Appendix D

Population Change criteria

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

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

#  {#section .Level-3}

Population Change Criteria Overview

![](media/image1.wmf){width="4.283464566929134in"
height="2.839763779527559in"}The population change criteria (PCC)
provide a novel performance test for evaluating whether a threatened
population has recovered and is no longer in danger of extinction. The
approach starts with the development of a viability curve, which
describes the relationships among population abundance, productivity,
and extinction risk (Figure D.1). The extinction risk experienced by a
population is a function of both the population's productivity and size
(Musick 1999, McElhany et al. 2000). We define productivity as the
number of returns produced per spawner, when the population is at low
density relative to carrying capacity. All else being equal, a
population with a high average productivity could persist at a lower
abundance than a population with a low average productivity. This is
because a population with high average productivity would have a higher
probability of returning to the original abundance if perturbed to low
abundance than a population with low average productivity. A
high-productivity population could be characterized as being more
resilient than a low-productivity population. The amount of
environmental variation affects the likelihood that a population will be
perturbed to low abundance and is another key parameter in the
estimation of extinction risk. With regard to population size, all else
being equal, the smaller a population is, the more likely it is to
fluctuate to extinction (Thomas 1990, Lande 1993). The viability curve
can be estimated using a population projection model that incorporates
abundance, productivity, environmental variability, and any other factor
considered relevant for estimating extinction risk.

Key issues with developing criteria from viability curves are
determining an appropriate form for the population projection model and
determining how to estimate parameters. As described in the first
section of this appendix, the projection model used for the PCC
viability curve is relatively simple and is well described in the
population dynamics literature. The next section of this appendix
("Projection Model and Minimum-Size Estimation Methods") describes the
distinguishing features of the PCC approach. These features involve the
method used to estimate productivity and the development of a population
performance test.

Projection Model and Minimum-Size Estimation Methods

Model Overview

We calculated a viability curve using a population projection model of
stochastic exponential growth with a ceiling and a lower critical
threshold (Figures D.2 through D.5).

$\begin{array}{r}
N_{t + 1} = 0\text{  if  }N_{t} \leq \text{QET} \\
N_{t + 1} = N_{t}e^{r}\text{  if  QET} < N_{t} \leq k \\
N_{t + 1} = \text{ke}^{r}\text{    if  }N_{t} > k \\
\text{where }r \approx \text{Normal}(\mu,\sigma)\text{.}
\end{array}$ Eq. 1

where

*N~t~* is the population size at time *t*,

*k* is the maximum size of the reproductive population (i.e.,
"ceiling"),

*r* is a stochastic parameter describing the per capita reproductive
rate, and

QET is the quasi-extinction threshold.

The parameter μ is the median per capita growth rate of a population
below *k,* and σ^2^ describes the environmental variability in growth
rate ("process variance"). The normal distribution of *r* is a
theoretical consequence of the central limit theorem applied to a
multiplicative survival process (Hilborn and Walters 1992). In the
nomenclature of recruitment models, this is a "stochastic hockey-stick"
model, as compared to a Ricker or Beverton-Holt model (Barrowman and
Myers 2000). The median annual growth rate, λ, for a population below
*k* is λ = e^μ^. We will refer to the median growth rate of a population
below *k* as the productivity of the population, and represent
productivity with the symbol γ. The Ricker and Beverton-Holt recruitment
models have a productivity parameter often symbolized as α, which
represents the "intrinsic productivity" or number of returns per spawner
if there was only one spawner (Hilborn and Walters 1992). Since the
interpretation and values of the parameters in the hockey-stick and the
other models differ, we have adopted a different symbol to avoid
confusion. If γ \> 1, the equilibrium mean abundance with this model is
near *k*. If γ \< 1, the equilibrium mean abundance is 0 (extinction).
Extinction risk using the model is estimated as the probability that a
population starting at some initial population size, *N~0~*, declines to
the QET within a given time horizon. The extinction risk is estimated by
simulating the population process with some given growth rate and
process variance to produce many population trajectories, then
calculating the fraction of simulated population trajectories that
declined to QET within the specified time period.

Because of the age structure of salmon populations, the population
dynamics model was applied to a four-year running sum of annual spawner
counts as described in Holmes (2001) and McClure et al. (in review).
Thus,

$N_{t} = \sum_{i = 0}^{3}S_{t - i}$,

where *N~t~* is as above and *S~t~* is the number of spawners in year
*t*. Both initial population sizes and QET are stipulated in the model
in terms of four-year sums, which is equivalent to an average annual
spawner count over four years of *N*/4.

Using this model, we identify the minimum population size for a given
productivity as the initial population size, *N~0~*, which just produces
an acceptable extinction risk (Figure D.2). The minimum size is found
using a simple search algorithm that tests the extinction risk
associated with a number of different potential initial population
sizes. If a population were to start out at a size smaller than the
minimum size, the extinction risk would be too high; and if the initial
population size were larger, the extinction risk would be lower than the
acceptable risk originally specified. The variance parameter of the
model, σ^2^, is an empirical estimate based on recent historical
abundance time-series data for the population or species (see below for
estimation approach). The population ceiling, *k*, is set as the initial
population size. Thus, we estimate the minimum population size under the
scenario that the minimum population size is also the population
ceiling. This effectively allows the minimum population size estimate to
also be an estimate of minimum carrying capacity. We can seldom estimate
with confidence the carrying capacity of a population, and this approach
provides a precautionary estimate of the minimum population size, since
a population constrained by a low ceiling has a higher extinction risk
than a population without a ceiling.

This is a very simplified model of salmonid dynamics, which does not
include many of the features associated with salmon biology, such as
ocean regime shifts, short-term temporal autocorrelations, complex
recruitment functions, etc. We addressed these issues in a variety of
ways, and the final criteria reflect consideration of more factors than
are reflected in Equation 1 alone.

![](media/image2.wmf){width="5.885416666666667in"
height="7.127777777777778in"}

Figure D.2 Simulated population trajectories illustrating the
relationship between population abundance, environmental variability,
and extinction risk. The lower line indicates the quasi-extinction
threshold (QET); populations that drop below this level are considered
functionally extinct.

![](media/image3.wmf)

Figure D.3 Conceptual drawing of recruitment function for projection
model to identify minimum population size. This is a hockey-stick model,
with a depensitory threshold. Below *QET* spawners, the population is
considered extinct. Above *k* spawners, the returns are constant. The
slope of the line at abundances between *QET* and *k* is an indication
of the productivity of the population (γ). This graph represents only
the deterministic skeleton of the model. Productivity is actually a
stochastic variable driven by environmental variation.

![](media/image4.wmf)

Figure D.4 Simulated trajectory showing the dynamics of the population
dynamics model. The upper dashed line represents *k* and the lower
dashed line represents *QET*. Once the population goes below *QET*, it
is considered functionally extinct, but the trajectory in the diagram
continues in order to show the future dynamics had a lower *QET* been
selected. Because this is a stochastic model, it is possible for a
population to temporarily exceed *k*, but *k* does constrain the upper
size of the population.

![](media/image5.wmf)

Figure D.5 Viability curves for populations with different values of
environmental variability. The acceptable risk is a 5% probability of
declining to a four-year average of 50 spawners in 100 years.

Specifying the Acceptable Risk

This criteria approach requires the specification of an acceptable
extinction risk. The acceptable risk can be stated as the probability
that the population will decline to QET individuals in "time horizon"
years. The probability and time horizon parameters are largely policy
decisions about acceptable risk, and options regarding these values are
presented in this document. The QET should have some biological meaning.
This is the population size below which depensitory (Allee) effects are
believed to be so strong that extinction risk greatly increases because
of processes in addition to environmental stochasticity, or that
uncertainty about population behavior becomes unacceptably high (Dennis
et al. 1991). This is an extremely difficult parameter to estimate, and
the consequences of this parameter estimation problem are discussed
below.

Setting QET

Some of the processes that may be important in setting the QET are
inbreeding depression, loss of genetic diversity, ecological Allee
effects, mate finding, and demographic stochasticity (Goodman 1987,
Lande 1998). Of these processes, we set QET at an abundance that avoids
potential negative effects from demographic stochasticity and the loss
of genetic diversity.

Demographic stochasticity refers to variability in fitness (family size)
among individuals, whereas environmental stochasticity refers to
environmental variability that affects the mean fitness of the entire
population (Lande 1998). The individual variability only tends to affect
extinction risk at very small population sizes, because at larger sizes
individual variations average out and environmental stochasticity
dominates. Demographic stochasticity can lead to increased extinction
risk of small populations, because even if the environment is constant,
chance variations in family size may result in reproductive failure of
all individuals in a single year. Risk from demographic stochasticity is
also influenced by chance variations in sex ratio (i.e., there is some
probably that only one gender will return). To inform our choice of QET,
we explored an individually based simulation model that identified an
abundance above which a population is expected to be relatively immune
from risks associated with demographic stochasticity caused by
variations in family size and chance fluctuations in sex ratio (McElhany
and Payne in prep). This model suggested that if a population stays
above 40 spawners in a given year, it is likely to experience little
additional extinction risk from demographic stochasticity over 100
years. This finding is similar to other studies of risks from
demographic stochasticity (Lande 1998).

