An#Evaluation#of#Biological# Escapement*Goalsfor ... 22, 2015 Suggested citation: Cunningham, C.J., D. Schindler, R. Hilborn. 2015. An evaluation of biological escapement goals for

An Evaluation of Biological

Escapement Goals for Sockeye Salmon

of Bristol Bay, Alaska

Draft Report

Prepared for:

Bristol Bay Science and Bristol Bay Regional Research Institute Seafood Development Dillingham, Alaska Association Anchorage, Alaska

February 2015

An Evaluation of Biological Escapement Goals for Sockeye Salmon of Bristol Bay, Alaska Curry J. Cunningham, Daniel Schindler, and Ray Hilborn

School of Fisheries and Aquatic Sciences University of Washington Box 355020, Seattle, WA 98195-5020 Draft Report prepared for: Bristol Bay Science and Research Institute PO Box 1464, Dillingham, Alaska 99576 and Bristol Bay Regional Seafood Development Association 800 E. Dimond Blvd, Suite 3-131 #158 Anchorage AK 99515-2028 February 22, 2015 Suggested citation: Cunningham, C.J., D. Schindler, R. Hilborn. 2015. An evaluation of biological escapement goals for sockeye salmon of Bristol Bay, Alaska. Unpublished draft report prepared for the Bristol Bay Science and Research Institute, Dillingham, and the Bristol Bay Regional Seafood Development Association, Alaska. 48 p.

i

Executive Summary

The Alaska Department of Fish and Game (ADF&G) manages the Bristol Bay sockeye salmon fishery to achieve escapement goals for 9 river systems that flow into 5 commercial fishing districts. Escapement goals are periodically evaluated using the latest escapement and recruitment data, along with a historical dataset spanning 50+ years on many of the river systems (Fair et al. 2012; Baker et al. 2006). The Bristol Bay fishery began in the late 1800s and after decades of intense fishing, there is little information in the data to understand the equilibrium or unfished abundance of salmon in the river systems and the relationship between escapement and recruitment at high levels of escapement – important information for determining escapement goals. In addition, the productivity of these salmon stocks and the escapement thought to maximize sustainable yield periodically shifts and the best way to deal with these shifts has not previously been established.

We used three alternative methods and a range of assumptions regarding likely unfished population size to evaluate the spawner-recruit relationships for Bristol Bay sockeye stocks and determine the Biological Escapement Goals (BEG), defined as those that would maximize long term expected harvests. We used estimates of population abundance of each system prior to industrial fishing, obtained from chemical analysis of lake sediments, to provide bounds on the unfished population size. In all cases we used the standard Ricker stock recruitment curve fitted to the spawner-recruit data but considered three specific cases.

• Case 1: we used data from all brood years (1963+) that are available in the brood tables from Cunningham et al. (2012).

• Case 2: we allowed for a change in the relationship between spawners and recruits that occurred in brood year 1980 and fitted two curves, one employing data before 1980 and one from 1980 and afterwards.

• Case 3: we used a model that assumes any brood year may be described one of two spawner-recruit relationships, a productive and an unproductive one, and that the timing of the switch from one relationship to another is estimated from the data rather than occurring at a fixed date as in Case 2. This model allowed us to estimate the probability of each river system seeing high or low sockeye productivity in the future.

For most stocks there is very strong evidence for there being changes in productivity, and thus either Case 2 or Case 3 seem most appropriate. In general we believe that Case 3 is the preferred model, as we may use estimates of optimal BEGs under high and low productivity regimes alongside simulation of future regime occupancy from estimated transition probabilities to develop BEG’s which are robust to changes in salmon survival and productivity. When setting escapement goals, managers need to balance the probability that the stock will encounter good or poor survival through their life history, with differences in escapement targets that are predicted to maximize future harvest opportunities.

ii

Table of Contents Executive Summary ........................................................................................................... i Table of Contents .............................................................................................................. ii List of Tables .................................................................................................................... iii List of Figures ................................................................................................................... iii Introduction ....................................................................................................................... 1 Methods .............................................................................................................................. 4

Bayesian Ricker Spawner-recruit Model ........................................................................ 4 PDO Breakpoint Ricker Model ....................................................................................... 7 Regime Transition Ricker Model .................................................................................... 8 Estimating BEGs for the Regime Transition Ricker Model ......................................... 10

Results .............................................................................................................................. 14 Kvichak ......................................................................................................................... 18 Naknek .......................................................................................................................... 25 Egegik ........................................................................................................................... 30 Ugashik ......................................................................................................................... 34 Igushik........................................................................................................................... 37 Wood ............................................................................................................................. 40

Discussion ........................................................................................................................ 44 Limitations .................................................................................................................... 47 Future work ................................................................................................................... 47

References ........................................................................................................................ 47

iii

List of Tables Table 1. Median of posterior estimates for Smsy’ (in millions) by river system, under

five different scenarios regarding change in productivity and two prior types on the unfished population size, including the log-normal correction. ......... 14

Table 2. Median of posterior estimates of MSY’ (in millions) for the different systems under five different scenarios regarding change in productivity and two priors on the unfished population size, including the log-normal correction. ......... 16

Table 3. Percent change in estimated MSY before and after 1980 from the Fixed Breakpoint model and between the low and high productivity regimes estimated by the Markov Regime model. ...................................................... 17

Table 4. Results of Markov Regime model, in millions of sockeye. .......................... 18

