APPENDIX G

EXPLORATION OF\
RECRUITS PER SPAWNER ANALYSIS

***Paul McElhany***

***National Marine Fisheries Service, Northwest Fisheries Science
Center***

***John Payne, University of Washington***

Overview

*All models are wrong, some are useful*.

---Box 1976

A generic approach for identifying a viable productivity-abundance
criterion involves estimating extinction risk using a population
dynamics model and determining the threshold where productivity and
abundance parameters just yield an acceptable risk. The results of this
sort of analysis can be plotted in a "viability curve," where every
point on the curve represents a productivity-abundance combination with
identical extinction risk (Figure G.1). A key issue in developing a
specific method from this generic approach is defining the form of the
population dynamics model used to estimate extinction risk. In Table
G.1, we describe a number of relatively simple population dynamics
models that have been applied to salmon and could potentially be used to
estimate extinction risk. Many of these models are discussed in Hilborn
and Walters (1992).

![](media/image1.wmf){width="4.090157480314961in"
height="2.4346456692913385in"}

All of these models except the "constant recruitment" model involve
projecting the number of recruits as a stochastic function of the number
of spawners. The constant recruitment model assumes that recruitment
fluctuates stochastically around some fixed value, regardless of the
number of spawners. In addition to a parameter describing the variation
in the spawner-recruit relationship, some of the models include
additional terms, such as carrying capacity or marine survival. It is
important to emphasize that recruitment curves describe an average
relationship between recruits and spawners from which individual years
will surely deviate, and to reiterate the point made in the opening
quote of this appendix, none of the models in Table G.1 describes the
true relationship between recruits and spawners. The challenge is
determining whether any of them may be useful for setting viability
criteria. We return to this point in the section below on model
selection.

Table G.1 Population dynamics models proposed for salmon populations.

  ------------- ------------------------------ --------------------------------------------------------------------------------------
  Model Number  Model Name                     Equation^a^

  Model 0       Random walk                    $$R = \text{Sexp}\left( \sigma_{0}Z \right)$$

  Model 1       Random walk with drift;        $$R = S\text{exp}\left( a_{1} + \sigma_{1}Z \right)$$
                stochastic exponential growth  
                or decline                     

  Model 2       Constant recruitment           ![](media/image2.wmf){width="1.1527777777777777in" height="0.2361111111111111in"}

  Model 3       Stochastic hockey stick;       $$R = \text{min}\left( S,b_{3} \right)\text{exp}\left( a_{3} + \sigma_{3}Z \right)$$
                stochastic exponential growth  
                with a ceiling                 

  Model 4       Ricker; stochastic logistic    ![](media/image3.wmf){width="1.7638888888888888in" height="0.2361111111111111in"}

  Model 5       Beverton-Holt                  ![](media/image4.wmf){width="1.5416666666666667in" height="0.7083333333333334in"}

  Model 6       Ricker juvenile production     ![](media/image5.wmf){width="1.94375in" height="0.25in"}
                with given marine survival     
  ------------- ------------------------------ --------------------------------------------------------------------------------------

> ^a^ In the equations,
>
> *S~t~* = the number of spawners
>
> *R* = the number of recruits
>
> *Z* = a unit normal random variable
>
> *σ~\#\ ~* = the standard deviation of the process error
>
> *a~\#~* and *b~\#~* = equation-specific parameters, with the *a~\#~*
> parameter relating in some way to "intrinsic productivity" and the
> *b~\#~* parameter relating in some way to "capacity"
>
> *c~6~* = a marine survival parameter; the *a~6~* and *b~6~* parameters
> in this equation relate to the production of juvenile outmigrants from
> spawners

Viability Curves

![](media/image6.wmf){width="5.125196850393701in"
height="3.8751968503937007in"}In Figures G.2 to G.5, we present several
viability curves associated with the recruitment functions in Table G.1.
The extinction risk associated with any particular parameter combination
for a given model is found by simulating a large number of population
trajectories and counting the fraction of trajectories that drop below
the quasi-extinction risk threshold within the given time horizon. The
intrinsic productivity axis in the curves refers to the number of
recruits per spawner at very low (approaching 0) abundance. Exactly how
the intrinsic productivity value relates to extinction risk depends on
the specific form of the population dynamics model. In all the models,
the intrinsic productivity provides an indication of population
resilience, which is the tendency of the population to return toward an
equilibrium value if perturbed to low abundance. The abundance axis in
the curves refers to the point estimate equilibrium abundance. The
initial population size for the population trajectories was the
equilibrium (or mean equilibrium) abundance value for the Beverton-Holt
and Ricker curves, and the carrying capacity for the hockey-stick curve.
The shape of the viability curve was found by a grid search of the
parameter space to identify productivity-abundance combinations with
equivalent risk. This meant varying the productivity, capacity, and
process error variance parameters (i.e., the *a*, *b* and σ^2^
parameters in Table G.1) and fixing all other parameters. In addition to
the equilibrium abundance, the figures show the viability curves in
terms of the "carrying capacity." The carrying capacity has different
biological interpretations for the different models, so they are not
directly comparable. However, the shape of these capacity curves is
informative.

A common feature of all the viability curves we have examined is that as
the intrinsic productivity parameter exceeds about 1.1, the number of
spawners needed for a viable population (i.e., a population that has a
risk of less than 5% of declining to a four-year average of 50 fish in
100 years) declines to a few hundred fish. For the example, Ricker and
Beverton-Holt curves in Figures G.3 and G.4, the viable equilibrium
abundance is less than 200 spawners and relatively constant as long as
the intrinsic productivity parameter is above 1. The parameter that
varies more substantially in these models is the carrying capacity
parameter, although it is a parameter we can never directly observe. If
a population can be demonstrated to have an intrinsic productivity
substantially above 1, the actual abundance of the population becomes
much less relevant. A resilient population will likely be viable, even
if it is very small.

![](media/image7.wmf){width="6.095669291338583in"
height="3.8811023622047243in"}

![](media/image8.wmf){width="4.5875in" height="4.101388888888889in"}

Figure G.4 Viability curves based on Ricker recruitment function. The
different curves are for different levels of environmental variability.
The viability curves were generated for a semelparous population where
the average percentages of individuals spawning at a given age are: age
1 = 0%, age 2 = 1%, age 3 = 19%, age 4 = 57%, and age 5 = 23%. This
life-history structure is typical of that observed for chinook salmon.
The solid lines show equilibrium abundance, and the dashed lines show
the value of the "capacity" parameter in the Ricker function. Every
point on a curve has the same extinction probability. In this example,
the extinction probability is a 5% probability of declining to a
four-year average of 50 spawners in 100 years.

![](media/image9.wmf)

Figure G.5 Viability curves based on Ricker recruitment function for
juvenile outmigrants with a cycle in marine survival. The different
curves are for different levels of environmental variability. The
viability curves were generated for a semelparous population where the
average percentages of individuals spawning at a given age are: age 1 =
0%, age 2 = 1%, age 3 = 19%, age 4 = 57%, and age 5 = 23%. This
life-history structure is typical of that observed for chinook salmon.
The intrinsic productivity refers to the production of juvenile
outmigrants. The ocean cycle in survival was a sine wave of 40 years'
length with a mean survival of 0.05 and an amplitude of 0.03, beginning
in year 0 of the cycle. The solid lines show equilibrium abundance of
spawners, and the dashed lines show the value of the "capacity"
parameter in the Ricker function for juveniles. Every point on a curve
has the same extinction probability. In this example, the extinction
probability is a 5% probability of declining to a four-year average of
50 spawners in 100 years.

Estimating Intrinsic Productivity

A key to evaluating a population's viability of using this approach is
to estimate the intrinsic productivity. One of the great challenges with
this general approach is determining which model, if any, might be
appropriate for estimating intrinsic productivity. We can potentially
look to existing abundance time series to determine which of the
potential models is the "best approximating model" for this purpose
(Burnham and Anderson 1998). Figure G.6 is an example of a spawner
abundance time series. With information about the age structure of the
population (and in some cases, numbers of hatchery spawners), it is
possible to estimate how many recruits were naturally produced from each
year's spawning (Figure G.7). To determine which of the proposed model
forms may be useful as approximating models for setting criteria,
parameters for each model were estimated from available time series and
the models were statistically compared (e.g., Figure G.8). Formal model
selection analysis has been relatively rare in fisheries management, and
models are often adopted without adequate consideration of the
alternatives.

![](media/image10.wmf)

Figure G.6 Lower Columbia Gorge tributary chum salmon spawner abundance.

Figure G.7 Lower Columbia Gorge tributary chum salmon recruits versus
spawners.