A number of theoretical and empirical studies relate extinction risk and
loss of genetic diversity (e.g., Soule 1980, Thomas et al. 1996, Keller
and Waller 2002). As one measure of genetic diversity, the rate of loss
of neutral alleles can inform our selection of QET, though it is
difficult to make direct links between the loss of neutral alleles and
population viability. Published studies on the loss of genetic diversity
in small population sizes suggest that at effective population sizes
below about 50, there is a relatively high probability of the loss of
neutral alleles due to genetic drift (Soule 1980). The effective
population size, N~e~, is a genetic term referring to number of
individuals required if the population had an "ideal" mating system
(Wright 1938). The effective size of a population is generally smaller
than the census count of the population (Waples 1990a and 1990b) and by
assuming an average generation time of five years and an effective
population size to census count ratio of 0.2, the Puget Sound Technical
Recovery Team developed a recommended QET of an average of 62.5 spawners
per year for four years (PS-TRT 2002).

Both the demographic stochasticity and genetic loss approaches suggest
that extinction risk is affected by deleterious processes in addition to
environmental stochasticity at population sizes below about 50 spawners
in a given year. Therefore, we used a QET value of 50 spawners per year
for estimating growth rate and abundance viability criteria. This annual
spawner count threshold translates to a QET of 200 in the four-year
running-sum model (Eq. 1).

Estimating Variance

After the acceptable risk statement is specified, the only parameter
used to derive the estimation of the minimum population size for a given
productivity is the estimate of environmental variance. Environmental
variance is the variance parameter describing the distribution of *r* in
equation 1. If we assume that perfect abundance counts are available and
that a population is not experiencing density dependence, the variance
parameter can be estimated from an abundance time series as (Dennis et
al. 1991):

${\hat{\sigma}}^{2} = \text{var}\left( \text{ln}\left( \frac{N_{t + 1}}{N_{t}} \right) \right)$.
Eq. 2

If the population is near some density-dependent carrying capacity, this
equation will tend to underestimate the environmental variance parameter
in equation 1. Because the recent historical time series used to
estimate the environmental variance typically contain large measurement
errors, we employed the slope method variance estimation technique
developed by Holmes (2001). This method helps correct for the large
upward bias in the variance estimate that is produced by measurement
error. The slope method equation is:

${\hat{\sigma}}^{2} = \text{ slope of }\text{var}\left( \text{ln}\left( \frac{N_{t + \tau}}{N_{t}} \right) \right)\text{ vs}\text{.}\text{ τ}$,
Eq. 3

where τ is the temporal lag between the values used for the variance
estimate. For our variance estimations, we estimated the slope based on
a maximum τ of 4.

The variance estimate is just that, an estimate. Because we assume,
based on theoretical and empirical considerations, that
ln(*N~t+1~*/*N~t~*) is normally distributed, we have an estimate of the
sampling distribution of σ^2^. The sampling distribution of the variance
of a normally distributed random variable is:

$\sigma^{2} \approx \frac{{\hat{\sigma}}^{2} \ast \text{df}}{Χ_{\text{df}}^{2}}$,
Eq. 4

where *df* is the sample degrees of freedom, and $Χ_{\text{df}}^{2}$ is
a chi square distribution with *df* degrees of freedom (Sokal and Rohlf
1981). If the variance is estimated using perfect abundance counts and
equation 2, the degrees of freedom is equal to the number of
*N~t+1~*/*N~t~* ratios minus 1. If four-year running sums are used, the
degrees of freedom would be the number of annual spawner counts minus 4.
Variance estimates calculated with the slope method have this same
distributional form, but the degrees of freedom are reduced (Holmes and
Fagan 2002). Although the slope method reduces bias in the variance
estimate associated with measurement error, it does so at a cost of
decreased precision. Holmes and Fagan (2002) have calculated tables for
determining the degrees of freedom associated with slope method variance
estimates.

It is likely that, because of unique circumstances, every population has
a unique environmental variance value. However, the variance estimate
for any particular population is often extremely uncertain because
available time-series data sets are short relative to the levels of
variability. If we assume that the populations within an evolutionarily
significant unit (ESU) tend to experience similar levels of
environmental variation, we can obtain a potentially more accurate and
precise estimate of the variance by "pooling" variance estimates from
multiple populations. If it is assumed that there is a single true
environmental variance value that is common to every population in an
ESU and that every population time series represents an independent
sample of that variance, the average of all the population estimates
provides an unbiased estimate of the true variance, and the sample
distribution has the degrees of freedom equal to the sum of the degrees
of freedom from each individual population estimate. Under the
assumption that all populations experience basically the same levels of
environmental variation, the differences in observed variance estimates
for individual populations represent a form of sampling error and do not
necessarily reflect true differences in variation.

In calculating the minimum population size, we are interested in the
natural levels of environmental variation that will be present no matter
what hatchery or harvest management strategy is employed. Hatcheries and
harvests have the potential to obscure estimates of natural
environmental variation if we simply look at number of spawners on the
spawning ground. Therefore, in our approach we have incorporated a way
of partitioning out the variance changes induced by hatcheries and
harvest (McClure et al, McElhany and Payne in prep). We single out
hatcheries and harvest for this variance correction process partially
because we can measure the effect, but primarily because we have an *a
priori* expectation that hatcheries and harvest will alter the level of
variation observed on the spawning ground since most harvest strategies
explicitly or implicitly seek to reduce variation in escapement and
hatcheries are likewise expected to affect observed levels of variance.
These variance estimation details are presented in Appendix E.

The variance estimation approach assumes that the historical time series
is not experiencing density dependence. If the historical time series
represents a population at carrying capacity, then the variance estimate
describes the variability in carrying capacity and survival. It is not
clear whether this variance estimate would be higher or lower than the
variance observed if a population were not experiencing density
dependence. If the carrying capacity is fairly stable, the variance
estimate calculated for a population near carrying capacity would tend
to underestimate the variance of the population abundance below carrying
capacity. The power to detect density dependence is generally pretty low
(Dennis and Taper 1994, Appendix G this document), which increases our
uncertainty about the variance estimate. Given that many populations are
declining, it seems reasonable to assume that they are below capacity
and are declining, because survivals are too low for replacement;
however, the populations could simply be tracking a declining capacity.

Using recent time series to estimate levels of environmental variation
for modeling future population dynamics carries the explicit assumption
that the recent past will be predictive of future levels of
environmental variation (stationarity assumption). Human actions can
affect environmental variation, and the future may not resemble the
past, but we cannot predict the magnitude or direction of potential
change. In general, the viability criteria are determined assuming that
the past is a good predictor of future behavior of salmon populations.
To the extent that this assumption is violated, the criterion will need
to be reevaluated. We obviously will not know the extent to which the
assumption is violated until the future happens. It is important to
actively test the model's assumptions.

PCC Targets

PCC Targets Overview

If the demographic model and viability curves are going to be employed
to establish viability criteria, it is necessary to somehow estimate
population productivity. The viability of a population is a function of
both the population size and productivity. Therefore, both population
size and productivity will need to be evaluated in the future to
determine whether currently listed populations have achieved viable
status.

The traditional fisheries approach to estimating productivity relies on
fitting recent time-series data to stock-recruitment functions such as
the Ricker, Beverton-Holt, or hockey-stick models (Hilborn and Walters
1992, Appendix G this document). However, there is generally very little
statistical power to estimate productivity with the stock-recruitment
model fitting approach (Hilborn and Walters 1992, Appendix G this
document). In fact, it is often impossible to even determine whether or
not a population has experienced density dependence near capacity over
the observed time period (Dennis and Taper 1994, Hooten 1995, Ray and
Hastings 1996, Shenk et al. 1998, McClure et al. in review). The
conclusion researchers tend to reach regarding whether or not a
population is at carrying capacity depends on prior assumptions and on
how the question is asked. If the null hypothesis (prior assumption) is
that the population *is not* experiencing density dependence, the
hypothesis is generally very difficult to disprove. If the null
hypothesis (prior assumption) is that the population *is* experiencing
density dependence, that hypothesis is also generally very hard to
disprove. Accurately and precisely estimating intrinsic productivity is
even more challenging than testing hypotheses about carrying capacity
because estimating intrinsic productivity requires extrapolation to
predict recruitment at very low (i.e., \< 1 fish) spawner abundances
(Hilborn and Walters 1992). There is seldom much data at these low
abundances to support the extrapolations. The extrapolations tend to
depend critically on the exact form of the recruitment function
employed, and there is often little statistical power to distinguish
among different possible recruitment functions (Appendix G). An
understanding of the limitations of recruitment curve fitting would be
greatly advanced if confidence intervals or probability distributions
were commonly reported for parameter estimates of intrinsic
productivity, and if formal model selection methods (e.g., Akaike's
Information Criterion (AIC)) were adopted. Although in some situations
data clearly convey a particular stock-recruitment relationship, they
tend to be the exception rather than the rule.