List of Figures Figure 1. Egegik log recruits per spawner plotted against number of spawners with

best-fit linear trend line. .................................................................................. 2 Figure 2. Prior probability on unfished (equilibrium) population size for Bristol Bay

river systems specified to the Bayesian estimation models. ........................... 6 Figure 3. Expected yield for the Egegik River across a range of potential spawning

abundances, estimated by the regime transition Ricker model assuming the Top 20% prior on the beta parameter. ........................................................... 12

Figure 4. Example calculation of Smsy and MSY from the mixture yield curve from the regime transition model for the Egegik River, assuming the top 20% prior on the beta parameter. ................................................................................... 13

Figure 5. Prior probability on unfished population size for Egegik using the ADF&G prior (blue line), the mean and variance from all of the paleo data (grey line) and the top 20% of the paleo data with fixed CV=0.5 (orange line). ............ 16

Figure 6. Kvichak spawner-recruit curve from the basic Bayesian Ricker model with top 20% prior. ................................................................................................ 19

Figure 7. Kvichak spawner-recruit curve from the basic Bayesian Ricker model with the mean prior. ............................................................................................... 20

Figure 8. 1980 breakpoint model for the Kvichak River assuming the top 20% prior. 21 Figure 9. Changes in productivity of the Kvichak River, as reflected in estimates of the

alpha parameter over time from the Markov transition model. ..................... 22 Figure 10. Spawner-recruit relationships for the high (green) and low (purple)

productivity regimes in the Kvichak system, estimated by the Markov Regime transition model with the top 20% prior. ......................................... 23

Figure 11. Posterior probability distributions for the equilibrium (unfished) population size parameter (𝛽) in the Markov Regime model with top 20% prior. ......... 24

iv

Figure 12. Naknek spawner-recruit curve from the basic Bayesian Ricker model with top 20% prior. ...................................................................................................... 25

Figure 13. ..... Naknek spawner-recruit curve from the basic Bayesian Ricker model with the mean prior. ............................................................................................... 26

Figure 14. 1980 breakpoint model for the Naknek River assuming the top 20% prior. .. 27 Figure 15. Changes in productivity of the Naknek River, as reflected in estimates of the


productivity regimes in the Naknek system, estimated by the Markov Regime transition model with the top 20% prior. ....................................................... 29

Figure 17. Temporal pattern in recruits per spawner for Egegik. ................................... 30 Figure 18. 1980 breakpoint model for the Egegik River assuming the top 20% prior. .. 31 Figure 19. Spawner-recruit relationships for the high (green) and low (purple)

productivity regimes in the Egegik system, estimated by the Markov Regime transition model with the top 20% prior. ....................................................... 32

Figure 20. Change in the alpha parameter over time from the Markov transition model for the Egegik River. ..................................................................................... 33

Figure 21. 1980 breakpoint model for the Ugashik River, assuming the top 20% prior. 34 Figure 22. Changes in productivity of the Ugashik River, as reflected in estimates of the


productivity regimes in the Ugashik system, estimated by the Markov Regime transition model with the top 20% prior. ......................................... 36

Figure 24. 1980 breakpoint model for the Igushik River, assuming the top 20% prior. 37 Figure 25. Changes in productivity of the Igushik River, as reflected in estimates of the


productivity regimes in the Igushik system, estimated by the Markov Regime transition model with the top 20% prior. ....................................................... 39

Figure 27. Wood spawner-recruit curve from the basic Bayesian Ricker model with top 20% prior. ...................................................................................................... 40

Figure 28. 1980 breakpoint model for the Wood River assuming the top 20% prior. ... 41 Figure 29. Spawner-recruit relationships for the high (green) and low (purple)

productivity regimes in the Wood system, estimated by the Markov Regime transition model with the top 20% prior. ....................................................... 42

Figure 30. Change in the alpha parameter over time from the Markov transition model for the Wood River. ....................................................................................... 43

Figure 31. Change in the beta parameter over time from the Markov transition model for the Wood River. ............................................................................................ 44

1

Introduction

Effective management of commercially exploited salmon populations depends on the ability to quantify the relationship between the number of spawning individuals and resultant recruitment over time. By representing the relationship between spawning abundance and recruitment with an appropriate mathematical model, the spawning abundance (Smsy) that is likely to provide the maximum potential for harvest, or maximum sustainable yield (MSY), may be estimated. For semelparous salmonid species that can be reliably enumerated and aged during the return migration to freshwater, spawning abundances represent the number of fish escaping commercial harvest to enter freshwater spawning grounds. The recruitment resulting from a particular brood year may be calculated by as the sum of fishery catches and upriver escapements in subsequent years that resulted from the spawning stock in that brood year.

Management of commercial fisheries for sockeye salmon (Oncorhynchus nerka) in Bristol Bay, Alaska operate based upon stock-specific escapement goal ranges. Fishing effort is actively regulated on a daily basis by the Alaska Department of Fish and Game (ADFG), in terminal fishing districts at the mouth of major tributaries of the Bristol Bay drainages, in an attempt to achieve an escapement within the range of the escapement goal. The approach permits a sufficiently large number of individuals to bypass the commercial fishery and propagate future generations (Clark et al. 2006). Biological escapement goals (BEGs) are set at the level that has the highest probability of producing maximum sustainable yield (Smsy), and the BEG range is traditionally the range of spawning abundances which are expected to produce 90% of MSY (Fair et al. 2012). Traditionally, BEG’s have been determined for Bristol Bay’s river systems by fitting a Ricker’s equation (Ricker 1954) to the observed spawner-recruit relationship described for a period of time deemed appropriate and for which reliable abundance and age-composition data are available to reconstruct brood tables (i.e., brood years 1963+), and using the analytical solution from Hilborn (1985) to find Smsy (see Fair et al. (2012)). The choice of which time periods from which to use spawner-recruit data has differed depending who conducted the analysis.