![](media/image11.wmf)

![](media/image12.wmf)

Figure G.8 Recruitment curves for lower Columbia Gorge tributary chum
salmon.

Comparing Models

A number of different approaches exist for evaluating the relative
utility of nonnested models. Among these approaches, the Aikake
Information Criterion (AIC) method (Akaike 1973) addresses the question
that is most relevant; i.e., how much do each of the models differ from
the "true" process (Burnham and Anderson 1998). The AIC is a standard
statistical measure of how well a model fits a data set, given a
parameter set and assumptions about the error distribution. It is useful
because it penalizes models in proportion to the number of parameters
they have, without which we would not be able to compare larger models
with smaller models. We used a version of the AIC that is corrected for
small sample sizes: AIC~c~ = 2(-ln(L)) + (2p + 2\*p\*(p+1))/(n-p-1)
(Burnham and Anderson 1998).

Other methods that can be used to select among models include likelihood
ratio tests and comparing R^2^ values associated with each model. In
addition, several tests have been developed specifically to detect
density dependence in abundance time series (e.g., Dennis and Taper
1994, Bulmer 1975, Pollard et al. 1987, Ruesink 2000, Shenk et al.
1998). All these methods are conceptually different from the AIC
approach and have fundamental theoretical limitations. In our analysis,
we have concentrated on the AIC evaluation.

We have estimated parameters for all available WLC spawner data sets for
models 0--5 in Table G.1, then calculated AICc difference values to
identify the best approximating models. The results are in Table G.2a-g.
There are no fixed thresholds for interpreting the AIC difference
values, but there are some general rules of thumb (Burnham and Anderson
1998). The AICc difference value for the best model is 0. If the AIC
difference between the best model and the model with the lowest AIC is
less than 2, then the second model provides a very good approximation
relative to the best model, and the models might be given equal
consideration. If the AIC difference value is greater than 10, the model
is not a very good approximating model relative to the best model and is
not very well supported by the data. If the AIC difference is between 2
and 10, the interpretation is less clear, and the biology of the
situation and the question being asked should drive how the model is
considered.

For 20 of 32 populations examined, the best approximating model
identified using the AIC method was the constant recruitment model
(Figure G.9). The Ricker model was identified as the best approximating
model for six populations, but for four of them the AIC difference value
for the constant recruitment model was less than 2, and either the
Beverton-Holt or the hockey-stick models were also within 2, so the
models are approximately equally good. For 12 of the 32 populations, at
least one of the density-dependent recruitment models (i.e.,
hockey-stick, Ricker, or Beverton-Holt) had a low AIC difference value
and could be considered a contender as the best approximating model
(Figure G.10). The constant recruitment model is interesting because it
is the only model examined that assumes there is no relationship between
the number of spawners and the number of recruits; the number of
recruits is assumed to fluctuate around a constant value. The constant
recruitment model is biologically implausible if extrapolated to very
low spawner numbers because at the extreme, zero spawners must yield
zero recruits. However, the fact that this model was selected as the
best approximating model suggests that there is little data in the range
of very low abundance to exclusively support one of the models that
explicitly includes a relationship between spawners and recruits.
Because there are few data at low abundance, there is very little
information from which to estimate the intrinsic productivity. This is
also reflected in the large confidence intervals on the intrinsic
productivity estimates of individual models. The ability to estimate
intrinsic productivity as evaluated by the model selection analysis does
not seem to improve with increased length of the time series (Figures
G.11 and G.12).

Table G.2.a Lower Columbia River late-fall (bright) chinook salmon
population parameter estimates and model comparison. Except as noted,
recruits are based on estimates of preharvest natural-origin fish, and
spawners are based on the estimate of natural-origin spawners plus half
of the hatchery- origin spawners (hatchery-origin spawners are assumed
to have lower reproductive success than natural-origin spawners.)
Exceptions to these spawner and recruit definitions occur because of
data limitations.

+----------+------------+------+------------+-------+----+----------+
| Po       | Model^a^   | a    | b          | σ^2^  | AI | Relative |
| pulation |            |      |            |       | Cc | AICc     |
+----------+------------+------+------------+-------+----+----------+
| Lewis    | Random     |      |            | 0.86  | 3  | 10.82    |
| River    | walk       |      |            |       | 7. |          |
|          |            |      |            | (     | 50 |          |
|          |            |      |            | 0.61- |    |          |
|          |            |      |            | 1.11) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Constant   |      | 19,769     | 0.48  | 2  | 0.56     |
|          |            |      |            |       | 7. |          |
|          | r          |      | (15,0      | (     | 24 |          |
|          | ecruitment |      | 86-24,451) | 0.23- |    |          |
|          |            |      |            | 0.73) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Random     | 1.5  |            | 0.73  | 3  | 9.44     |
|          | walk with  |      |            |       | 6. |          |
|          | trend      | (1   |            | (     | 13 |          |
|          |            | .08- |            | 0.35- |    |          |
|          |            | 2.1) |            | 0.98) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Ho         | 2.4  | 19,769     | 0.48  | 3  | 3.73     |
|          | ckey-stick |      |            |       | 0. |          |
|          |            | (    | (15,0      | (     | 41 |          |
|          |            | 1.8- | 86-26,012) | 0.23- |    |          |
|          |            | 3.6) |            | 0.73) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | **Ricker** | **6  | **22,890** | **0   | *  | **0.00** |
|          |            | .9** |            | .48** | *2 |          |
|          |            |      | **(19,769  |       | 6. |          |
|          |            | **(3 | -33,816)** | **(0. | 69 |          |
|          |            | .3-1 |            | 23-0. | ** |          |
|          |            | 5)** |            | 61)** |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Bev        | 25   | 21,329     | 0.48  | 3  | 4.30     |
|          | erton-Holt |      |            |       | 0. |          |
|          |            | (5.1 | (16,6      | (     | 99 |          |
|          |            | -25) | 47-32,255) | 0.23- |    |          |
|          |            |      |            | 0.73) |    |          |
+----------+------------+------+------------+-------+----+----------+
| Sandy    | **Random   |      |            | **0   | *  | **0.00** |
| River^b^ | walk**     |      |            | .73** | *1 |          |
|          |            |      |            |       | 7. |          |
|          |            |      |            | **(0. | 21 |          |
|          |            |      |            | 48-0. | ** |          |
|          |            |      |            | 86)** |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Constant   |      | 753        | 0.73  | 2  | 5.40     |
|          |            |      |            |       | 2. |          |
|          | r          |      | (          | (     | 62 |          |
|          | ecruitment |      | 506-1,247) | 0.35- |    |          |
|          |            |      |            | 0.98) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Random     | 0.94 |            | 0.73  | 2  | 4.15     |
|          | walk with  |      |            |       | 1. |          |
|          | trend      | (    |            | (     | 37 |          |
|          |            | 0.6- |            | 0.35- |    |          |
|          |            | 1.4) |            | 0.73) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Ho         | 1.15 | 918        | 0.61  | 2  | 10.90    |
|          | ckey-stick |      |            |       | 8. |          |
|          |            | (    | (          | (     | 11 |          |
|          |            | 0.6- | 588-1,494) | 0.35- |    |          |
|          |            | 2.7) |            | 0.73) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Ricker     | 1.08 | 2,564      | 0.73  | 2  | 11.13    |
|          |            |      |            |       | 8. |          |
|          |            | (    | (          | (     | 34 |          |
|          |            | 0.65 | 671-2,564) | 0.23- |    |          |
|          |            | -12) |            | 0.73) |    |          |
+----------+------------+------+------------+-------+----+----------+
|          | Bev        | 1.4  | 2,564      | 0.73  | 2  | 11.20    |
|          | erton-Holt |      |            |       | 8. |          |
|          |            | (    | (          | (     | 41 |          |
|          |            | 0.75 | 835-2,564) | 0.35- |    |          |
|          |            | -25) |            | 0.86) |    |          |
+----------+------------+------+------------+-------+----+----------+

> ^a^ The *a*, *b*, and σ^2^ parameters for each model are described in
> Table G.1. The 95% confidence intervals on the parameter estimated are
> shown in parentheses. The AICc best approximating model for each
> population is highlighted in dark gray, and any models with an AICc
> difference \<2 are highlighted in light gray.
>
> ^b^ Recruits based on natural-origin escapement, not preharvest.

Table G.2.b Lower Columbia River spring chinook salmon population
parameter estimates and model comparison. Except as noted, recruits are
based on estimates of preharvest natural-origin fish, and spawners are
based on the estimate of natural-origin spawners plus half of the
hatchery-origin spawners (hatchery-origin spawners are assumed to have
lower reproductive success than natural- origin spawners.) Exceptions to
these spawner and recruit definitions occur because of data limitations.