As an alternative to fitting stock-recruitment functions, we have relied
on estimates of the population growth rate (observed λ) as a measure of
population productivity (γ). The observed growth rate of a population is
a precautionary estimate of population productivity, in that the
productivity is unlikely to be lower than the observed growth rate, but
it may very well be higher. If a population is below carrying capacity,
it can grow as a result of increased survival, in which case λ is, by
definition, an appropriate estimate of γ (Table D.1). If a population is
near carrying capacity, population growth requires an increase in
capacity. The γ value for a population tracking an increase in capacity
may be expected to be at least equal to its observed growth rate, though
it may be higher.

Table D.1 Possible relationships between median annual growth rate and
intrinsic productivity.

+------------+-----------+--------------+-----------------------------+
| Observed   | Carrying  | Intrinsic    | Interpretation              |
| Median     | Capacity  | Productivity |                             |
| Annual     | (*k*)     | (γ)          |                             |
| Growth     |           |              |                             |
| Rate (λ)   |           |              |                             |
+------------+-----------+--------------+-----------------------------+
| λ \< 1     | *N* \<    | γ = λ \<1    | Population below carrying   |
|            | *k*       |              | capacity and declining      |
|            |           |              | because of low survival.    |
+------------+-----------+--------------+-----------------------------+
|            | *N* =     | γ \>= 1;     | Population tracking a       |
|            | *k*;      |              | declining carrying          |
|            |           | γ may be \>  | capacity.                   |
|            | *k*       | λ            |                             |
|            | declining |              |                             |
+------------+-----------+--------------+-----------------------------+
| λ = 1      | *N* \<    | γ = λ = 1    | Population below carrying   |
|            | *k*       |              | capacity and productivity   |
|            |           |              | just at replacement.        |
+------------+-----------+--------------+-----------------------------+
|            | *N* =     | γ \>= 1;     | Population has relatively   |
|            | *k*;      |              | high intrinsic productivity |
|            |           | γ may be \>  | and is fluctuating around   |
|            | *k*       | λ            | capacity.                   |
|            | stable    |              |                             |
+------------+-----------+--------------+-----------------------------+
| λ \>1      | *N* \<    | γ = λ \> 1   | Population below capacity,  |
|            | *k*       |              | improvement in survival     |
|            |           |              | produces productivity       |
|            |           |              | greater than 1. Population  |
|            |           |              | will stabilize (λ = 1) once |
|            |           |              | it reaches capacity.        |
+------------+-----------+--------------+-----------------------------+
|            | *N* =     | γ \> 1;      | Population has relatively   |
|            | *k*;      |              | high intrinsic productivity |
|            |           |              | and is tracking an          |
|            | *k*       |              | increasing capacity.        |
|            | i         |              |                             |
|            | ncreasing |              |                             |
+------------+-----------+--------------+-----------------------------+

It is possible to calculate in advance the growth rate associated with a
particular change in population size over a specified period of time
using the equation

$\lambda = \text{exp}\left( \frac{\text{ln}\left( \frac{\varphi}{\iota} \right)}{y} \right)$.
Eq. 5,

where *ι* is the initial population size, *φ* is the final population
size, and *y* is the number of years between observations. For example,
if a population increased from a four-year average annual abundance of
1,000 spawners to 1,800 in 20 years, the point estimate of λ (= γ) would
be 1.033. In addition, the spawner abundance at the end of the 20 years
would be 1,800. This ability to estimate productivity associated with a
given increase in population size allows for the calculation of the PCC
(Figures D.6 and D.7). With PCC, we ask, "Given the current population
size, how big does the population need to be in *Y* years to have
demonstrated a productivity and abundance that gives an acceptable
risk?" This future population size that gives an acceptable risk we
refer to as the target size for the population in *Y* years. The target
size of a population is a function of the current size of the
population, the environmental variance of the population, the acceptable
risk statement, and the number of years in which to reach the target.
The target size is found using a search algorithm that examines the
extinction risk associated with a number of different potential target
sizes before identifying the target size with the specified acceptable
risk.

The PCC targets may be expressed equivalently as either a target
abundance in a given number of years when starting from a given initial
abundance (i.e., *φ* in Eq. 5) or as a population growth rate when
starting from a given abundance (i.e.,$\hat{\lambda}$ in Eq. 5). In this
appendix, we report both abundance and growth rate, but in presenting
criteria tend to focus on the growth rate targets. Expressing the target
as a growth rate emphasizes the key parameter driving the extinction
risk evaluation, which is
productivity.![](media/image6.wmf){width="5.508333333333334in"
height="5.945833333333334in"}

![](media/image7.wmf){width="5.763888888888889in"
height="7.432638888888889in"}

A computer program for calculating PCC based on user input is available
at
<http://research.nwfsc.noaa.gov/cbd/trt/trt_wlc/viability_report.htm>.

Parameter Uncertainty in Setting Criteria

There are a number of important assumptions and uncertainties associated
with this approach to setting viability criteria. One major source of
uncertainty is model uncertainty. Any model is a simplification of
reality that attempts to capture the key elements of the problem in
order to address specific questions. The appropriateness of the model
construct we have used is discussed in the section "Model Uncertainty."
In this section, we discuss incorporating uncertainty surrounding
parameter estimation in the criteria. In applying the criteria, three
parameters are estimated from time series of abundance: σ^2^, γ (= λ),
and population abundance. The other biologically informed model
parameter, QET, is not estimated from the salmon time series.

Because there is natural variability in the system and only relatively
short time series are available, there is some probability that the
point estimates generated for σ^2^ and λ will not reflect the true
parameter values. This uncertainty is captured in the parameters'
sampling distributions. The sampling distributions of σ^2^ and λ can be
estimated based on the model assumption that ln(*N~t+1~*/*N~t~*) is
normally distributed. The sampling distribution of σ^2^ is given in
Equation 4 and is a function of the point estimate of the
variance,${\hat{\sigma}}^{2}$, and the degrees of freedom for the
estimate, which is a direct function of the number of years of data used
to calculate the variance estimate. The sampling distribution of λ is:

$\begin{array}{r}
\lambda \approx e^{\mu}, \\
\mu \approx \mu - \text{tinv}\left( \text{df} \right)\sqrt{\frac{\sigma^{2}}{b}}
\end{array}$, Eq. 6

$\mu = \text{mean}\left( \text{ln}\left( \frac{N_{t + 1}}{N_{t}} \right) \right)$,
Eq. 7

$\sigma^{2} = \text{var}\left( \text{ln}\left( \frac{N_{t + 1}}{N_{t}} \right) \right)$,
Eq. 8

where *tinv*(*df*) is the inverse t-distribution with *df* degrees of
freedom, *df* is the degrees of freedom associated with the variance
estimate, and *b* is the number of N~*t*+1~/N*~t~* ratios used to
calculate$\hat{\mu}$. If the four-year running sum approach is used, *b*
= number of years of spawner counts ­minus 4. Note that the time series
used to estimate ${\hat{\sigma}}^{2}$, does not need to be identical to
the time series used to estimate $\hat{\mu}$, and the *df* associated
with the sampling distribution is functionally independent of the *b*
parameter. This allows the use of the variance estimate and degrees of
freedom associated with the pooled variance estimate in determining the
sampling distribution of λ (see Appendix E). The *b* parameter will be a
function of the number of years needed to achieve the target.