However, the time-series of spawning abundance and recruitment data available for analysis often does not show a sufficiently large decline in the observed number of recruits per spawner, at high spawning abundance, necessary to inform estimates of the equilibrium population size parameter (𝛽) of the Ricker-type (Eq. 1) spawner-recruit relationship (Ricker 1954).

(1) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−=β

α ttt

SSR 1exp

We can estimate the parameters of the Ricker curve by transforming the equation

to produce a linear equation:

2

(2) ln RtSt

!

"#

$

%&=α −

αβSt

which is a linear regression with intercept α and slope –α/β.

Figure 1 shows the spawner-recruit relationship for the Egegik stock as well as the best-fit linear regression that indicates an increase in recruits per spawner with increasing number of spawners. This implies that the optimal escapement goal is much larger than any escapement that has been observed.

Figure 1. Egegik log recruits per spawner plotted against number of spawners with best-fit linear trend line.

The primary reasons for the apparent lack of density dependent compensation is (1) the lack of data prior to intensive fishing and (2) a major increase in productivity of some stocks since the late 1970s which gave rise to both higher escapements and higher recruits per spawner. For Bristol Bay sockeye salmon stocks, the collection of catch, escapement, and age composition data of sufficient precision to permit reconstruction of spawner-recruit relationships, began long after the advent of the commercial fishery, thus limiting the upper extent of observed spawning abundances. Without high spawning abundances, it becomes more difficult to accurately estimate system-specific equilibrium population size. In the absence of a definite estimate for equilibrium population size (i.e., the Ricker 𝛽 parameter), there appears to be no evidence for a decrease in the number of recruits per spawner at higher spawning sizes (compensation), and thus there exists a large amount of uncertainty surrounding the estimates of Smsy and MSY, often resulting in unrealistically high expectations for both.

In order to deal with the uncertainty surrounding estimation of equilibrium population size and its effect on defining BEG’s based on estimates of Smsy, we utilized

-‐0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 500 1,000 1,500 2,000 2,500 3,000

Log(recruits/spawne

r)

Spawners (in thousands)

3

Bayesian methods for fitting the spawner-recruit relationship, incorporating prior information on system-specific equilibrium population size available from paleolimnological evaluation of sediments from rearing lakes.

A second difficulty associated with establishing biological escapement goals by fitting a Ricker-type model is that we implicitly assume that the spawner-recruit relationship does not change over time and there is a single underlying average relationship. However, there is significant interest in accounting for the differential productivity of salmon populations, associated with observed variation in climate regimes (Hare et al. 1999, Mantua and Hare 2002). Most estimates of escapement goals for Bristol Bay stocks (Baker et al. 2006 for instance) have partitioned many of the spawner-recruit time series in the late 1970s when productivity increased markedly coinciding with the observed shift from cold to warm-phases of the Pacific Decadal Oscillation, and estimated Ricker parameters by fitting to data from the two time periods separately. However, methods such as these that arbitrarily define breakpoints within the data, obscure the process which is driving changes in population productivity over time and do not provide inference about the probability of future regime shifts.

Contrary to the assumption of constant spawner-recruit relationships are the pronounced shifts in in both abundance (Burkenroad 1946) and productivity (Gilbert 1997) observed for many aquatic species. These shifts suggest that the productivity of fish populations over time may be correlated with changes in the environment which they inhabit (Burkenroad 1953). With mounting evidence suggesting that the productivity of fish population may routinely shift between states of differing productivity (Vert-pre et al. 2013), it becomes imperative to estimate the frequency and magnitude of these changes over time in our analyses of spawner-recruit relationships. By accounting for the temporal variation in observed spawner-recruit relationships resulting from differences marine and freshwater survival, we can evaluate both the alternative Smsy goals predicted to optimize harvest as well as the resultant MSY expected under alternative regime states. This will permit development robust escapement goals that take into account these alternative states and the probability of their occurrence in the future, thus producing greater yield across production regimes.

The purpose of this analysis was to evaluate spawner-recruit relationships for the major Bristol Bay river systems and determine the spawning stock size which is likely to produce maximum sustainable yield (Smsy) and the resultant yield (MSY), along with uncertainty in the estimation process. To overcome the problem of poorly defined equilibrium population sizes in the absence of fishing (the Ricker 𝛽 parameter), we incorporate information on the historical abundance of salmon populations reconstructed from paleolimnological analysis of nitrogen isotopes in lake-bottom sediments (Rogers et al. 2013). This method provides a range of historical run sizes before the advent of industrial fishing, which represent the best available prior information on the distribution of the 𝛽 parameter in Ricker equation (Eq. 1).