+----------+-------------+-------+--------+--------+------+---------+
| Po       | Model^a^    | a     | b      | σ^2^   | AICc | R       |
| pulation |             |       |        |        |      | elative |
|          |             |       |        |        |      | AICc    |
+----------+-------------+-------+--------+--------+------+---------+
| Cowlitz  | Random walk |       |        | 0.73   | 3    | 23.23   |
| River^b^ |             |       |        |        | 5.68 |         |
|          |             |       |        | (0.61  |      |         |
|          |             |       |        | -0.86) |      |         |
+----------+-------------+-------+--------+--------+------+---------+
|          | *           |       | *      | **     | *    | *       |
|          | *Constant** |       | *315** | 0.35** | *12. | *0.00** |
|          |             |       |        |        | 45** |         |
|          | **re        |       | *      | **(    |      |         |
|          | cruitment** |       | *(262- | 0.23-0 |      |         |
|          |             |       | 351)** | .35)** |      |         |
+----------+-------------+-------+--------+--------+------+---------+
|          | Random walk | 1.15  |        | 0.73   | 3    | 25.50   |
|          | with trend  |       |        |        | 7.95 |         |
|          |             | (0.85 |        | (0.48  |      |         |
|          |             | -1.5) |        | -0.86) |      |         |
+----------+-------------+-------+--------+--------+------+---------+
|          | H           | 3.3   | 315    | 0.35   | 1    | 3.17    |
|          | ockey-stick |       |        |        | 5.62 |         |
|          |             | (2.1  | (26    | (0.23  |      |         |
|          |             | -3.6) | 2-351) | -0.35) |      |         |
+----------+-------------+-------+--------+--------+------+---------+
|          | Ricker      | 2.1   | 422    | 0.48   | 2    | 14.15   |
|          |             |       |        |        | 6.60 |         |
|          |             | (1.5  | (33    | (0.35  |      |         |
|          |             | -3.9) | 3-511) | -0.48) |      |         |
+----------+-------------+-------+--------+--------+------+---------+
|          | Be          | 25    | 333    | 0.35   | 1    | 3.34    |
|          | verton-Holt |       |        |        | 5.79 |         |
|          |             | (4.   | (28    | (0.23  |      |         |
|          |             | 5-25) | 0-422) | -0.35) |      |         |
+----------+-------------+-------+--------+--------+------+---------+

> ^a^ The *a*, *b*, and σ^2^ parameters for each model are described in
> Table G.1. The 95% confidence intervals on the parameter estimated are
> shown in parentheses. The AICc best approximating model for each
> population is highlighted in dark gray, and any models with an AICc
> difference \<2 are highlighted in light gray.
>
> ^b^ Recruits based on natural-origin escapement, not preharvest.

#  

Table G.2.c Lower Columbia River fall chinook salmon population
parameter estimates and model comparison. Except as noted, recruits are
based on estimates of preharvest natural-origin fish, and spawners are
based on the estimate of natural-origin spawners plus half of the
hatchery-origin spawners (hatchery-origin spawners are assumed to have
lower reproductive success than natural -origin spawners). Exceptions to
these spawner and recruit definitions occur because of data limitations.

+------------+-----------+------+---------+----------+------+--------+
| Population | Model^a^  | a    | b       | σ^2^     | AICc | Re     |
|            |           |      |         |          |      | lative |
|            |           |      |         |          |      | AICc   |
+------------+-----------+------+---------+----------+------+--------+
| Big White  | Random    |      |         | 1.62     | 5    | 20.18  |
| Salmon     | walk      |      |         |          | 8.53 |        |
|            |           |      |         | (1.      |      |        |
|            |           |      |         | 24-1.87) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **901** | **0.73** | *    | **     |
|            | onstant** |      |         |          | *38. | 0.00** |
|            |           |      | *       | **(0.48  | 35** |        |
|            | **recr    |      | *(732-1 | -0.86)** |      |        |
|            | uitment** |      | ,238)** |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 3    |         | 1.11     | 5    | 12.07  |
|            | walk with |      |         |          | 0.42 |        |
|            | trend     | (    |         | (0.      |      |        |
|            |           | 1.8- |         | 86-1.24) |      |        |
|            |           | 5.1) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 6.9  | 1,069   | 0.73     | 4    | 2.04   |
|            | key-stick |      |         |          | 0.39 |        |
|            |           | (    | (732    | (0.      |      |        |
|            |           | 3.6- | -1,743) | 48-0.86) |      |        |
|            |           | 9.9) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 6.9  | 1,575   | 0.73     | 4    | 2.66   |
|            |           |      |         |          | 1.00 |        |
|            |           | (4.5 | (1,069  | (0.      |      |        |
|            |           | -11) | -2,248) | 48-0.86) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 19   | 1,238   | 0.73     | 4    | 2.19   |
|            | rton-Holt |      |         |          | 0.54 |        |
|            |           | (6.6 | (901    | (0.      |      |        |
|            |           | -25) | -2,080) | 48-0.86) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Coweeman   | Random    |      |         | 2.12     | 6    | 31.30  |
| River      | walk      |      |         |          | 2.37 |        |
|            |           |      |         | (1.      |      |        |
|            |           |      |         | 74-2.37) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **994** | **0.61** | *    | **     |
|            | onstant** |      |         |          | *31. | 0.00** |
|            |           |      | *       | **(0.48  | 07** |        |
|            | **recr    |      | *(734-1 | -0.73)** |      |        |
|            | uitment** |      | ,254)** |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 6    |         | 1.11     | 4    | 15.14  |
|            | walk with |      |         |          | 6.21 |        |
|            | trend     | (    |         | (0.      |      |        |
|            |           | 3.6- |         | 48-1.36) |      |        |
|            |           | 9.3) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 11   | 1,254   | 0.61     | 3    | 1.90   |
|            | key-stick |      |         |          | 2.97 |        |
|            |           | (7.8 | (864    | (0.      |      |        |
|            |           | -25) | -1,774) | 35-0.61) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 13   | 1,644   | 0.61     | 3    | 0.08   |
|            |           |      |         |          | 1.15 |        |
|            |           | (9.6 | (1,254  | (0.      |      |        |
|            |           | -20) | -2,294) | 35-0.61) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 25   | 1,384   | 0.61     | 3    | 1.69   |
|            | rton-Holt |      |         |          | 2.76 |        |
|            |           | (13  | (994    | (0.      |      |        |
|            |           | -25) | -2,684) | 35-0.73) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Cowlitz    | Random    |      |         | 1.24     | 4    | 12.16  |
| River      | walk      |      |         |          | 4.25 |        |
|            |           |      |         | (0.      |      |        |
|            |           |      |         | 48-1.74) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Constant  |      | 1,377   | 0.86     | 3    | 4.78   |
|            |           |      |         |          | 6.87 |        |
|            | re        |      | (872    | (0.      |      |        |
|            | cruitment |      | -2,051) | 48-0.98) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 0.6  |         | 1.11     | 4    | 12.65  |
|            | walk with |      |         |          | 4.74 |        |
|            | trend     | (0   |         | (0.      |      |        |
|            |           | .6-0 |         | 48-1.49) |      |        |
|            |           | .96) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 1    | 1,545   | 0.86     | 3    | 7.64   |
|            | key-stick |      |         |          | 9.72 |        |
|            |           | (0   | (1,040  | (0.      |      |        |
|            |           | .75- | -2,388) | 35-0.98) |      |        |
|            |           | 1.8) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | *         | **3  | **      | **0.61** | *    | **     |
|            | *Ricker** | .9** | 2,051** |          | *32. | 0.00** |
|            |           |      |         | **(0.23  | 08** |        |
|            |           | *    | **(     | -0.73)** |      |        |
|            |           | *(1. | 1,545-3 |          |      |        |
|            |           | 5-8. | ,566)** |          |      |        |
|            |           | 4)** |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 25   | 1,545   | 0.86     | 4    | 8.33   |
|            | rton-Holt |      |         |          | 0.42 |        |
|            |           | (1.8 | (1,040  | (0.      |      |        |
|            |           | -25) | -3,735) | 35-0.98) |      |        |
+------------+-----------+------+---------+----------+------+--------+

Table G.2.c cont.