Because there is uncertainty in the parameter estimates, the true
probability of extinction is not simply the fraction of time the
population with point estimate σ^2^ and γ values is expected to go
extinct. There is some probability that the true σ^2^ value is higher
than ${\hat{\sigma}}^{2}$and/or that the true μ is lower than
$\hat{\mu}$, in which case the probability of extinction would be higher
than that estimated by the parameter point estimates. Likewise, there is
some probability that the true σ^2^ value is lower than
${\hat{\sigma}}^{2}$and/or that the true μ is higher than$\hat{\mu}$, in
which case the probability of extinction would be lower than that
estimated by the parameter point estimates. To account for this
uncertainty, we calculated the population prediction intervals to
establish the PCC targets (Figures D.8 through
D.10).![](media/image8.wmf){width="5.611805555555556in"
height="6.159722222222222in"}

To calculate the population prediction interval, we simulate the
population process ![](media/image9.png){width="6.430708661417323in"
height="3.5744094488188978in"}![](media/image13.wmf){width="6.5in"
height="6.75in"}trajectories that drop below QET within the specified
period of time (e.g., 100 years). Rather than parameterize the model
simply using the point estimates, the γ and σ^2^ parameters are drawn
independently and randomly from the appropriate sampling distributions.
This approach has been referred to in the literature as population
prediction intervals, parametric bootstrapping, or simply a type of
Monte Carlo simulation. Figures D.6 and D.8 compare extinction risks
calculated with point estimates and risks calculated using population
prediction intervals. When we incorporate the uncertainty associated
with parameter estimation into our assessment of extinction risk, we
generally require larger target population size for a given acceptable
level of risk. Original guidance from NMFS identified an acceptable
population extinction risk of a 5% probability of extinction in 100
years for a VSP. ![](media/image14.wmf){width="5.847222222222222in"
height="6.513194444444444in"}

In order to evaluate the status of a population relative to the
criteria, it is also necessary to estimate its abundance at the initial
and target time periods. The time series of abundance is not informative
regarding the accuracy of the abundance estimates. To access uncertainty
about the abundance estimates, it is necessary to know something about
the measurement and sampling error associated with the count method. The
WLC-TRT has not yet evaluated the errors associated with different
abundance estimates; we assume that the initial and target abundances
are measured precisely and without bias. As future studies evaluate the
accuracy of abundance counts, the target sizes may need to be adjusted
to achieve the same level of certainty about the population extinction
risk.

The QET is a biological parameter that is not estimated from salmon
data. The only way we can incorporate uncertainty about QET into our
criteria assessment is through sensitivity analysis (Figures D.11 and
D.12). In sensitivity analysis, we explore the effect of changing the
assumption about QET on the proposed criteria. As the γ value increases,
the effect of QET declines.

Hatcheries and PCC Targets

In assessing viability, we are concerned with the question of whether a
population would be naturally self-sustaining. Hatchery-origin fish that
spawn with natural-origin fish have the potential to "mask" the
productivity of the wild population (McClure et al. in review). The
equation for estimating the growth rate used to calculate the PCC target
of a population with hatchery-origin fish is:

$\lambda = \text{exp}\left( \text{mean}\left( \text{ln}\left( \frac{N_{t + 1}}{N_{t} + \text{hN}_{t}} \right) \right) \right) = \text{exp}\left( \frac{\text{ln}\left( \frac{\varphi}{\iota} \ast (1 - \eta)^{b} \right)}{y} \right)$,
Eq. 9

where *N~t~* is the number of natural-origin spawners in year *t*,
*hN~t~* describes the effective number of hatchery-origin fish spawning
in year *t* as a function of *N~t~*, *φ* is the target number of
natural-origin spawners, *ι* is the current number of natural-origin
spawners, *η* is the effective proportion of the spawning population of
hatchery origin, and *y* is the number of years between observations.
The effective proportion of hatchery-origin spawners may be different
from the census count proportion of hatchery-origin spawners if
hatchery-origin fish have a different reproductive success than
natural-origin spawners. The fraction of hatchery-origin spawners is the
fraction anticipated over the target period. Figure D.13 shows the
effect of changing the fraction of hatchery-origin spawners. A
relatively small fraction of hatchery-origin spawners can have a big
impact on the target size needed to demonstrate a given level of
productivity. To evaluate the productivity of a population with
hatchery-origin spawners, it is necessary to have an accurate estimate
of the effective fraction of hatchery-origin fish.

![](media/image15.wmf){width="5.9875in" height="6.884722222222222in"}

Ocean Cycles

The population dynamics model described in Equation 1 assumes no
temporal autocorrelation in productivity. However, salmon are recognized
as experiencing decade-scale periods of higher- or lower-than-average
productivity as a result of long-term cycles in ocean conditions (Mantua
et al. 1997, Anderson 1998, Beamish et al. 1999, Hare et al. 1999).
These long-period "regime shifts" are difficult to model because they
are difficult to predict. However, they can have significant
consequences for setting and evaluating performance of viability
criteria. It is important to not conclude that population is viable
during a period of high marine survival if it can be anticipated that
the population is likely to go extinct during the next period of low
marine survival. Likewise, we would not want to conclude that a
population is not viable during a period of low ocean survival if it can
be anticipated that the long-term prospects for the population are good,
given that it is likely to soon enter a period of higher ocean survival.
These issues are illustrated in Figure D.14. We partially address this
concern about ocean cycles by including juvenile outmigrant (JOM)
criteria, which attempt to separate out the freshwater and marine
survivals. However, we also considered marine cycles in setting adult
abundance viability criteria.

![](media/image16.wmf){width="6.5in" height="3.3244094488188978in"}

Given that it is difficult to predict patterns of marine survival, we
took the approach of modifying the target criteria as a function of how
the marine survival over the target period compared to the long-term
average marine survival (Figure D.15). The modification, applied to the
calculation of λ over the target period, is as follows:

$\lambda = \text{exp}\left( \frac{\text{ln}\left( \frac{\varphi}{\iota} \right)}{y} - \frac{\text{ln}\left( \frac{\nu}{\theta} \right)}{y} \right)$,
Eq. 10

where *ν* is the marine survival index observed over the target period,
*θ* is the long-term average value of the same marine survival index and
all other symbols are as in Equation 5. A basic assumption of this
approach is that the target values calculated without the correction
represent the minimum sizes based on some long-term average growth rate.
When we apply the correction, we assume that the observed growth rate
differs from the long-term average growth rate in an amount that is
proportional to the difference between the observed marine index and the
long-term average marine index. Since there is logically a direct
relationship between ocean survival and productivity throughout the life
cycle, this a reasonable assumption.

In developing the viability criteria, we applied this correction
asymmetrically; that is, the modification is only used to increase the
target during periods of high ocean survival, not to reduce the target
during periods of low ocean survival. This is a precautionary
application. If we observe a population with a marine survival over the
target period that is higher than long-term average, we are relatively
certain that at some future time the marine survival will decrease; thus
we should stipulate a higher target during the "good" ocean years. The
converse is not necessarily true. If we observe a lower than long-term
average marine survival over the target period, it is not clear that
marine survival will improve. This is because human activities---such as
those that affect global warming---may have permanently reduced ocean
productivity for salmon, or the condition of fish as they leave
freshwater may be the cause of the low marine survivals. For these
reasons, we did not lower the abundance target during periods of low
ocean survival.

We have not yet identified the appropriate index (assuming one exists)
to use for this marine survival modification to the target criteria.
Several candidates exist, including measures of marine survival
estimates from hatchery-marked fish or physical indexes such as the
Pacific Decadal Oscillation (PDO) or El Niño-Southern Oscillation
(ENSO), which are correlated with salmon marine survival. Although many
features of this marine index approach are conceptually attractive,
whether it can be satisfactorily implemented remains to be seen.

![](media/image17.wmf){width="5.996527777777778in"
height="6.804166666666666in"}

Figure D.15 Population change criteria showing the effect of marine
survival modification. The marine ratio is the marine survival index
observed over the target period divided by the long-term average marine
index. The criteria are based on population prediction intervals. The
variance is 0.05, with 20 degrees of freedom, and the acceptable risk is
a 20% probability of declining to a four-year average of 50 spawners in
100 years. Panel B shows an expansion of the lower portion of the x axis
of panel A. The "replacement" curve is for reference purposes; it
indicates where the target size equals the initial size.

Model Uncertainty

We address model uncertainty by evaluating how well the criteria
performed when confronted with simulated time series abundance data that
was generated using processes other than those used to set the criteria
(McElhany and Payne in prep.)(Figure D.16). For example, we generated a
large number of trajectories with different recruitment functions (e.g.,
Ricker, Beverton-Holt), short-lag autocorrelations, decadal-scale regime
shifts, and changes in population carrying capacity. We then calculated
viability criteria using the early part of the simulated time series,
determined the conclusion we would reach about the population after
applying the criteria to the next segment of the time series, and
finally looked at the long-term fate of the simulated population to
determine whether our conclusions were correct. For every scenario
tested we generated a table like Table D.2 to examine the rate at which
the criteria lead to certain types of errors. The criteria tested by
McElhany and Payne are not identical to the criteria presented in this
report (for example, the marine index modification is a recent addition
to the criteria), but the criteria in the drafts are very similar, and
the general conclusions are appropriate to both. In general, the
criteria were robust to the exact function of the population dynamics
model (e.g., Ricker versus hockey-stick recruitment function, presence
of short-lag autocorrelation, etc.). As expected, the criteria lead to
the wrong conclusion most often when the population is starting at
carrying capacity and has a high intrinsic productivity. Under these
conditions, a population has a relatively low risk of extinction, and
the criteria tended to be overly precautionary by not recognizing the
populations as viable. Given the low current abundance of most
populations, it is anticipated that most populations will need to grow
to be considered viable, and this overly precautionary scenario will be
the exception rather than the rule.