The temporal variation in Bristol Bay spawner-recruit relationships was addressed using three models of increasing complexity. First is a Ricker spawner-recruit model that assumes no temporal changes in productivity. Second, a Ricker spawner-recruit model

4

allowing for a single shift in production occurring in brood year 1980, intended to evaluate the influence of an environmental shift resulting from the transition between cool and warm phases of the Pacific Decadal Oscillation (Hare et al. 1999, Mantua and Hare 2002). Third, a spawner-recruit model which assumes that salmon production has alternated between two alternative productivity regimes, with transition between regimes treated as a 1st order Markov process where the probability of shifting from one regime to another is described by a transition probability matrix. These three alternative models for quantifying sockeye salmon spawner-recruit relationships for Bristol Bay, Alaska were applied to brood table data available for the Kvichak, Naknek, Egegik, Ugashik, Igushik, and Wood River stocks. Methods

Bayesian Ricker Spawner-‐recruit Model

In order to estimate biological escapement goals for sockeye salmon stocks spawning in major river systems of the Bristol Bay drainage, data on spawning abundance and resultant recruitment for brood years 1963 – 2005 were obtained. Data were available from run reconstructions by Cunningham et al. (2012), which partitioned Bristol Bay catches in mixed-stock commercial fishing districts and estimated interception rates for non-natal fishing districts during return migrations using age and genetic composition of catch information, to reconstruct annual returns by stock and age class. The underlying spawner-recruit relationship was estimated by fitting a Ricker spawner-recruit model assuming lognormally distributed process uncertainty (Eq. 3) to these data using Bayesian methods (Gelman and Shalizi 2012).

(3) Rt = St exp α 1− St

β

"

#$

%

&'

(

)*

+

,-*exp ε( )

ε ~ N 0,σ( )

St is the number of individuals spawning in year t, and Rt is the adult production from spawning in year t that may return anywhere from 3 to 7 years later. The estimated parameters include the maximal productivity in the absence of density-dependent compensation (𝛼), the equilibrium population size (𝛽), and the standard deviation for the lognormal error distribution (𝜎). The Ricker model was fit to spawner-recruit data using Bayesian methods for two reasons: 1) a Bayesian analysis allows the uncertainty in parameter estimates (𝛼,𝛽,𝜎!) and estimates of derived quantities of interest (Smsy and MSY) to be directly quantified, and 2) Bayesian methods allow incorporation of prior information (Gelman et al. 2004) about the parameter describing equilibrium population size (β), which is often poorly defined by the available time series of data.

The prior probability distribution for the equilibrium population size parameter (𝛽) was derived from paleolimnological data collected from nursery lakes of the Bristol

5

Bay river systems (Schindler et al. 2005, 2006, Rogers et al. 2013). Analysis of changes in isotopic concentrations (𝛿!"𝑁) among layers of a core sample taken of lake-bottom sediments, permits a historical record of salmon abundance prior to the industrialized fishery to be reconstructed (Schindler et al. 2005, Schindler et al. 2006). By comparing nitrogen isotope concentrations (𝛿!"𝑁) in sockeye rearing lakes of interest with returning salmon, and reference lakes above impassible barriers to salmon migration, the concentration of marine derived nitrogen which provides a metric for salmon abundance may be separated from terrestrial sources of the nitrogen isotope. Lead isotope ( 𝑃𝑏!"# ) reactivity provides a means for determining the age of sediments in each layer of a core sample, thus providing a way to date salmon abundances reconstructed from sediment cores.

Two alternative priors for the 𝛽 parameter were derived from the paleolimnological data. The first was a mildly informative prior (Top 20 prior), which was normally distributed with mean equal to the average of the top 20% of reconstructed salmon abundances prior to commercial fishing (1750 – 1890). A CV=0.5 was assumed which provides significant uncertainty in the 𝛽 prior (Fig. 2). The second prior (Mean prior) for the 𝛽 parameter based upon paleolimnological information was considerably more informative, and was normally distributed with mean and standard deviation equal to that observed for reconstructed abundance across the historical 1750 – 1890 time series (Fig. 2). Uninformative priors were assumed for the standard deviation of the lognormal error distribution (𝜎~𝑢𝑛𝑖𝑓𝑜𝑟𝑚 0.001,2 ), and the basal productivity parameter of the Ricker equation (𝛼~𝑢𝑛𝑖𝑓𝑜𝑟𝑚[0.001,5]).

While using uniform priors on parameters is often considered to be uninformative, the approach used by Fair et al. (2012) illustrates difficulties in this assumption. Fair et al. (2012) used a Ricker curve estimated in the form (equation 4).

(4) 𝐿𝑜𝑔 !!= 𝑙𝑜𝑔 𝛼 − 𝛽𝑆

In this form 1/ β is proportion to the unfished equilibrium population size, and by assuming that their β was uniformly distributed they assigned very high prior probability to small equilibrium stock sizes. Therefore, Fair et al. (2012) have a very informative prior on unfished stock size and thus Smsy. No justification for this prior is given and it is far from uninformative about Smsy.

The analytical solution for Smsy described by Hilborn (1985) was used to define biological escapement goals and to calculate the expected MSY for a deterministic curve (Eq. 5).

(5) Smsy = β * 0.5− 0.07α( )

MSY = Smsy*exp α 1− Smsyβ

"

#$

%

&'

(

)*

+

,-− Smsy

6

Figure 2. Prior probability on unfished (equilibrium) population size for Bristol Bay river systems specified to the Bayesian estimation models. The prior probability distributions defined by the top 20% of sockeye abundances reconstructed from paleolimnological data and assuming a CV=0.5 are in red, and those defined by the true mean and variance of the paleolimnological time-series (1750 – 1890) are in blue.