+------------+-----------+------+---------+----------+------+--------+
| East Fork  | Random    |      |         | 0.86     | 3    | 14.33  |
| Lewis      | walk      |      |         |          | 7.28 |        |
| River      |           |      |         | (0.      |      |        |
|            |           |      |         | 61-1.11) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **573** | **0.48** | *    | **     |
|            | onstant** |      |         |          | *22. | 0.00** |
|            |           |      | **(460  | **(0.35  | 95** |        |
|            | **recr    |      | -686)** | -0.61)** |      |        |
|            | uitment** |      |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.8  |         | 0.61     | 3    | 8.01   |
|            | walk with |      |         |          | 0.96 |        |
|            | trend     | (    |         | (0.      |      |        |
|            |           | 1.4- |         | 35-0.73) |      |        |
|            |           | 2.4) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 3    | 573     | 0.48     | 2    | 2.15   |
|            | key-stick |      |         |          | 5.10 |        |
|            |           | (    | (4      | (0.      |      |        |
|            |           | 1.8- | 60-686) | 23-0.48) |      |        |
|            |           | 3.9) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 5.1  | 630     | 0.48     | 2    | 3.09   |
|            |           |      |         |          | 6.03 |        |
|            |           | (    | (5      | (0.      |      |        |
|            |           | 2.7- | 17-799) | 23-0.61) |      |        |
|            |           | 9.9) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 15   | 630     | 0.48     | 2    | 3.03   |
|            | rton-Holt |      |         |          | 5.98 |        |
|            |           | (3.3 | (517    | (0.      |      |        |
|            |           | -25) | -1,025) | 35-0.61) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Elochoman  | Random    |      |         | 1.36     | 5    | 12.38  |
| River      | walk      |      |         |          | 1.04 |        |
|            |           |      |         | (1.      |      |        |
|            |           |      |         | 11-1.62) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **626** | **0.86** | *    | **     |
|            | onstant** |      |         |          | *38. | 0.00** |
|            |           |      | **(432  | **(0.61  | 66** |        |
|            | **recr    |      | -819)** | -0.98)** |      |        |
|            | uitment** |      |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.8  |         | 1.24     | 5    | 12.78  |
|            | walk with |      |         |          | 1.44 |        |
|            | trend     | (0.9 |         | (0.      |      |        |
|            |           | 4-3) |         | 86-1.62) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 9.9  | 626     | 0.86     | 4    | 3.31   |
|            | key-stick |      |         |          | 1.97 |        |
|            |           | (1   | (432    | (0.      |      |        |
|            |           | .8-1 | -1,109) | 61-0.86) |      |        |
|            |           | 0.5) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 4.5  | 1013    | 0.73     | 3    | 0.85   |
|            |           |      |         |          | 9.51 |        |
|            |           | (    | (723    | (0.      |      |        |
|            |           | 2.7- | -1,496) | 48-0.86) |      |        |
|            |           | 6.9) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 25   | 626     | 0.86     | 4    | 3.22   |
|            | rton-Holt |      |         |          | 1.88 |        |
|            |           | (3.6 | (432    | (0.      |      |        |
|            |           | -25) | -1,496) | 61-0.86) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Grays      | **Random  |      |         | **1.49** | *    | **     |
| River      | walk**    |      |         |          | *55. | 0.00** |
|            |           |      |         | **(1.24  | 62** |        |
|            |           |      |         | -1.62)** |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Constant  |      | 371     | 1.49     | 6    | 4.49   |
|            |           |      |         |          | 0.11 |        |
|            | re        |      | (1      | (1.      |      |        |
|            | cruitment |      | 41-716) | 24-1.74) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.3  |         | 1.36     | 5    | 2.26   |
|            | walk with |      |         |          | 7.87 |        |
|            | trend     | (    |         | (1.      |      |        |
|            |           | 0.7- |         | 11-1.62) |      |        |
|            |           | 2.1) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 2.1  | 716     | 1.24     | 5    | 1.65   |
|            | key-stick |      |         |          | 7.27 |        |
|            |           | (1   | (371    | (0.      |      |        |
|            |           | .08- | -1,176) | 98-1.36) |      |        |
|            |           | 9.6) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 3    | 601     | 1.24     | 5    | 1.53   |
|            |           |      |         |          | 7.14 |        |
|            |           | (    | (371    | (0.      |      |        |
|            |           | 1.2- | -1,865) | 86-1.36) |      |        |
|            |           | 7.2) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 3.9  | 716     | 1.24     | 5    | 1.89   |
|            | rton-Holt |      |         |          | 7.50 |        |
|            |           | (1.3 | (371    | (0.      |      |        |
|            |           | -12) | -3,359) | 98-1.36) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Kalama     | Random    |      |         | 1.36     | 5    | 12.57  |
| River      | walk      |      |         |          | 1.46 |        |
|            |           |      |         | (0.      |      |        |
|            |           |      |         | 98-1.74) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | *       | **0.86** | *    | **     |
|            | onstant** |      | *7369** |          | *38. | 0.00** |
|            |           |      |         | **(0.48  | 88** |        |
|            | **recr    |      | **(     | -0.98)** |      |        |
|            | uitment** |      | 4,917-9 |          |      |        |
|            |           |      | ,821)** |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.5  |         | 1.36     | 5    | 13.66  |
|            | walk with |      |         |          | 2.54 |        |
|            | trend     | (0.8 |         | (0.      |      |        |
|            |           | 5-3) |         | 86-1.62) |      |        |
+------------+-----------+------+---------+----------+------+--------+

Table G.2.c cont.

+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 6    | 7,369   | 0.86     | 4    | 3.31   |
|            | key-stick |      |         |          | 2.19 |        |
|            |           | (    | (4,917  | (0.      |      |        |
|            |           | 1.8- | -9,821) | 48-0.98) |      |        |
|            |           | 9.3) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 3.9  | 9,821   | 0.86     | 4    | 5.60   |
|            |           |      |         |          | 4.49 |        |
|            |           | (    | (7,369- | (0.      |      |        |
|            |           | 2.4- | 14,724) | 61-0.98) |      |        |
|            |           | 9.3) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 25   | 7,369   | 0.86     | 4    | 3.70   |
|            | rton-Holt |      |         |          | 2.59 |        |
|            |           | (6.9 | (4,917- | (0.      |      |        |
|            |           | -25) | 11,047) | 61-0.98) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Mill Creek | Random    |      |         | 1.11     | 4    | 4.91   |
|            | walk      |      |         |          | 5.87 |        |
|            |           |      |         | (0.      |      |        |
|            |           |      |         | 86-1.36) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **      | **0.86** | *    | **     |
|            | onstant** |      | 2,465** |          | *40. | 0.00** |
|            |           |      |         | **(0.48  | 96** |        |
|            | **recr    |      | **(     | -1.11)** |      |        |
|            | uitment** |      | 1,389-3 |          |      |        |
|            |           |      | ,542)** |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.5  |         | 1.11     | 4    | 5.65   |
|            | walk with |      |         |          | 6.60 |        |
|            | trend     | (0   |         | (0.      |      |        |
|            |           | .92- |         | 73-1.24) |      |        |
|            |           | 2.4) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 5.1  | 2,465   | 0.86     | 4    | 3.31   |
|            | key-stick |      |         |          | 4.27 |        |
|            |           | (1.2 | (1,748  | (0.      |      |        |
|            |           | -11) | -4,618) | 48-0.98) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 3.3  | 3,183   | 0.86     | 4    | 3.65   |
|            |           |      |         |          | 4.61 |        |
|            |           | (    | (2,465  | (0.      |      |        |
|            |           | 1.5- | -5,695) | 48-1.11) |      |        |
|            |           | 7.2) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 12   | 2,824   | 0.86     | 4    | 3.16   |
|            | rton-Holt |      |         |          | 4.12 |        |
|            |           | (1.8 | (2,107- | (0.      |      |        |
|            |           | -25) | 10,001) | 61-0.98) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Washougal  | Random    |      |         | 1.11     | 4    | 9.86   |
| River      | walk      |      |         |          | 4.61 |        |
|            |           |      |         | (0.      |      |        |
|            |           |      |         | 73-1.36) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **      | **0.73** | *    | **     |
|            | onstant** |      | 2,692** |          | *34. | 0.00** |
|            |           |      |         | **(0.35  | 75** |        |
|            | **recr    |      | **(     | -0.86)** |      |        |
|            | uitment** |      | 2,000-3 |          |      |        |
|            |           |      | ,383)** |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.2  |         | 1.11     | 4    | 12.32  |
|            | walk with |      |         |          | 7.07 |        |
|            | trend     | (    |         | (0.      |      |        |
|            |           | 0.7- |         | 73-1.36) |      |        |
|            |           | 1.8) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 8.4  | 2,692   | 0.73     | 3    | 3.31   |
|            | key-stick |      |         |          | 8.06 |        |
|            |           | (1.4 | (2,000  | (0.      |      |        |
|            |           | -12) | -3,729) | 35-0.86) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 8.1  | 4,075   | 0.73     | 3    | 2.59   |
|            |           |      |         |          | 7.34 |        |
|            |           | (3.6 | (3,037  | (0.      |      |        |
|            |           | -16) | -5,458) | 35-0.86) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 25   | 2,692   | 0.73     | 3    | 3.93   |
|            | rton-Holt |      |         |          | 8.68 |        |
|            |           | (25  | (2,000  | (0.      |      |        |
|            |           | -25) | -4,075) | 35-0.86) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Wind       | Random    |      |         | 1.99     | 4    | 14.79  |
| River^bb^  | walk      |      |         |          | 4.95 |        |
|            |           |      |         | (1       |      |        |
|            |           |      |         | .24-2.5) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Constant  |      | 208     | 1.11     | 3    | 6.36   |
|            |           |      |         |          | 6.52 |        |
|            | re        |      | (1      | (0.      |      |        |
|            | cruitment |      | 12-351) | 61-1.49) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.4  |         | 1.99     | 4    | 17.72  |
|            | walk      |      |         |          | 7.87 |        |
|            |           | (    |         | (1       |      |        |
|            | with      | 0.6- |         | .24-2.5) |      |        |
|            | trend     | 3.9) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 20   | 255     | 1.11     | 4    | 9.91   |
|            | key-stick |      |         |          | 0.07 |        |
|            |           | (2.1 | (1      | (0.      |      |        |
|            |           | -20) | 12-446) | 48-1.49) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | *         | **   | **780** | **0.61** | *    | **     |
|            | *Ricker** | 18** |         |          | *30. | 0.00** |
|            |           |      | *       | **(0.23  | 16** |        |
|            |           | **(9 | *(494-1 | -0.86)** |      |        |
|            |           | .9-2 | ,018)** |          |      |        |
|            |           | 5)** |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+