Minimum Targets

The PPC approach is appropriate once the initial population size is
above a certain level, but it does not work well at extremely small
initial sizes. For example, we cannot use the approach to set a target
for a currently extirpated population. The analysis requires evaluating
the term *targetSize*/*initalSize*. Since *initialSize* for an
extirpated population is 0, the term is undefined, and no target size
can be identified. Even if we have a non-zero initial size, so that the
equations are solvable, there is still a difficulty at small population
size. If the initial size is one fish and the population increases to 50
fish over 20 years, the growth rate for the population is large (λ =
1.28, or a 28% increase per year), and because of the large growth rate,
a population size of 50 may exceed the minimum size requirement for an
acceptable risk (this is a function of the variance and QET). However,
50 fish may not be considered adequate target abundance for a number of
reasons. One primary reason is because the proportional error rates in
abundance estimates tend to be higher at small abundance (Holmes and
Fagan 2002). Therefore, an estimate of productivity made at small
population size is more likely to be wrong than an estimate made at
higher population size. Consequently, we developed a set of minimum
targets that should be met no matter how low the initial estimate of
abundance. These minimum targets are based on setting a minimum initial
population size that will serve as the basis for target criteria for all
populations starting below the minimum initial size. Because of the
uncertainty concerns, we have explored a number of values as the minimum
initial size. If a population is below the minimum default value and
achieves the targets for a population with an initial size of the
minimum default value, the population will actually have a higher point
estimate productivity than would be required if the criteria algorithm
were simply applied at the low abundance.

Alternative Methods of Estimating Productivity

The population change criteria provide a precautionary and statistically
defensible approach to estimating the intrinsic productivity of a
population. However, in some cases it may not be necessary to directly
observe population growth in order to conclude that a population has a
productivity-size combination with an acceptably low risk level. If a
population demonstrates a productivity-size combination above the
appropriate viability curve, the population would be considered viable.

As discussed in Appendix G, fitting recruitment models to abundance data
generally provides poor estimates of intrinsic productivity, but in
particular cases data may support the use of this method. Appendix H
describes a particular two life-stage recruits-per-spawner model.
Information available for harvested populations may provide additional
data to evaluate the productivity of a population. Given certain
assumptions about natural levels of post-harvest mortality, it may be
possible to estimate something about the "resilience" of a population
(though not necessarily its intrinsic productivity). Calculations
involving harvest would need to have an accurate method of assessing the
harvest rate actually experienced by a particular population. In
addition, an accurate accounting of hatchery fish in the system would be
required to estimate natural productivity.

To be used to evaluate the viability of a population, any alternative
method of estimating population productivity would need to meet
reasonable standards of statistical rigor. The potential use of
alternative methods to estimate productivity does not really aid in
specifying, *a priori,* a particular point on the viability curve to use
as a target. Rather, the alternative methods may be used to
retrospectively evaluate whether or not a population should be
considered viable.

Application of Population Change Criteria to Healthy Populations

The PCC approach is only applicable for evaluating whether or not a
population that has been depressed below its historical abundance has
improved in status and should be considered viable. If a population has
not been depressed below its historical abundance, it would not be
expected to grow in the future. If a population is not growing, the PCC
approach assumes that the population productivity is 1. Abundance
targets associated with a productivity of 1 are often larger than
estimates of historical abundance. We would intuitively categorize a
population that is stable at about its historical abundance as "healthy"
because we are assuming, perhaps unconsciously, that the population
productivity is actually greater than 1, and that the population is not
growing because it is constrained by carrying capacity. If a population
is stable at about historical abundance, we may not require further
evidence about its productivity to conclude that it is viable.
Alternatively, we may be able to apply one of the alternative methods
for estimating productivity described in the previous section.

Most Willamette/Lower Columbia (WLC) populations are substantially below
historical abundance and are not considered currently healthy, hence the
Endangered Species Act (ESA) listing. Even the most abundant population,
the Lewis River bright chinook salmon population, at its most recent
four-year annual average of 8,900 spawners, is well below the historical
estimate of equilibrium abundance based on habitat productivity
viability analysis (HPVA) of 43,000 spawners. Even given the
uncertainties associated with the ecosystem diagnosis and treatment
(EDT) estimates, it seems likely that there is, at least theoretically,
potential for the population to grow.

Evaluation Time Period

Power analyses indicate that at least 12 years of data are required
before λ estimates have any meaning (Holmes 2001, Holmes and Fagan 2002,
McElhany and Payne in prep, McClure et al. 2003). While we have shown
10-year observation periods for illustration purposes, 10 years is
really too short; 15 to 20 years is more appropriate, both in terms of
estimating growth rate and averaging over a longer portion of any marine
survival cycles (Figures D.6, D.7, and D.17). However tempting it may be
to conclude that a population is okay if it achieves the target
abundance before 15 to 20 years, it is crucial to recognize that such a
conclusion would be statistically unsound. The criteria are based on
variability patterns, and it is necessary to wait and see if the
population is still above the target size after the target time. Even a
declining population may momentarily exceed the target size, and it is
the long-term behavior of the population that is relevant.

An important question in applying these criteria is when to start
evaluating population status. One strategy is to simply start with the
current population size and look forward. Alternatively, we can
stipulate that any time series of acceptable length that meets the
criteria and includes the most recent year's data would qualify as
viable. While the later option may be possible in some populations, for
many of them there is simply no credible historical time series
available: starting from the present and looking forward is the only
option. Given the sensitivity of the criteria to small changes in the
fraction of hatchery-origin spawners, it becomes even more unlikely that
historical data are adequate. However, it is possible to include data
before 2002 in assessing the status of populations if the data are of
sufficient quality.

It is not possible to entirely stipulate the criteria in advance because
they depend on evaluating marine survivals over some future period.
Although the projected fraction of hatchery-origin spawners can be
estimated, it too will need to be actually evaluated to determine if the
abundance target is adequate. As part of an adaptive management
protocol, the variance estimates should also be updated as more data
become available.

![](media/image18.wmf){width="6.000694444444444in"
height="6.799305555555556in"}

Figure D.17 Target size as a function of the number of years to reach
the target for a number of different initial population sizes. The
criteria are based on population prediction intervals. The variance is
0.05 with 20 degrees of freedom, and the acceptable risk is a 20%
probability of declining to a four-year average of 50 spawners in 100
years. Panel B shows an expansion of the lower portion of the y axis of
panel A.

PCC Criteria in the WLC

Current Abundance and Hatchery Fraction

PCC targets (either growth rate or abundance) assume a variety of
conditions, which can be found in Tables D.3 and D.4. An appropriate
target could be determined from Table D.3 or D.4 if the current
population size (Table D.5) and the other model parameters are known.
The current population sizes for many WLC populations are found in Table
D.5. The table also contains the recent fraction of hatchery-origin
spawners for some populations, which could be used in conjunction with
Table D.3, assuming that the current fraction of hatchery-origin
spawners will continue into the future. However, hatchery production is
under human control, and the future fraction of hatchery-origin spawners
will reflect future policy decisions.

Variance Estimates

The key empirical parameter for setting the criteria is the estimate of
environmental variance. Variance estimates for populations in the WLC
domain are summarized in Appendix E. The Lower Columbia ESUs have
average variance point estimates of about 0.05; a value of 0.05 was used
to generate criteria for these populations. In general, the variance
estimates (and targets) will need to be evaluated as more data become
available.

Final PCC Recommendations

This appendix is intended to describe and illustrate the PCC approach by
example. The final WLC-TRT recommendations regarding the PCC criteria
are located in the main text of this document. The final recommendations
include a discussion of when it would be appropriate to use the PCC
approach as viability criteria and when other methods should be used.

Table D.3 Sensitivity analysis of PCC targets. Targets are expressed as
observed, median, annual population growth rates, assessed on a
four-year running sum.