−20 0 20 40

Kvichak Top20Mean

−5 0 5 10

Naknek

−10 0 10 20 30

Egegik

−5 0 5 10 15 20

Ugashik

−4 −2 0 2 4 6 8 10

Wood

−1 0 1 2 3

Igushik

Unfished population size in millions

Prio

r pro

babi

lity

7

However, given the assumption of lognormal stochastic error in the Ricker spawner-recruit relationship it is necessary to account for the difference between the mode and the mean of the expected distribution for model parameters. Therefore, we have also employed the lognormal correction derived parameters Smsy and MSY (Hilborn 1985) (Eq. 6). 𝛼!, 𝛽!, 𝑆𝑚𝑠𝑦!, and 𝑀𝑆𝑌′ indicate the deterministic equivalent values for estimated and derived model parameters. It is important to note that these are not corrections for bias in α and β, but rather a calculation of what the values of α and β would be so that the deterministic calculations in equation (5) would provide the correct answer given the stochastic recruitment process.

(6)

α ' =α +σ2

2

β ' = β α 'α

!

"#

$

%&

Smsy ' = β '* 0.5− 0.07α '( )

MSY ' = Smsy '* exp α ' 1− Smsy 'β '

(

)*

+

,-

!

"#

$

%&− Smsy '

Fair et al. (2012) did not make this correction, apparently mistakenly believing that it is a correction for bias in the estimates of the parameters, rather than correcting for the fact that the lognormal nature of the process errors means that the average return will be higher than that predicted by their spawner recruit equation, and thus the Smsy will be higher than that predicted from equation 5. By not correcting Smsy, estimates of optimal biological escapement goals are likely to be biased low.

PDO Breakpoint Ricker Model

As an alternative to the stationary Ricker approximation of spawner-recruit relationships for Bristol Bay sockeye stocks, we also evaluated differences in both model parameters (𝛼 and 𝛽) and derived parameters (Smsy and MSY), before and after the shift from cold to warm phases of the Pacific Decadal Oscillation (PDO) that occurred in the late 1970’s. The PDO describes a long-term pattern in climate variation within the Pacific Ocean defined by specific spatio-temporal patterns in sea surface temperature, pressure, and currents (Mantua and Hare 2002). The PDO is defined by a long (20 – 30 year) period which correlates with catches and presumably productivity patterns for Alaskan salmon (Hare et al. 1999). A shift from the “cold” to “warm” phases of the PDO during the late 1970’s provides a good breakpoint for splitting the spawner-recruit time series’, and evaluating whether shifts in marine climate patterns have influenced the estimated value of Ricker basal productivity (𝛼), equilibrium (unfished) population size (𝛽), or error (𝜎) parameters, as well as estimates of potential yield (MSY) and optimal biological escapement goals (Smsy).

In order to test whether the PDO shift influenced productivity patterns for Bristol Bay sockeye and resultant estimates of optimal management goals (Smsy), we created a

8

Ricker breakpoint model where separate model parameters were estimated for each of the two periods in the time series (Eq. 7).

(7)

Rt = St exp αr 1−Stβr

"

#$

%

&'

(

)**

+

,--*exp ε( )

ε ~ N 0,σ r( )

r =1, t <=19802, t >1980./0

Separate estimates for Smsy and MSY were also found for the periods before and after the 1980 brood year (Eq. 8), and the lognormal correction for MSY and Smsy described in equation 6 was applied to parameters for each period. In this way we implicitly assume that the spawner-recruit relationship is not time-invariant and that its form has shifted once during the time period of our analysis, at a time that correlates with large-scale marine climate pattern changes associated with the PDO.

(8)

Smsyr = βr * 0.5− 0.07αr( )

MSYr = Smsyr *exp αr 1−Smsyrβr

"

#$

%

&'

(

)**

+

,--− Smsyr

Regime Transition Ricker Model

The third type of spawner-recruit model employed to evaluate Bristol Bay sockeye salmon production patterns was a conditional mixture model. Like the breakpoint Ricker model described above, this model also assumes that the form of the spawner-recruit relationship has changed historically, and estimates separate model and derived parameters for each of two productivity regimes. However, instead of specifying a specific break point based on a priori expectations of the likely timing of productivity regime shifts associated with the PDO, this model treats transition between hidden productivity states (regimes) as a 1st-order Markov process where state transition probabilities are estimated directly from the data for each river system.

By treating occupancy of differing productivity states across time as a 1st-order Markov process, successive hidden states are conditional on the state in the previous brood year. This hidden Markov Ricker model treats the expected recruitment in each year (t) as a mixture of two separate spawner-recruit relationships, with the probability that any particular brood year (t) belongs to one productivity regime depending on which regime (state) was observed in the previous brood year (t-1) (Eq. 9). More generally, this means that whether recruitment resulting from a particular brood year is likely to follow expectations from the high or low productivity Ricker relationship, depends upon recruitment in the previous brood year followed the high or low relationship. This dependency permits estimation of the conditional probability of remaining within a

9

productivity regime or transitioning to another, and allows predictions to be generated for future regime occupancy.

(9)

Rt = St *exp αt 1−Stβt

!

"#

$

%&

'

())

*

+,,*exp εt( )

εt ~ N 0,σ t( )αt = α̂λt

βt = β̂λtσ t = σ̂ λt

λt ~ Cat γ t,r( )γ t=1,r = prγ t,r = πλt−1,r

In evaluating the support from the spawner-recruit data for differences in productivity amongst regimes (states), the 𝛼!, 𝛽! ,𝜎! parameters will again be treated as varying across time between productivity states. However, instead of treating these parameters as consistent across time until the fixed breakpoint (Eq. 7), elements of a transition probability matrix (𝜋!,!) were estimated which govern the likelihood of regime (state) occupancy, conditioned upon the previous brood year. In this formulation 𝑖 represents the state from which one is transitioning and 𝑗 represents the productivity state to which one is transitioning (Eq. 10).

