factor for the trailrace is 1.

The predator activity coefficients are input in the model under BEHAVIOR button submenu pred coef. They reflect the predator activity in river passage.

Activity Coefficient Variability

To estimate the range of the activity coefficient, confidence limits in mortality estimated from a Monte Carlo simulation by Rieman et al. (1991) are applied. To equate mortality confidence interval to an upper and lower range of the activity coefficient, we define the survival of smolts while in the reservoir in terms of the equation

(94)

This yields the solution

(95)

where

• S(*) = smolt density
• t = representative time in the reservoir
• P = predator density
• r = predation rate.

(96)

and so the total mortality m within a reservoir can be approximated

(97)

A ratio of two activity coefficients is thus related to a ratio of two mortality estimates with all other factors equal. It follows then that the range of estimates in mortality can be related to the range of estimates in the predator activity coefficient. The equation is

(98)

The upper and lower limits of the activity coefficient can be estimated from the 95% confidence limits of mortality given by Rieman et al. (1991). They estimated total salmonid mortality in John Day reservoir at 14% with a 95% confidence interval from 9% to 19%. Upper and lower estimates of the activity coefficient are

(99)

²This non-steady state assumption used for a particular release does not conflict with the steady-state assumptions for total density since as fish leave the downstream end they are replaced with fish at the upstream end of the region.

Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
Copyright © 1996, Columbia Basin Research. All rights reserved.