<table style="width:100%;">
<colgroup>
<col style="width: 7%" />
<col style="width: 7%" />
<col style="width: 6%" />
<col style="width: 4%" />
<col style="width: 6%" />
<col style="width: 4%" />
<col style="width: 4%" />
<col style="width: 4%" />
<col style="width: 4%" />
<col style="width: 6%" />
<col style="width: 6%" />
<col style="width: 4%" />
<col style="width: 6%" />
<col style="width: 4%" />
<col style="width: 6%" />
<col style="width: 9%" />
<col style="width: 7%" />
</colgroup>
<tbody>
<tr class="odd">
<td rowspan="2"><p><strong>Current</strong></p>
<p><strong>Size</strong></p></td>
<td rowspan="2"><strong>Standard<sup>a</sup></strong></td>
<td rowspan="2"><strong>Time Period 40<sup>b</sup></strong></td>
<td colspan="2"><strong>Variance<sup>c</sup></strong></td>
<td colspan="3"><strong>Variance Degrees of
Freedom<sup>d</sup></strong></td>
<td colspan="3"><strong>Hatchery Fraction<sup>e</sup></strong></td>
<td rowspan="2"><strong>QET<sup>f</sup> 100</strong></td>
<td colspan="3"><strong>Extinction Risk<sup>g</sup></strong></td>
<td rowspan="2"><strong>Acceptable Time Horizon<br />
200<sup>h</sup></strong></td>
<td rowspan="2"><strong>Marine<br />
Index<br />
Long-Term<sup>i</sup></strong></td>
</tr>
<tr class="even">
<td><strong>.01</strong></td>
<td><strong>.1</strong></td>
<td><strong>5</strong></td>
<td><strong>10</strong></td>
<td><strong>40</strong></td>
<td><strong>5%</strong></td>
<td><strong>10%</strong></td>
<td><strong>30%</strong></td>
<td><strong>1</strong></td>
<td><strong>25</strong></td>
<td><strong>60</strong></td>
</tr>
<tr class="odd">
<td>100</td>
<td>12%</td>
<td>7%</td>
<td>4%</td>
<td>18%</td>
<td>16%</td>
<td>13%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>&gt;21%</td>
<td>14%</td>
<td>16%</td>
<td>7%</td>
<td>4%</td>
<td>13%</td>
<td>13%</td>
</tr>
<tr class="even">
<td>150</td>
<td>11%</td>
<td>6%</td>
<td>4%</td>
<td>17%</td>
<td>15%</td>
<td>12%</td>
<td>10%</td>
<td>15%</td>
<td>20%</td>
<td>&gt;21%</td>
<td>13%</td>
<td>15%</td>
<td>6%</td>
<td>2%</td>
<td>12%</td>
<td>12%</td>
</tr>
<tr class="odd">
<td>200</td>
<td>11%</td>
<td>6%</td>
<td>3%</td>
<td>16%</td>
<td>15%</td>
<td>12%</td>
<td>10%</td>
<td>15%</td>
<td>20%</td>
<td>&gt;21%</td>
<td>12%</td>
<td>14%</td>
<td>5%</td>
<td>1%</td>
<td>12%</td>
<td>12%</td>
</tr>
<tr class="even">
<td>500</td>
<td>9%</td>
<td>5%</td>
<td>3%</td>
<td>14%</td>
<td>12%</td>
<td>10%</td>
<td>9%</td>
<td>13%</td>
<td>19%</td>
<td>&gt;21%</td>
<td>10%</td>
<td>13%</td>
<td>4%</td>
<td>-1%</td>
<td>11%</td>
<td>10%</td>
</tr>
<tr class="odd">
<td>1,000</td>
<td>8%</td>
<td>4%</td>
<td>2%</td>
<td>13%</td>
<td>11%</td>
<td>9%</td>
<td>8%</td>
<td>12%</td>
<td>18%</td>
<td>&gt;21%</td>
<td>9%</td>
<td>12%</td>
<td>2%</td>
<td>-2%</td>
<td>10%</td>
<td>10%</td>
</tr>
<tr class="even">
<td>1,500</td>
<td>7%</td>
<td>4%</td>
<td>1%</td>
<td>12%</td>
<td>10%</td>
<td>8%</td>
<td>7%</td>
<td>12%</td>
<td>17%</td>
<td>&gt;21%</td>
<td>8%</td>
<td>12%</td>
<td>2%</td>
<td>-3%</td>
<td>9%</td>
<td>9%</td>
</tr>
<tr class="odd">
<td>2,000</td>
<td>7%</td>
<td>4%</td>
<td>1%</td>
<td>12%</td>
<td>10%</td>
<td>7%</td>
<td>7%</td>
<td>12%</td>
<td>17%</td>
<td>&gt;21%</td>
<td>8%</td>
<td>12%</td>
<td>2%</td>
<td>-3%</td>
<td>9%</td>
<td>8%</td>
</tr>
<tr class="even">
<td>2,500</td>
<td>7%</td>
<td>3%</td>
<td>1%</td>
<td>12%</td>
<td>10%</td>
<td>8%</td>
<td>6%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>8%</td>
<td>11%</td>
<td>2%</td>
<td>-3%</td>
<td>9%</td>
<td>9%</td>
</tr>
<tr class="odd">
<td>3,000</td>
<td>6%</td>
<td>3%</td>
<td>1%</td>
<td>12%</td>
<td>9%</td>
<td>7%</td>
<td>6%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>7%</td>
<td>11%</td>
<td>1%</td>
<td>-3%</td>
<td>9%</td>
<td>8%</td>
</tr>
<tr class="even">
<td>3,500</td>
<td>7%</td>
<td>3%</td>
<td>0%</td>
<td>11%</td>
<td>9%</td>
<td>7%</td>
<td>6%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>7%</td>
<td>10%</td>
<td>1%</td>
<td>-4%</td>
<td>9%</td>
<td>8%</td>
</tr>
<tr class="odd">
<td>4,000</td>
<td>6%</td>
<td>3%</td>
<td>0%</td>
<td>11%</td>
<td>9%</td>
<td>7%</td>
<td>6%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>7%</td>
<td>11%</td>
<td>1%</td>
<td>-4%</td>
<td>9%</td>
<td>8%</td>
</tr>
<tr class="even">
<td>4,500</td>
<td>6%</td>
<td>3%</td>
<td>0%</td>
<td>11%</td>
<td>9%</td>
<td>7%</td>
<td>6%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>7%</td>
<td>10%</td>
<td>1%</td>
<td>-4%</td>
<td>9%</td>
<td>8%</td>
</tr>
<tr class="odd">
<td>5,000</td>
<td>6%</td>
<td>3%</td>
<td>0%</td>
<td>11%</td>
<td>8%</td>
<td>7%</td>
<td>6%</td>
<td>11%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>7%</td>
<td>10%</td>
<td>1%</td>
<td>-4%</td>
<td>9%</td>
<td>8%</td>
</tr>
<tr class="even">
<td>6,000</td>
<td>6%</td>
<td>3%</td>
<td>0%</td>
<td>11%</td>
<td>9%</td>
<td>6%</td>
<td>5%</td>
<td>10%</td>
<td>16%</td>
<td>&gt;21%</td>
<td>7%</td>
<td>10%</td>
<td>0%</td>
<td>-4%</td>
<td>9%</td>
<td>7%</td>
</tr>
<tr class="odd">
<td>7,000</td>
<td>6%</td>
<td>3%</td>
<td>0%</td>
<td>11%</td>
<td>8%</td>
<td>6%</td>
<td>5%</td>
<td>10%</td>
<td>15%</td>
<td>&gt;21%</td>
<td>6%</td>
<td>10%</td>
<td>0%</td>
<td>-5%</td>
<td>8%</td>
<td>7%</td>
</tr>
<tr class="even">
<td>8,000</td>
<td>5%</td>
<td>2%</td>
<td>0%</td>
<td>10%</td>
<td>8%</td>
<td>6%</td>
<td>5%</td>
<td>10%</td>
<td>15%</td>
<td>&gt;21%</td>
<td>6%</td>
<td>11%</td>
<td>0%</td>
<td>-5%</td>
<td>8%</td>
<td>7%</td>
</tr>
<tr class="odd">
<td>9,000</td>
<td>5%</td>
<td>2%</td>
<td>0%</td>
<td>10%</td>
<td>8%</td>
<td>6%</td>
<td>5%</td>
<td>10%</td>
<td>15%</td>
<td>&gt;21%</td>
<td>6%</td>
<td>10%</td>
<td>0%</td>
<td>-5%</td>
<td>8%</td>
<td>7%</td>
</tr>
<tr class="even">
<td>10,000</td>
<td>5%</td>
<td>2%</td>
<td>-1%</td>
<td>10%</td>
<td>8%</td>
<td>6%</td>
<td>5%</td>
<td>10%</td>
<td>15%</td>
<td>&gt;21%</td>
<td>6%</td>
<td>10%</td>
<td>0%</td>
<td>-5%</td>
<td>8%</td>
<td>7%</td>
</tr>
</tbody>
</table>

> ^a^ This column describes the targets assuming standard conditions:
> for these analyses, they were a 20-year observation period, process
> variance of 0.05, 20 degrees of freedom for the variance estimate, 0
> hatchery-origin spawners, a QET four-year average of 50 spawners per
> year, and an acceptable extinction risk of 5% in 100 years. The other
> target columns show target calculated by varying one of the standard
> assumptions and keeping all others the same.
>
> ^b^ Time Period 40 assumes the observation period is 40 years.
>
> ^c^ Variance 0.01 and 0.1 assume difference process variance values.
>
> ^d^ Variance Degrees of Freedom columns assume different variance
> degrees of freedom values.
>
> ^e^ Hatchery Fraction columns assume different fractions of
> hatchery-origin spawners in the population.
>
> ^f^ QET 100 shows targets assuming a QET of a four-year average of 100
> spawners per year.
>
> ^g^ Extinction Risk columns assume an acceptable extinction risk of \#
> percent in 100 years.
>
> ^h^ Acceptable Time Horizon 200 assumes an acceptable extinction risk
> of 5% in 200 years.
>
> ^i^ Marine Index Long-Term assumes the marine survival over the
> observation period was twice the long-term average.
>
> Table D.4 Identical to Table D.3, except the targets are expressed as
> observed four-year average spawner abundances.