(10) π i, j =pi=1, j=1 pi=1, j=2pi=2, j=1 pi=2, j=2

!

"

##

$

%

&&

The diagonal elements of the transition probability matrix represent the probability of remaining in the same state from one brood year to the next and the off-diagonal elements of 𝜋!,! describe the probability of state transitions. All elements of the transition probability matrix were treated as free parameters within the model, with row elements summing to 1. Priors for the transition probability matrix were mildly informative with the row elements of 𝜋!,! assumed to be represented by separate multinomial distributions and therefore separate Dirichlet priors for each row (𝛼 = 2,2). This Dirichlet prior describes a broad normal distribution with an equal expected value of 0.5 for transitioning to the other state or remaining in the same state. γ t,r is the matrix of state occupancy probabilities, based upon the transition probability matrix (𝜋!,!). The categorical distribution is used to draw an integer value 1 or 2, for each sample from the posterior based on the γ t,r probability of observing each regime in each year t.

10

In order to generate expectations for the probability of future state (regime) occupancy, the stationary distribution for the transition probability matrix was determined. The stationary distribution may be approximated with suitable precision by repeated matrix multiplication of the transition probability matrix (𝜋!,!) by itself. Initial testing of the hidden Markov model indicated that the stationary distribution was achieved within 10 iterations of the matrix multiplication, but to ensure convergence the transition matrix multiplication was iterated 20 times during each MCMC sample. In this way predictions for future regime occurrence could be estimated from the available data.

The Gibbs sampler implement in JAGS was used to generate MCMC samples from the posterior distributions for model and derived parameters. For each model type, river system, and prior choice 3 chains were run for 2-million iterations, with a 50% burn-in. 1/10 samples were thinned and kept for evaluation to ensure no posterior correlation was observed. Gelman-Rubin test statistics were used to compared intra and inter-chain variance and ensure convergence to the stationary distribution (Gelman et al. 2004).

Estimating BEGs for the Regime Transition Ricker Model

The primary benefit of the hidden Markov regime transition spawner-recruit model is that in addition to estimating Smsy and MSY for both the high and low productivity regimes observed historically, it also provides a way to estimate the future probability of occupying each of these regimes. In addition results of the Markov regime transition model provide the necessary basis for simulating future regime occupancy and evaluate optimal escapement strategies across alternative patterns in future production. Based upon the transition probability matrix (𝜋!,!) and the estimated joint posterior distribution for model parameters, the potential yield across a range of spawning abundances was calculated, accounting for estimation uncertainty, stochastic recruitment, and future regime occupancy, in four steps (Eq. 11). First, 1,000 random samples for model parameters (𝛼! ,𝛽! ,𝜎! ,𝜋!,!) were drawn from the joint posterior. These 1,000 random samples represent alternative states of nature and account for estimation uncertainty. In addition these samples preserve posterior correlation in estimated parameter values because they are drawn from the joint posterior. Second, for each of these 1,000 samples s, a 100-year time-series of future regime states 𝜑!,! was generated based upon the estimated values for elements of the transition probability matrix (𝜋!,!,!). Each 100-year time-series was created by first generating a random (1,2) integer for the initial regime, then stepping forward one year at a time and determining the state in each year y (𝜑!,!) by drawing a random uniform [0,1] deviate (𝜏) and comparing it to the probability of remaining in the current state i (𝜋!,!,!) and the probability of transitioning to the other state (production regime) j (𝜋!,!,!) (Eq. 11).

(11)

τ ~ unif 0,1( )

ϕ y.s =i,τ ≤ πϕy−1. s ,i,s

j,τ > πϕy−1. s ,i,s

#$%

&%

11

This process was repeated until 100-year patterns in future production regimes (𝜑!,!,!) were simulated independently for each of the 1,000 posterior samples or alternative state of nature (s). In this way the future dynamics of stock-specific production regime occupancy were simulated based upon the regime transition tendency exhibited by each population in the past. Third, for each of 100 trial spawning abundances across a suitably large interval, the expected yield for each year y and posterior sample s was calculated (𝑌!,!) by determining expected recruitment (𝑅!,!) including lognormally-distributed stochastic recruitment deviations (Eq. 12).

(12)

Yy,s = Ry,s − Sy,s

Ry,s = Sy,s *exp αy,s 1−Sy,sβy,s

"

#$$

%

&''

(

)**

+

,--*exp εy,s( )

εy,s ~ N 0,σ̂ y,s( )αy,s ~ h x | α̂r, β̂r,σ̂ r( )βy,s ~ h x | α̂r, β̂r,σ̂ r( )σ y,s ~ h x | α̂r, β̂r,σ̂ r( )r =ϕ y.s

This process resulted in 100,000 predictions for yield (100 years by 1,000

posterior samples) for each trial spawning abundance, accounting uncertainty in future regime occupancy, estimation, and stochastic recruitment. Figure 3 displays the expected yield for the Egegik River across 1,000 states of nature and 100-year time series, when the Top 20% prior is used for spawner-recruit relationship estimation.