Table G.2.c cont.

+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 25   | 255     | 1.11     | 4    | 10.31  |
|            | rton-Holt |      |         |          | 0.46 |        |
|            |           | (25  | (1      | (0.      |      |        |
|            |           | -25) | 12-494) | 48-1.49) |      |        |
+------------+-----------+------+---------+----------+------+--------+
| Clackamas  | Random    |      |         | 0.86     | 6    | 22.66  |
| Riverc^c^  | walk      |      |         |          | 8.84 |        |
|            |           |      |         | (0.      |      |        |
|            |           |      |         | 73-1.11) |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | **C       |      | **447** | **0.61** | *    | **     |
|            | onstant** |      |         |          | *46. | 0.00** |
|            |           |      | **(387  | **(0.48  | 18** |        |
|            | **recr    |      | -568)** | -0.61)** |      |        |
|            | uitment** |      |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Random    | 1.02 |         | 0.86     | 7    | 25.01  |
|            | walk      |      |         |          | 1.19 |        |
|            |           | (0   |         | (0.      |      |        |
|            | with      | .75- |         | 73-1.11) |      |        |
|            | trend     | 1.4) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Hoc       | 2.7  | 508     | 0.61     | 4    | 1.91   |
|            | key-stick |      |         |          | 8.09 |        |
|            |           | (    | (3      | (0.      |      |        |
|            |           | 1.5- | 87-568) | 35-0.61) |      |        |
|            |           | 3.9) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Ricker    | 3.3  | 568     | 0.61     | 4    | 1.57   |
|            |           |      |         |          | 7.76 |        |
|            |           | (    | (5      | (0.      |      |        |
|            |           | 2.1- | 08-750) | 35-0.61) |      |        |
|            |           | 5.1) |         |          |      |        |
+------------+-----------+------+---------+----------+------+--------+
|            | Beve      | 20   | 508     | 0.61     | 4    | 2.66   |
|            | rton-Holt |      |         |          | 8.85 |        |
|            |           | (4.2 | (4      | (0.      |      |        |
|            |           | -25) | 47-629) | 48-0.61) |      |        |
+------------+-----------+------+---------+----------+------+--------+

> ^a^ The *a*, *b*, and σ^2^ parameters for each model are described in
> Table G.1. The 95% confidence intervals on the parameter estimated are
> shown in parentheses. The AICc best approximating model for each
> population is highlighted in dark gray, and any models with an AICc
> difference \<2 are highlighted in light gray.
>
> ^b^ Recruits based on natural-origin escapement, not preharvest.
>
> ^c^ Recruits based on natural-origin escapement, not preharvest.
> Spawners based on total spawners, and the fraction of hatchery-origin
> is unknown.

Table G.2.d Lower Columbia River winter steelhead population parameter
estimates and model comparison. Except as noted, recruits are based on
estimates of preharvest natural-origin fish, and spawners are based on
the estimate of natural-origin spawners plus half of the hatchery-origin
spawners (hatchery-origin spawners are assumed to have lower
reproductive success than natural-origin spawners.) Exceptions to these
spawner and recruit definitions occur because of data limitations.

+---------------+----------+-------+--------+--------+------+-------+
| Population    | Model^a^ | a     | b      | σ^2^   | AICc | Rel   |
|               |          |       |        |        |      | ative |
|               |          |       |        |        |      | AICc  |
+---------------+----------+-------+--------+--------+------+-------+
| East Fork     | Random   |       |        | 0.98   | 1    | 3.19  |
| Lewis         | walk     |       |        |        | 4.83 |       |
| River^b^b     |          |       |        | (0.73  |      |       |
|               |          |       |        | -1.11) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | **Co     |       | **86** | **     | *    | **0   |
|               | nstant** |       |        | 0.23** | *11. | .00** |
|               |          |       | **(82  |        | 64** |       |
|               | **recru  |       | -88)** | **(    |      |       |
|               | itment** |       |        | 0.23-0 |      |       |
|               |          |       |        | .23)** |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Random   | 0.6   |        | 0.48   | 2    | 9.51  |
|               | walk     |       |        |        | 1.15 |       |
|               | with     | (0.6  |        | (0.23  |      |       |
|               | trend    | -0.6) |        | -0.61) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Hock     | 0.6   | 86     | 0.23   | Infi | Inf   |
|               | ey-stick |       |        |        | nity | inity |
|               |          | (0.6- | (      | (0.23  |      |       |
|               |          | 1.08) | 82-90) | -0.23) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Ricker   | 1.02  | 90     | 0.23   | Infi | Inf   |
|               |          |       |        |        | nity | inity |
|               |          | (0.65 | (      | (0.23  |      |       |
|               |          | -1.4) | 84-95) | -0.23) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Bever    | 3.9   | 95     | 0.23   | Infi | Inf   |
|               | ton-Holt |       |        |        | nity | inity |
|               |          | (1.0  | (8     | (0.23  |      |       |
|               |          | 2-25) | 4-132) | -0.23) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
| Clackamas     | Random   |       |        | 1.24   | 12   | 66.06 |
| River         | walk     |       |        |        | 6.10 |       |
|               |          |       |        | (1.11  |      |       |
|               |          |       |        | -1.36) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | **Co     |       | **4    | **     | *    | **0   |
|               | nstant** |       | ,152** | 0.48** | *60. | .00** |
|               |          |       |        |        | 04** |       |
|               | **recru  |       | **(3,  | **(    |      |       |
|               | itment** |       | 696-5, | 0.35-0 |      |       |
|               |          |       | 063)** | .61)** |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Random   | 2.7   |        | 0.73   | 8    | 22.53 |
|               | walk     |       |        |        | 2.57 |       |
|               | with     | (2.4  |        | (0.61  |      |       |
|               | trend    | -3.3) |        | -0.73) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Hock     | 7.5   | 4,152  | 0.48   | 6    | 2.36  |
|               | ey-stick |       |        |        | 2.41 |       |
|               |          | (3.3  | (      | (0.35  |      |       |
|               |          | -7.8) | 3,696- | -0.61) |      |       |
|               |          |       | 5,063) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Ricker   | 6.6   | 5,063  | 0.48   | 6    | 6.10  |
|               |          |       |        |        | 6.14 |       |
|               |          | (4.5  | (      | (0.35  |      |       |
|               |          | -9.6) | 4,152- | -0.61) |      |       |
|               |          |       | 5,518) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Bever    | 25    | 5,063  | 0.48   | 6    | 2.89  |
|               | ton-Holt |       |        |        | 2.93 |       |
|               |          | (8.   | (      | (0.35  |      |       |
|               |          | 1-25) | 4,152- | -0.61) |      |       |
|               |          |       | 6,429) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
| Kalama River  | Random   |       |        | 0.61   | 4    | 5.90  |
|               | walk     |       |        |        | 0.55 |       |
|               |          |       |        | (0.48  |      |       |
|               |          |       |        | -0.86) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | **Co     |       | **1    | **     | *    | **0   |
|               | nstant** |       | ,108** | 0.48** | *34. | .00** |
|               |          |       |        |        | 65** |       |
|               | **recru  |       | **(    | **(    |      |       |
|               | itment** |       | 952-1, | 0.35-0 |      |       |
|               |          |       | 419)** | .61)** |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Random   | 0.88  |        | 0.61   | 4    | 7.40  |
|               | walk     |       |        |        | 2.04 |       |
|               | with     | (0.7- |        | (0.48  |      |       |
|               | trend    | 1.15) |        | -0.73) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Hock     | 2.4   | 1,108  | 0.48   | 3    | 2.85  |
|               | ey-stick |       |        |        | 7.50 |       |
|               |          | (0.85 | (952-  | (0.35  |      |       |
|               |          | -2.4) | 1,574) | -0.61) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Ricker   | 2.1   | 1,263  | 0.48   | 3    | 2.05  |
|               |          |       |        |        | 6.70 |       |
|               |          | (1.15 | (      | (0.35  |      |       |
|               |          | -3.3) | 1,108- | -0.61) |      |       |
|               |          |       | 1,729) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Bever    | 5.1   | 1,419  | 0.48   | 3    | 2.15  |
|               | ton-Holt |       |        |        | 6.79 |       |
|               |          | (1.   | (      | (0.35  |      |       |
|               |          | 4-25) | 1,108- | -0.61) |      |       |
|               |          |       | 2,817) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+