<table>
<colgroup>
<col style="width: 5%" />
<col style="width: 6%" />
<col style="width: 5%" />
<col style="width: 5%" />
<col style="width: 5%" />
<col style="width: 6%" />
<col style="width: 5%" />
<col style="width: 5%" />
<col style="width: 6%" />
<col style="width: 6%" />
<col style="width: 6%" />
<col style="width: 5%" />
<col style="width: 5%" />
<col style="width: 4%" />
<col style="width: 5%" />
<col style="width: 7%" />
<col style="width: 6%" />
</colgroup>
<tbody>
<tr class="odd">
<td rowspan="2"><p><strong>Current</strong></p>
<p><strong>Size</strong></p></td>
<td rowspan="2"><strong>Standard<sup>a</sup></strong></td>
<td rowspan="2"><strong>Time Period 40<sup>b</sup></strong></td>
<td colspan="2"><strong>Variance<sup>c</sup></strong></td>
<td colspan="3"><strong>Variance Degrees of
Freedom<sup>d</sup></strong></td>
<td colspan="3"><strong>Hatchery Fraction<sup>e</sup></strong></td>
<td rowspan="2"><strong>QET<sup>f</sup> 100</strong></td>
<td colspan="3"><strong>Extinction Risk<sup>g</sup></strong></td>
<td rowspan="2"><strong>Acceptable Time Horizon<br />
200<sup>h</sup></strong></td>
<td rowspan="2"><strong>Marine<br />
Index<br />
Long-Term<sup>i</sup></strong></td>
</tr>
<tr class="even">
<td><strong>.01</strong></td>
<td><strong>.1</strong></td>
<td><strong>5</strong></td>
<td><strong>10</strong></td>
<td><strong>40</strong></td>
<td><strong>5%</strong></td>
<td><strong>10%</strong></td>
<td><strong>30%</strong></td>
<td><strong>1</strong></td>
<td><strong>25</strong></td>
<td><strong>60</strong></td>
</tr>
<tr class="odd">
<td>100</td>
<td>600</td>
<td>1,200</td>
<td>200</td>
<td>1,400</td>
<td>1,100</td>
<td>700</td>
<td>500</td>
<td>1,060</td>
<td>&gt;2,000</td>
<td>&gt;2,000</td>
<td>800</td>
<td>1,000</td>
<td>300</td>
<td>200</td>
<td>700</td>
<td>700</td>
</tr>
<tr class="even">
<td>150</td>
<td>800</td>
<td>1,400</td>
<td>300</td>
<td>1,800</td>
<td>1,500</td>
<td>900</td>
<td>700</td>
<td>1,459</td>
<td>2,797</td>
<td>&gt;3,000</td>
<td>1,000</td>
<td>1,400</td>
<td>400</td>
<td>200</td>
<td>1,000</td>
<td>1,000</td>
</tr>
<tr class="odd">
<td>200</td>
<td>1,000</td>
<td>1,700</td>
<td>300</td>
<td>2,100</td>
<td>1,800</td>
<td>1,200</td>
<td>900</td>
<td>1,835</td>
<td>3,754</td>
<td>&gt;4,000</td>
<td>1,200</td>
<td>1,700</td>
<td>500</td>
<td>200</td>
<td>1,200</td>
<td>1,200</td>
</tr>
<tr class="even">
<td>500</td>
<td>1,900</td>
<td>3,000</td>
<td>700</td>
<td>4,300</td>
<td>3,200</td>
<td>2,200</td>
<td>1,900</td>
<td>3,613</td>
<td>7,618</td>
<td>&gt;10,000</td>
<td>2,300</td>
<td>3,600</td>
<td>900</td>
<td>400</td>
<td>2,500</td>
<td>2,500</td>
</tr>
<tr class="odd">
<td>1,000</td>
<td>3,400</td>
<td>4,600</td>
<td>1,300</td>
<td>7,400</td>
<td>5,400</td>
<td>3,800</td>
<td>3,200</td>
<td>6,283</td>
<td>13,768</td>
<td>&gt;20,000</td>
<td>3,900</td>
<td>6,500</td>
<td>1,500</td>
<td>700</td>
<td>4,600</td>
<td>4,400</td>
</tr>
<tr class="even">
<td>1,500</td>
<td>4,700</td>
<td>6,000</td>
<td>1,900</td>
<td>9,600</td>
<td>7,400</td>
<td>5,400</td>
<td>4,500</td>
<td>8,938</td>
<td>19,358</td>
<td>&gt;30,000</td>
<td>5,400</td>
<td>9,000</td>
<td>2,100</td>
<td>1,000</td>
<td>6,200</td>
<td>6,000</td>
</tr>
<tr class="odd">
<td>2,000</td>
<td>6,000</td>
<td>7,200</td>
<td>2,300</td>
<td>12,200</td>
<td>9,000</td>
<td>6,300</td>
<td>5,500</td>
<td>11,721</td>
<td>23,737</td>
<td>&gt;40,000</td>
<td>6,800</td>
<td>12,000</td>
<td>2,600</td>
<td>1,200</td>
<td>8,000</td>
<td>7,100</td>
</tr>
<tr class="even">
<td>2,500</td>
<td>7,100</td>
<td>8,500</td>
<td>2,800</td>
<td>14,700</td>
<td>11,100</td>
<td>8,200</td>
<td>6,800</td>
<td>14,191</td>
<td>28,397</td>
<td>&gt;50,000</td>
<td>8,100</td>
<td>13,600</td>
<td>3,200</td>
<td>1,400</td>
<td>10,100</td>
<td>9,400</td>
</tr>
<tr class="odd">
<td>3,000</td>
<td>8,200</td>
<td>9,900</td>
<td>3,300</td>
<td>17,100</td>
<td>12,700</td>
<td>9,000</td>
<td>7,800</td>
<td>16,699</td>
<td>33,955</td>
<td>&gt;60,000</td>
<td>9,400</td>
<td>15,600</td>
<td>3,600</td>
<td>1,700</td>
<td>11,700</td>
<td>10,300</td>
</tr>
<tr class="even">
<td>3,500</td>
<td>9,700</td>
<td>10,900</td>
<td>3,800</td>
<td>19,200</td>
<td>14,800</td>
<td>11,000</td>
<td>9,000</td>
<td>18,349</td>
<td>39,406</td>
<td>&gt;70,000</td>
<td>10,500</td>
<td>17,000</td>
<td>4,100</td>
<td>1,900</td>
<td>13,500</td>
<td>11,700</td>
</tr>
<tr class="odd">
<td>4,000</td>
<td>10,600</td>
<td>11,300</td>
<td>4,300</td>
<td>21,700</td>
<td>16,000</td>
<td>11,300</td>
<td>10,100</td>
<td>21,297</td>
<td>42,670</td>
<td>&gt;80,000</td>
<td>11,300</td>
<td>20,200</td>
<td>4,600</td>
<td>2,100</td>
<td>14,900</td>
<td>13,800</td>
</tr>
<tr class="even">
<td>4,500</td>
<td>11,400</td>
<td>12,500</td>
<td>4,600</td>
<td>24,000</td>
<td>17,500</td>
<td>12,800</td>
<td>10,800</td>
<td>23,032</td>
<td>47,254</td>
<td>&gt;90,000</td>
<td>12,800</td>
<td>21,800</td>
<td>5,100</td>
<td>2,300</td>
<td>16,800</td>
<td>14,600</td>
</tr>
<tr class="odd">
<td>5,000</td>
<td>12,800</td>
<td>14,500</td>
<td>5,100</td>
<td>25,000</td>
<td>18,400</td>
<td>14,500</td>
<td>12,100</td>
<td>24,806</td>
<td>51,380</td>
<td>&gt;100,000</td>
<td>14,600</td>
<td>23,400</td>
<td>5,500</td>
<td>2,500</td>
<td>18,800</td>
<td>15,900</td>
</tr>
<tr class="even">
<td>6,000</td>
<td>14,800</td>
<td>15,900</td>
<td>6,000</td>
<td>30,300</td>
<td>22,800</td>
<td>16,100</td>
<td>14,100</td>
<td>29,057</td>
<td>61,153</td>
<td>&gt;120,000</td>
<td>17,000</td>
<td>28,800</td>
<td>6,100</td>
<td>2,900</td>
<td>22,300</td>
<td>18,300</td>
</tr>
<tr class="odd">
<td>7,000</td>
<td>17,200</td>
<td>17,400</td>
<td>6,800</td>
<td>35,100</td>
<td>24,700</td>
<td>18,100</td>
<td>15,900</td>
<td>32,254</td>
<td>69,359</td>
<td>&gt;140,000</td>
<td>18,800</td>
<td>32,100</td>
<td>7,300</td>
<td>3,300</td>
<td>25,500</td>
<td>21,900</td>
</tr>
<tr class="even">
<td>8,000</td>
<td>17,500</td>
<td>18,700</td>
<td>7,700</td>
<td>36,900</td>
<td>27,900</td>
<td>20,600</td>
<td>18,300</td>
<td>37,051</td>
<td>80,045</td>
<td>&gt;160,000</td>
<td>20,800</td>
<td>40,100</td>
<td>8,200</td>
<td>3,800</td>
<td>28,000</td>
<td>25,100</td>
</tr>
<tr class="odd">
<td>9,000</td>
<td>20,900</td>
<td>21,700</td>
<td>8,500</td>
<td>40,700</td>
<td>30,800</td>
<td>22,700</td>
<td>20,100</td>
<td>39,393</td>
<td>85,742</td>
<td>&gt;180,000</td>
<td>23,400</td>
<td>39,600</td>
<td>8,900</td>
<td>4,100</td>
<td>32,100</td>
<td>25,900</td>
</tr>
<tr class="even">
<td>10,000</td>
<td>21,700</td>
<td>23,600</td>
<td>9,200</td>
<td>45,100</td>
<td>34,400</td>
<td>24,700</td>
<td>21,900</td>
<td>45,669</td>
<td>93,802</td>
<td>&gt;200,000</td>
<td>25,300</td>
<td>43,200</td>
<td>9,500</td>
<td>4,600</td>
<td>34,800</td>
<td>28,400</td>
</tr>
</tbody>
</table>