Fourth, the average yield at each of the 100 trial spawning abundance was calculated, and generalized additive model (GAM) was used to approximate this relationship and interpolate the expected mean yield (Fig. 4). From the curve approximating average yield across both future retime states and alternative states of nature, both the maximum sustainable yield (MSY) and spawning abundance which will produce maximal yield (Smsy) were calculated (Fig. 4). In addition, the range of spawning stock sizes which are likely to produce yield at or greater than 90% of maximum yield across regimes (90% MSY) were calculated. This range (Smsy Lower to Smsy Upper) is the escapement range that is expected to produce ≥ 90% of maximum yield, given the estimated differences in productivity of regimes and their future probability of occurrence.

The four steps in the process of simulating future yield and calculating

expectations for management quantities based on the average of expected yields, was repeated for each stock and type of prior imposed on the equilibrium abundance parameter (𝛽) in the Markov regime transition model.

12

Figure 3. Expected yield for the Egegik River across a range of potential spawning abundances, estimated by the regime transition Ricker model assuming the Top 20% prior on the beta parameter. The shaded region describes the 50% credible interval for the yield prediction, with the dotted line representing the average of predicted yields and the solid like representing the median of yield predictions.

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Egegik River Prior: Top20

Spawning Abundance (millions)

Mix

ture

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ld (m

illion

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13

Figure 4. Example calculation of Smsy and MSY from the mixture yield curve from the regime transition model for the Egegik River, assuming the top 20% prior on the beta parameter. The blue dots are the calculated average yield at 100 trial spawning abundances. The red line and shaded region indicate expected yield as approximated by the GAM model. Dark green lines indicate the spawning stock size that is expected to produce maximal yield (Smsy) and the realized yield (MSY). Light green lines describe the spawning stock size range (Smsy Lower to Smsy Upper), within which 90% or greater of potential yield (90% MSY) will be achieved. Values listed at the top right are in millions of sockeye.

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ons)

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MSY: 8.121Smsy: 3.52890% MSY: 7.309Smsy Lower: 2.191Smsy Upper: 5.085

14

Results

Table 1 summarizes the Smsy estimates using priors for the unfished equilibrium population size (𝛽) parameter based on the top 20% of salmon abundances reconstructed from the paleo lake coring data and the mean of the lake coring data, as well as the Smsy estimated using the method of Fair et al. (2012) but employing the lognormal correction. Table 1. Median of posterior estimates for Smsy’ (in millions) by river system, under five different scenarios regarding change in productivity and two prior types on the unfished population size, including the log-normal correction. Before and after 1980 indicate Smsy predictions based upon the Ricker breakpoint model. Regimes 1 and 2 represent the low and high productivity regimes respectively, estimated by the regime transition model. The final row shows the Smsy estimates from Fair et al. (2012) when corrected for the lognormal returns.

First we see that Naknek, Igushik and Wood all provide estimates that are generally in the range of historical escapements and are not too sensitive to the prior assumed for the equilibrium abundance parameter (𝛽), or the model used to represent the spawner-recruit relationship. Of the two prior distributions we used, the mean prior consistently provides lower estimates of Smsy independent of the model employed, because the these priors on beta are both lower and have a lower variance for all systems.

Results from the regime transition model indicate that Smsy for the low productivity regime (Regime 1) is higher than that estimated for the high productivity regime (Regime 2), independent of the river system and prior choice. The estimated Smsy difference between productivity regimes is most pronounced for the Kvichak and Ugashik systems, when the prior based on the top 20% of salmon abundances reconstructed from the paleolimnological data is used. When the Smsy estimates for the low and high productivity regimes are compared, we see a 53% reduction in the estimated optimal spawning stock size (Smsy) for the Kvichak and a 65% reduction for the Ugashik, as the systems move from low to high productivity periods. These Smsy differences between high and low productivity regimes, are significantly higher than the

Top$20%$Priors Kvichak Naknek Egegik Ugashik Igushik WoodSingle'Ricker 12.5''''''' 2.1''''''''' 4.4''''''''' 3.8''''''''' 0.4''''''''' 1.7'''''''''Before'1980 12.8''''''' 1.8''''''''' 3.8''''''''' 5.0''''''''' 0.5''''''''' 1.4'''''''''After'1980 11.1''''''' 2.0''''''''' 3.2''''''''' 2.0''''''''' 0.4''''''''' 1.7'''''''''Regime'1 19.8''''''' 2.1''''''''' 4.4''''''''' 6.1''''''''' 0.6''''''''' 1.1'''''''''Regime'2 9.9''''''''' 1.8''''''''' 3.4''''''''' 1.3''''''''' 0.4''''''''' 1.6'''''''''Mean$PriorsSingle'Ricker 8.8''''''''' 1.5''''''''' 2.8''''''''' 2.4''''''''' 0.4''''''''' 1.1'''''''''Before'1980 9.2''''''''' 1.2''''''''' 2.2''''''''' 2.8''''''''' 0.4''''''''' 1.1'''''''''After'1980 7.8''''''''' 1.4''''''''' 2.1''''''''' 1.4''''''''' 0.3''''''''' 0.9'''''''''Regime'1 13.6''''''' 1.5''''''''' 2.5''''''''' 3.4''''''''' 0.4''''''''' 1.1'''''''''Regime'2 6.4''''''''' 1.4''''''''' 2.3''''''''' 1.2''''''''' 0.3''''''''' 0.8'''''''''Fair'et'al.'2012 19.0''''''' 2.0''''''''' 5.9''''''''' 3.3''''''''' 0.3''''''''' 1.7'''''''''

15

average difference in Smsy between regimes across river systems of 27% when the top 20% prior on unfished population size (𝛽) was used, and 31% when the mean prior was used. From these results it is clear that BEG’s based on Smsy will be lower, the more time the system is expected to remain in the higher productivity regime in the future.