Table G.2.d cont.

+---------------+----------+-------+--------+--------+------+-------+
| North Fork    | Random   |       |        | 1.11   | 2    | 9.18  |
| Toutle        | walk     |       |        |        | 3.54 |       |
|               |          |       |        | (0.35  |      |       |
|               |          |       |        | -1.74) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | **Co     |       | *      | **     | *    | **0   |
|               | nstant** |       | *173** | 0.35** | *14. | .00** |
|               |          |       |        |        | 36** |       |
|               | **recru  |       | *      | **(    |      |       |
|               | itment** |       | *(131- | 0.23-0 |      |       |
|               |          |       | 214)** | .48)** |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Random   | 1.8   |        | 0.86   | 2    | 10.71 |
|               | walk     |       |        |        | 5.07 |       |
|               | with     | (1.1  |        | (0.23  |      |       |
|               | trend    | -3.3) |        | -1.24) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Hock     | 9.6   | 173    | 0.35   | 2    | 7.00  |
|               | ey-stick |       |        |        | 1.36 |       |
|               |          | (1.   | (14    | (0.23  |      |       |
|               |          | 5-18) | 2-225) | -0.48) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Ricker   | 4.5   | 235    | 0.61   | 2    | 12.11 |
|               |          |       |        |        | 6.47 |       |
|               |          | (2.   | (18    | (0.23  |      |       |
|               |          | 1-11) | 3-359) | -0.73) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Bever    | 25    | 183    | 0.35   | 2    | 7.27  |
|               | ton-Holt |       |        |        | 1.63 |       |
|               |          | (2.   | (16    | (0.23  |      |       |
|               |          | 4-25) | 2-370) | -0.61) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
| South Fork    | **Random |       |        | **     | *    | **0   |
| Toutle        | walk**   |       |        | 0.48** | *10. | .00** |
|               |          |       |        |        | 71** |       |
|               |          |       |        | **(    |      |       |
|               |          |       |        | 0.23-0 |      |       |
|               |          |       |        | .73)** |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Constant |       | 1,526  | 0.35   | 1    | 1.36  |
|               |          |       |        |        | 2.07 |       |
|               | rec      |       | (      | (0.23  |      |       |
|               | ruitment |       | 1,224- | -0.35) |      |       |
|               |          |       | 1,828) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Random   | 0.94  |        | 0.48   | 1    | 6.59  |
|               | walk     |       |        |        | 7.31 |       |
|               | with     | (0.7  |        | (0.23  |      |       |
|               | trend    | -1.3) |        | -0.61) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Hock     | 1.8   | 1,526  | 0.35   | 3    | 21.36 |
|               | ey-stick |       |        |        | 2.07 |       |
|               |          | (0.   | (      | (0.23  |      |       |
|               |          | 7-20) | 1,299- | -0.35) |      |       |
|               |          |       | 1,903) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Ricker   | 3.9   | 1,677  | 0.35   | 3    | 22.36 |
|               |          |       |        |        | 3.07 |       |
|               |          | (0.83 | (      | (0.23  |      |       |
|               |          | -9.9) | 1,526- | -0.35) |      |       |
|               |          |       | 3,186) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Bever    | 25    | 1,526  | 0.35   | 3    | 21.51 |
|               | ton-Holt |       |        |        | 2.22 |       |
|               |          | (1.1  | (      | (0.23  |      |       |
|               |          | 5-25) | 1,375- | -0.35) |      |       |
|               |          |       | 3,186) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
| Sandy River   | Random   |       |        | 0.23   | 4.92 | 8.98  |
|               | walk     |       |        |        |      |       |
|               |          |       |        | (0.23  |      |       |
|               |          |       |        | -0.35) |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | **Co     |       | **2    | **     | *    | **0   |
|               | nstant** |       | ,696** | 0.23** | *-4. | .00** |
|               |          |       |        |        | 07** |       |
|               | **recru  |       | **(2,  | **(    |      |       |
|               | itment** |       | 616-2, | 0.23-0 |      |       |
|               |          |       | 855)** | .23)** |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Random   | 0.96  |        | 0.23   | 7.64 | 11.71 |
|               | walk     |       |        |        |      |       |
|               | with     | (     |        | (0.23  |      |       |
|               | trend    | 0.83- |        | -0.35) |      |       |
|               |          | 1.08) |        |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Hock     | 1.3   | 2,775  | 0.23   | 0.01 | 4.08  |
|               | ey-stick |       |        |        |      |       |
|               |          | (1.15 | (      | (0.23  |      |       |
|               |          | -2.7) | 2,616- | -0.23) |      |       |
|               |          |       | 2,855) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Ricker   | 2.7   | 2,775  | 0.23   | 0.07 | 4.14  |
|               |          |       |        |        |      |       |
|               |          | (2.4  | (      | (0.23  |      |       |
|               |          | -3.9) | 2,696- | -0.23) |      |       |
|               |          |       | 3,014) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+
|               | Bever    | 25    | 2,855  | 0.23   | 0.27 | 4.33  |
|               | ton-Holt |       |        |        |      |       |
|               |          | (7.   | (      | (0.23  |      |       |
|               |          | 8-25) | 2,696- | -0.23) |      |       |
|               |          |       | 3,173) |        |      |       |
+---------------+----------+-------+--------+--------+------+-------+

> ^a^ The *a*, *b*, and σ^2^ parameters for each model are described in
> Table G.1. The 95% confidence intervals on the parameter estimated are
> shown in parentheses. The AICc best approximating model for each
> population is highlighted in dark gray, and any models with an AICc
> difference \<2 are highlighted in light gray.
>
> ^b^ Recruits based on natural-origin escapement, not preharvest.

Table G.2.e Lower Columbia River summer steelhead population parameter
estimates and model comparison. Recruits are based on estimates of
preharvest natural-origin fish, and spawners are based on the estimate
of natural-origin spawners plus half of the hatchery-origin spawners
(hatchery-origin spawners are assumed to have lower reproductive success
than natural-origin spawners.)