^a^ This column describes the targets assuming standard conditions: for
these analyses, they were a 20-year observation period, process variance
of 0.05, 20 degrees of freedom for the variance estimate, 0
hatchery-origin spawners, a QET four-year average of 50 spawners per
year, and an acceptable extinction risk of 5% in 100 years. The other
target columns show target calculated by varying one of the standard
assumptions and keeping all others the same.

^b^ Time Period 40 assumes the observation period is 40 years.

^c^ Variance 0.01 and 0.1 assume difference process variance values.

^d^ Variance Degrees of Freedom columns assume different variance
degrees of freedom values.

^e^ Hatchery Fraction columns assume different fractions of
hatchery-origin spawners in the population.

^f^ QET 100 shows targets assuming a QET of a four-year average of 100
spawners per year.

^g^ Extinction Risk columns assume an acceptable extinction risk of \#
percent in 100 years.

^h^ Acceptable Time Horizon 200 assumes an acceptable extinction risk of
5% in 200 years.

^i^ Marine Index Long-Term assumes the marine survival over the
observation period was twice the long-term average.

Table D.5 Recent average abundance and fraction of hatchery origin for
WLC populations.^a^

+---------------+-------------------+----------+--------+------------+
| ESU           | Population^b^     | Year     | C      | Hatchery   |
|               |                   |          | urrent | Fraction   |
|               |                   |          | Size   |            |
+---------------+-------------------+----------+--------+------------+
| Columbia chum | Grays River       | 19       | 874    | 0          |
| salmon        |                   | 97--1998 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Lower gorge       | 19       | 542    | 0          |
|               | tributaries       | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Upper gorge       | 19       | 100    |            |
|               | tributaries       | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
| Upper         | Mollala River     | 1997     | 574    | 24         |
| Willamette    |                   |          |        |            |
|               |                   |          |        |            |
| steelhead     |                   |          |        |            |
+---------------+-------------------+----------+--------+------------+
|               | North Santiam     | 1997     | 2,214  | 29         |
|               | River             |          |        |            |
+---------------+-------------------+----------+--------+------------+
|               | South Santiam     | 1997     | 900    | 0          |
|               | River             |          |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Calapooia River   | 1997     | 236    | 0          |
+---------------+-------------------+----------+--------+------------+
| Upper         | Clackamas River   | 19       | 1,453  |            |
| Willamette    |                   | 97--2000 |        |            |
|               |                   |          |        |            |
| chinook       |                   |          |        |            |
| salmon        |                   |          |        |            |
+---------------+-------------------+----------+--------+------------+
|               | McKenzie          | 19       | 1,904  | 24         |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
| Lower         | North Fork Toutle | 19       | 176    | 0          |
| Columbia      | River winter      | 97--2000 |        |            |
|               |                   |          |        |            |
| steelhead     |                   |          |        |            |
+---------------+-------------------+----------+--------+------------+
|               | South Fork Toutle | 19       | 463    | 2          |
|               | River winter      | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Coweeman River    | 19       | 487    | 50         |
|               | winter            | 98--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Kalama River      | 19       | 554    | 0          |
|               | winter            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Clackamas River   | 19       | 465    | 39         |
|               | winter            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Sandy River       | 19       | 1,005  |            |
|               | winter            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Hood River winter | 19       | 850    | 52         |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Kalama River      | 19       | 419    | 38         |
|               | summer            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | East Fork Lewis   | 19       | 287    | 33         |
|               | summer            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Washougal River   | 19       | 158    | 8          |
|               | summer            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Wind River summer | 19       | 368    | 10         |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Hood River summer | 19       | 866    | 82         |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
| Lower         | Grays River fall  | 19       | 127    | 37         |
| Columbia      |                   | 97--2000 |        |            |
|               |                   |          |        |            |
| chinook       |                   |          |        |            |
| salmon        |                   |          |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Elochoman River   | 19       | 754    | 69         |
|               | fall              | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Mill, etc. fall   | 19       | 491    | 47         |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Lower Cowlitz     | 19       | 1,702  | 67         |
|               | fall              | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Coweeman          | 19       | 425    | 0          |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Kalama River fall | 19       | 2,995  | 67         |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Salmon Creek late | 19       | 235    | 0          |
|               | fall              | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Washougal River   | 19       | 3,231  | 57         |
|               | fall              | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Sandy River fall  | 19       | 220    | 3          |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Upper gorge       | 19       | 159    | 17         |
|               | tributaries fall  | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Big White Salmon  | 19       | 234    | 21         |
|               | fall              | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Sandy late        | 19       | 839    | 3          |
|               |                   | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | North Fork Lewis  | 19       | 7,293  | 13         |
|               | bright            | 97--2000 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Upper Cowlitz     | 19       | 365    |            |
|               | spring            | 97--1999 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Kalama River      | 19       | 105    | 0          |
|               | spring            | 97--1999 |        |            |
+---------------+-------------------+----------+--------+------------+
|               | Lewis River       | 19       | 300    | 0          |
|               | spring            | 97--1999 |        |            |
+---------------+-------------------+----------+--------+------------+

^a^ The averages are standardized for the years 1997--2020: if data were
missing over these years, the average was based on the existing data.

^b^ This list does not include all WLC populations. Some populations are
extirpated and have a current abundance of 0. For populations not in
this table, there are no available abundance data.

Literature Cited

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River salmon. Sustainable Fisheries Conference Proceedings.

Barrowman, N., and R. Myers. 2000. Still more spawner-recruitment
curves: The hockey stick and its generalizations. Can. J. Fish. Aquat.
Sci. 57(4): 665--676.

Beamish, R. J., D. J. Noakes, G. A. MacFarlane, L. Klyshatorin, V. V.
Ivanov, and V. Kurashov. 1999. The regime concept and natural trends in
the production of Pacific salmon. Can. J. Fish. Aquat. Sci. 56:
516--526.

Dennis, B., P. L. Munholland, and J. M. Scott. 1991. Estimation of
growth and extinction parameters for endangered species. Ecol. Monogr.
61: 115--143.

Dennis, B., and M. L. Taper. 1994. Density dependence in time series
observations of natural populations: Estimation and testing. Ecol.
Monogr. 64: 205--224.

Goodman, D. 1987. The demography of chance extinction. *In* M. E. Soulé
(ed.), Viable populations for conservation. Cambridge University Press,
Cambridge. p. 11--34.

Hare, S. R., N. J. Mantua, and R. C. Francis. 1999. Inverse production
regimes: Alaska and west coast Pacific salmon. Fisheries 24: 6--14.

Hilborn, R., and C. J. Walters. 1992. Quantitative fisheries stock
assessment: Choice, dynamics, and uncertainty. Chapman and Hall, New
York.

Holmes, E. 2001. Estimating risks in declining populations with poor
data. Proc. Nat. Acad. Sci. 98(9): 5072--5077.

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