+----------+--------------+-------+--------+--------+-----+---------+
| Po       | Model^a^     | a     | b      | σ^2^   | A   | R       |
| pulation |              |       |        |        | ICc | elative |
|          |              |       |        |        |     | AICc    |
+----------+--------------+-------+--------+--------+-----+---------+
| Kalama   | Random walk  |       |        | 1.49   | 76  | 26.72   |
| River    |              |       |        |        | .34 |         |
|          |              |       |        | (1.24  |     |         |
|          |              |       |        | -1.87) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | **Constant** |       | *      | **     | **4 | *       |
|          |              |       | *906** | 0.73** | 9.6 | *0.00** |
|          | **r          |       |        |        | 3** |         |
|          | ecruitment** |       | **(    | **(    |     |         |
|          |              |       | 687-1, | 0.61-0 |     |         |
|          |              |       | 343)** | .86)** |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Random walk  | 0.6   |        | 1.24   | 68  | 19.09   |
|          | with trend   |       |        |        | .72 |         |
|          |              | (0.6  |        | (0.86  |     |         |
|          |              | -0.6) |        | -1.49) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Hockey-stick | 0.8   | 906    | 0.73   | 52  | 2.79    |
|          |              |       |        |        | .42 |         |
|          |              | (0.6  | (687-  | (0.48  |     |         |
|          |              | -2.7) | 1,343) | -0.86) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Ricker       | 0.65  | 1,343  | 0.86   | 55  | 6.06    |
|          |              |       |        |        | .69 |         |
|          |              | (0.6  | (906-  | (0.61  |     |         |
|          |              | -1.3) | 1,781) | -0.98) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | B            | 5.1   | 1,125  | 0.73   | 52  | 2.85    |
|          | everton-Holt |       |        |        | .48 |         |
|          |              | (0.8  | (906-  | (0.48  |     |         |
|          |              | 3-25) | 1,562) | -0.86) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
| W        | Random walk  |       |        | 0.61   | 20  | 7.45    |
| ashougal |              |       |        |        | .84 |         |
| River    |              |       |        | (0.48  |     |         |
|          |              |       |        | -0.61) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | **Constant** |       | *      | **     | **1 | *       |
|          |              |       | *178** | 0.35** | 3.3 | *0.00** |
|          | **r          |       |        |        | 9** |         |
|          | ecruitment** |       | *      | **(    |     |         |
|          |              |       | *(151- | 0.23-0 |     |         |
|          |              |       | 218)** | .48)** |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Random walk  | 0.8   |        | 0.48   | 21  | 8.48    |
|          | with trend   |       |        |        | .87 |         |
|          |              | (0.6- |        | (0.23  |     |         |
|          |              | 1.04) |        | -0.61) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Hockey-stick | 1.8   | 178    | 0.35   | 17  | 3.93    |
|          |              |       |        |        | .32 |         |
|          |              | (0.7  | (15    | (0.23  |     |         |
|          |              | -8.7) | 1-272) | -0.48) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Ricker       | 1.8   | 205    | 0.35   | 19  | 6.23    |
|          |              |       |        |        | .62 |         |
|          |              | (0.   | (17    | (0.23  |     |         |
|          |              | 92-3) | 8-299) | -0.48) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | B            | 7.2   | 205    | 0.35   | 17  | 3.77    |
|          | everton-Holt |       |        |        | .16 |         |
|          |              | (1.   | (16    | (0.23  |     |         |
|          |              | 1-25) | 4-489) | -0.35) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
| Wind     | Random walk  |       |        | 0.48   | 13  | 11.19   |
| River    |              |       |        |        | .88 |         |
|          |              |       |        | (0.35  |     |         |
|          |              |       |        | -0.61) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Constant     |       | 486    | 0.35   | 11  | 8.81    |
|          | recruitment  |       |        |        | .50 |         |
|          |              |       | (41    | (0.23  |     |         |
|          |              |       | 9-587) | -0.35) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | **Random     | **0   |        | **     | **  | *       |
|          | walk with    | .65** |        | 0.23** | 2.6 | *0.00** |
|          | trend**      |       |        |        | 9** |         |
|          |              | **(   |        | **(    |     |         |
|          |              | 0.6-0 |        | 0.23-0 |     |         |
|          |              | .7)** |        | .23)** |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Hockey-stick | 0.65  | 855    | 0.23   | 7   | 5.12    |
|          |              |       |        |        | .82 |         |
|          |              | (0.6- | (48    | (0.23  |     |         |
|          |              | 0.75) | 6-855) | -0.23) |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | Ricker       | 0.75  | 1,290  | 0.23   | 8   | 5.50    |
|          |              |       |        |        | .20 |         |
|          |              | (0.65 | (      | (0.23  |     |         |
|          |              | -0.9) | 1,089- | -0.23) |     |         |
|          |              |       | 1,290) |        |     |         |
+----------+--------------+-------+--------+--------+-----+---------+
|          | B            | 1.02  | 1,290  | 0.23   | 9   | 6.93    |
|          | everton-Holt |       |        |        | .63 |         |
|          |              | (0.85 | (      | (0.23  |     |         |
|          |              | -1.3) | 1,290- | -0.23) |     |         |
|          |              |       | 1,290) |        |     |         |
+----------+--------------+-------+--------+--------+-----+---------+

^a^ The *a*, *b*, and σ^2^ parameters for each model are described in
Table G.1. The 95% confidence intervals on the parameter estimated are
shown in parentheses. The AICc best approximating model for each
population is highlighted in dark gray, and any models with an AICc
difference \< 2 are highlighted in light gray.

**Table G.2.f Columbia River chum salmon population parameter estimates
and model comparison. Recruits are based on estimates of preharvest
natural-origin fish, and spawners are based on the estimate of
natural-origin spawners plus half of the hatchery-origin spawners
(hatchery-origin spawners are assumed to have lower reproductive success
than natural-origin spawners.)**

+---------+-------------+-------+---------+---------+------+--------+
| Pop     | Model^a^    | a     | b       | σ^2^    | AICc | Re     |
| ulation |             |       |         |         |      | lative |
|         |             |       |         |         |      | AICc   |
+---------+-------------+-------+---------+---------+------+--------+
| Grays   | Random walk |       |         | 1.11    | 8    | 14.88  |
| Riverb  |             |       |         |         | 4.72 |        |
|         |             |       |         | (0.8    |      |        |
|         |             |       |         | 6-1.49) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Constant    |       | 402     | 0.86    | 6    | 0.00   |
|         |             |       |         |         | 9.85 |        |
|         | recruitment |       | (3      | (0.6    |      |        |
|         |             |       | 19-569) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Random walk | 1.3   |         | 1.11    | 8    | 15.52  |
|         | with trend  |       |         |         | 5.37 |        |
|         |             | (0.94 |         | (0.7    |      |        |
|         |             | -1.8) |         | 3-1.36) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | H           | 25    | 402     | 0.86    | 7    | 2.59   |
|         | ockey-stick |       |         |         | 2.44 |        |
|         |             | (1.1  | (319    | (0.6    |      |        |
|         |             | 5-25) | -1,069) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Ricker      | 2.1   | 819     | 0.98    | 8    | 12.45  |
|         |             |       |         |         | 2.29 |        |
|         |             | (1.4  | (569    | (0.7    |      |        |
|         |             | -4.2) | -1,152) | 3-1.24) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Be          | 25    | 485     | 0.86    | 7    | 2.53   |
|         | verton-Holt |       |         |         | 2.38 |        |
|         |             | (1.   | (402    | (0.6    |      |        |
|         |             | 8-25) | -1,152) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
| Hardy   | Random walk |       |         | 1.49    | 14   | 51.33  |
| Creek   |             |       |         |         | 5.19 |        |
|         |             |       |         | (1.1    |      |        |
|         |             |       |         | 1-1.99) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Constant    |       | 180     | 0.73    | 9    | 0.00   |
|         |             |       |         |         | 3.86 |        |
|         | recruitment |       | (1      | (0.6    |      |        |
|         |             |       | 49-212) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Random walk | 1.4   |         | 1.49    | 14   | 51.35  |
|         | with trend  |       |         |         | 5.21 |        |
|         |             | (0.96 |         | (1.1    |      |        |
|         |             | -2.1) |         | 1-1.87) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | H           | 25    | 180     | 0.86    | 10   | 9.77   |
|         | ockey-stick |       |         |         | 3.63 |        |
|         |             | (7.   | (1      | (0.6    |      |        |
|         |             | 8-25) | 49-244) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Ricker      | 4.5   | 338     | 1.11    | 12   | 33.70  |
|         |             |       |         |         | 7.56 |        |
|         |             | (2.4  | (2      | (0.8    |      |        |
|         |             | -7.8) | 44-433) | 6-1.49) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Be          | 25    | 212     | 0.86    | 10   | 9.71   |
|         | verton-Holt |       |         |         | 3.57 |        |
|         |             | (     | (1      | (0.6    |      |        |
|         |             | 9-25) | 49-244) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
| Lower   | Random walk |       |         | 2.12    | 22   | 100.53 |
| Gorge   |             |       |         |         | 7.65 |        |
|         |             |       |         | (1.     |      |        |
|         |             |       |         | 36-2.5) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Constant    |       | 474     | 0.73    | 12   | 0.00   |
|         |             |       |         |         | 7.12 |        |
|         | recruitment |       | (3      | (0.6    |      |        |
|         |             |       | 83-565) | 1-0.98) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Random walk | 1.5   |         | 2.12    | 22   | 100.85 |
|         | with trend  |       |         |         | 7.97 |        |
|         |             | (0.96 |         | (1.     |      |        |
|         |             | -2.4) |         | 36-2.5) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | H           | 25    | 383     | 1.24    | 17   | 45.10  |
|         | ockey-stick |       |         |         | 2.21 |        |
|         |             | (2    | (3      | (0.7    |      |        |
|         |             | 5-25) | 83-474) | 3-1.49) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Ricker      | 6.3   | 929     | 1.62    | 20   | 73.42  |
|         |             |       |         |         | 0.54 |        |
|         |             | (3.   | (656    | (0.9    |      |        |
|         |             | 3-16) | -1,658) | 8-1.99) |      |        |
+---------+-------------+-------+---------+---------+------+--------+
|         | Be          | 25    | 474     | 1.24    | 17   | 46.03  |
|         | verton-Holt |       |         |         | 3.15 |        |
|         |             | (2    | (3      | (0.7    |      |        |
|         |             | 5-25) | 83-565) | 3-1.49) |      |        |
+---------+-------------+-------+---------+---------+------+--------+

> ^a^ The a, b, and σ2 parameters for each model are described in Table
> G.1. The 95% confidence intervals on the parameter estimated are shown
> in parentheses. The AICc best approximating model for each population
> is highlighted in dark gray, and any models with an AICc difference \<
> 2 are highlighted in light gray.

Table G.2.g Upper Willamette River spring chinook salmon population
parameter estimates and model comparison. Except as noted, recruits are
based on estimates of preharvest natural-origin fish, and spawners are
based on the estimate of natural-origin spawners plus half of the
hatchery-origin spawners (hatchery-origin spawners are assumed to have
lower reproductive success than natural-origin spawners.) Exceptions to
these spawner and recruit definitions occur because of data limitations.

+------------+-----------+----------+----------+-------+-----+-------+
| Population | Model^a^  | a        | b        | σ^2^  | A   | Rel   |
|            |           |          |          |       | ICc | ative |
|            |           |          |          |       |     | AICc  |
+------------+-----------+----------+----------+-------+-----+-------+
| Clackamas  | Random    |          |          | 0.73  | 92  | 11.27 |
| River^c^   | walk      |          |          |       | .86 |       |
|            |           |          |          | (     |     |       |
|            |           |          |          | 0.61- |     |       |
|            |           |          |          | 0.86) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Constant  |          | 1,238    | 0.86  | 102 | 20.41 |
|            |           |          |          |       | .00 |       |
|            | re        |          | (1,03    | (     |     |       |
|            | cruitment |          | 6-1,440) | 0.73- |     |       |
|            |           |          |          | 0.98) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Random    | 1.2      |          | 0.73  | 92  | 11.31 |
|            | walk with |          |          |       | .90 |       |
|            | trend     | (0       |          | (     |     |       |
|            |           | .99-1.5) |          | 0.61- |     |       |
|            |           |          |          | 0.86) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Hoc       | 1.5      | 2,250    | 0.61  | 82  | 0.43  |
|            | key-stick |          |          |       | .03 |       |
|            |           | (        | (1,84    | (     |     |       |
|            |           | 1.2-2.1) | 5-2,655) | 0.48- |     |       |
|            |           |          |          | 0.73) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Ricker    | 2.1      | 2,048    | 0.61  | 81  | 0.00  |
|            |           |          |          |       | .59 |       |
|            |           | (        | (1,84    | (     |     |       |
|            |           | 1.5-2.7) | 5-2,858) | 0.48- |     |       |
|            |           |          |          | 0.73) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Beve      | 2.1      | 3,465    | 0.61  | 82  | 1.08  |
|            | rton-Holt |          |          |       | .67 |       |
|            |           | (        | (2,45    | (     |     |       |
|            |           | 1.5-3.3) | 3-5,287) | 0.48- |     |       |
|            |           |          |          | 0.73) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
| McKenzie   | Random    |          |          | 0.86  | 65  | 24.05 |
| River^c^   | walk      |          |          |       | .10 |       |
|            |           |          |          | (     |     |       |
|            |           |          |          | 0.61- |     |       |
|            |           |          |          | 0.98) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Constant  |          | 2,242    | 0.48  | 41  | 0.00  |
|            |           |          |          |       | .05 |       |
|            | re        |          | (1,98    | (     |     |       |
|            | cruitment |          | 4-2,760) | 0.35- |     |       |
|            |           |          |          | 0.61) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Random    | 0.94     |          | 0.86  | 67  | 26.28 |
|            | walk with |          |          |       | .34 |       |
|            | trend     | (0       |          | (     |     |       |
|            |           | .75-1.3) |          | 0.61- |     |       |
|            |           |          |          | 0.98) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Hoc       | 3        | 2,242    | 0.48  | 43  | 2.57  |
|            | key-stick |          |          |       | .62 |       |
|            |           | (        | (1,98    | (     |     |       |
|            |           | 1.3-9.6) | 4-2,760) | 0.35- |     |       |
|            |           |          |          | 0.61) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Ricker    | 2.7      | 2,760    | 0.61  | 50  | 9.56  |
|            |           |          |          |       | .62 |       |
|            |           | (        | (2,24    | (     |     |       |
|            |           | 1.8-4.5) | 2-3,278) | 0.35- |     |       |
|            |           |          |          | 0.61) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+
|            | Beve      | 25       | 2,242    | 0.48  | 44  | 3.57  |
|            | rton-Holt |          |          |       | .62 |       |
|            |           | (5.1-25) | (1,98    | (     |     |       |
|            |           |          | 4-3,019) | 0.35- |     |       |
|            |           |          |          | 0.61) |     |       |
+------------+-----------+----------+----------+-------+-----+-------+

^a^ The *a*, *b*, and σ^2^ parameters for each model are described in
Table G.1. The 95% confidence intervals on the parameter estimated are
shown in parentheses. The AICc best approximating model for each
population is highlighted in dark gray, and any models with an AICc
difference \<2 are highlighted in light gray.

^c^ Recruits based on natural-origin escapement, not preharvest.
Spawners based on total spawners, and the fraction of hatchery-origin is
unknown.

![](media/image13.wmf){width="5.970866141732284in"
height="4.17007874015748in"}

![](media/image15.wmf)

Figure G.11 Best model as a function of the number of years of
spawner-recruit data (see Table G.2).

![](media/image16.wmf)

Figure G.12 Best model and near-best models as a function of the number
of years of spawner-recruit data (see Table G.2). Models considered near
best had AIC difference values less than 2.

Model Section Using Simulated Data

Although the observed data do not provide much information about the
exact value of intrinsic productivity in a population, the selection of
the constant recruitment model does suggest on the surface that
productivity is greater than 1 and that the population is simply showing
random fluctuations around a carrying capacity. However, this conclusion
may be overly optimistic. We simulated a number of population
trajectories using a hockey-stick model with an intrinsic productivity
of 1. The populations were started substantially below the ceiling, so
the trajectories were basically a random walk with an upper bound. We
then calculated the recruits-per-spawner values from the trajectories,
calculated parameters for the six models, and applied the AIC model
selection approach (Figures G.11). In most of these examples, the best
approximating model was either the constant recruitment model or the
random-walk model. We speculate that the constant recruitment model is
commonly selected as the best model because a short time series that
samples a random-walk process appears as a cloud of points on a
spawner-recruit graph. In the absence of data at very low (or very high)
spawner numbers, the data are likely to fit a constant recruitment
model. This is particularly true if the there is any sort of population
ceiling that leads to a flattening of the spawner-recruit cloud.

![](media/image17.wmf)

Figure G.11 Frequency of models selected as the best approximating model
for simulated population trajectories. We simulated 100 25-year
population trajectories with a hockey-stick model with an initial
population size of 5,000, carrying capacity of 50,000, intrinsic
productivity of 1, with log-normal process error (normal distribution
mean = 0 and variance = 0.6). The variance of 0.6 is similar to that
observed for Willamette/Lower Columbia populations (see Table G.2). The
viability curves were generated for a semelparous population where the
average percentages of individuals spawning at a given age are: age 1 =
0%, age 2 = 1%, age 3 = 19%, age 4 = 57%, and age 5 = 23%. This
life-history structure is typical to that observed for chinook salmon.

Conclusions

Analysis of both the WLC populations and simulated trajectories suggest
that the adult recruits-per-spawner data typically available for salmon
populations will be inadequate to estimate intrinsic productivity. The
lack of data at small spawner abundances make recruit curve--based
productivity estimates highly uncertain.

Recruitment models are a foundation of harvest modeling and have been
proposed as metrics for viability criteria. Before applying these
models, it is important to have a solid understanding of the
uncertainties involved in parameter estimation and model selection.

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