Differences in estimates of Smsy generated by the breakpoint model, before and after 1980, are also most pronounced for the Ugashik system, with higher Smsy in the pre-1980 period. Estimates for the Ugashik system indicate that the optimal spawning abundance (Smsy) decreased by between 51 and 61% after 1980, depending on whether the mean or top 20% prior is assumed. Despite the average post-1980 decrease in estimated Smsy across systems of 11-15% depending on prior choice, the spawner-recruit relationship for the Naknek system indicates that the optimal spawning abundance (Smsy) was between 10% and 15% higher after 1980, when the top 20% or mean prior was employed.

Despite the prior that assumes small unfished stock sizes are far more likely, when corrected for lognormal returns the Fair et al. (2012) method consistently provides the highest estimates of Smsy when compared to the Ricker model under either prior assumption. This is because the prior used by Fair et al. (2012) is actually fairly flat over the range of unfished stock sizes that are consistent with the data. Figure 5 shows the three priors for Egegik, the ADF&G prior, the mean of all the paleo data, and the top 20% of the paleo data.

When we look at potential yield (Table 2) predicted by the different model types, we see slightly larger potential yields when the top 20% prior is used. Results from the breakpoint model indicate that Naknek, Egegik and Wood show major increases in the potential yield after 1980 compared to before 1980. The most pronounced of these is Egegik with a 113% or 161% increase in MSY after 1980 depending on prior choice. Conversely, Kvichak, Ugashik, and Igushik show declines in potential yield after 1980, when the fixed transition coinciding with the PDO shift is assumed.

In the Markov Regime models, the low productivity Regime 1 has much less potential yield than the high productivity Regime 2, but the regimes are not synchronized across systems. Therefore, the average yield for Bristol Bay as a whole would be between the Regime 1 and Regime 2 values, for any particular brood year. On average across systems, 166% or 192% greater MSY is predicted in the high productivity regime (2), leading to a average increase in potential yield of 4.37 or 3.63 million sockeye, depending on whether the top 20% or mean prior on 𝛽 is specified. The observation that MSY for Regime 2 is greater than or equal to MSY for Regime 1 is a structural certainty of the model, and therefore expected. However, the magnitude of the differences in predicted MSY between regimes were quite surprising, ranging from a minimum of a 41% increase (1.14 million) in potential yield from low to high productivity regimes for Naknek river (top 20% prior), to a maximum 331% increase (4.35 million) in MSY for the Wood river (Table 3).

16

Figure 5. Prior probability on unfished population size for Egegik using the ADF&G prior (blue line), the mean and variance from all of the paleo data (grey line) and the top 20% of the paleo data with fixed CV=0.5 (orange line).

Table 2. Median of posterior estimates of MSY’ (in millions) for the different systems under five different scenarios regarding change in productivity and two priors on the unfished population size, including the log-normal correction. The final row shows the estimated MSY from Fair et al. 2012 when corrected for the lognormal returns.

0

0.005

0.01

0.015

0.02

0.025

3 5 7 9 11 13 15 17 19

Prior p

robability

Unfished population size in millions

F&G 20% All

Top$20%$Priors Kvichak Naknek Egegik Ugashik Igushik Wood TotalSingle'Ricker 11.6''''''' 3.5''''''''' 11.9''''''' 7.7''''''''' 0.7''''''''' 3.4''''''''' 38.9'''''''Before'1980 18.2''''''' 3.0''''''''' 6.0''''''''' 12.2''''''' 1.0''''''''' 2.2''''''''' 42.6'''''''After'1980 8.2''''''''' 3.9''''''''' 12.9''''''' 4.5''''''''' 0.7''''''''' 4.3''''''''' 34.5'''''''Regime'1 6.5''''''''' 2.8''''''''' 3.8''''''''' 3.6''''''''' 0.3''''''''' 1.4''''''''' 18.4'''''''Regime'2 15.5''''''' 3.9''''''''' 13.1''''''' 5.7''''''''' 1.0''''''''' 5.4''''''''' 44.6'''''''Mean$PriorsSingle'Ricker 8.6''''''''' 3.2''''''''' 8.5''''''''' 5.4''''''''' 0.7''''''''' 3.0''''''''' 29.4'''''''Before'1980 13.4''''''' 2.6''''''''' 3.9''''''''' 7.3''''''''' 0.8''''''''' 2.0''''''''' 30.0'''''''After'1980 6.3''''''''' 3.6''''''''' 10.3''''''' 3.9''''''''' 0.6''''''''' 4.4''''''''' 29.2'''''''Regime'1 6.4''''''''' 2.5''''''''' 2.5''''''''' 2.1''''''''' 0.3''''''''' 1.3''''''''' 15.0'''''''Regime'2 10.9''''''' 3.6''''''''' 10.2''''''' 5.4''''''''' 1.0''''''''' 5.7''''''''' 36.8'''''''Fair'et'al.'2012 16.2''''''' 3.4''''''''' 14.5''''''' 6.5''''''''' 0.7''''''''' 3.3''''''''' 44.6'''''''

17

Table 3. Percent change in estimated MSY before and after 1980 from the Fixed Breakpoint model and between the low and high productivity regimes estimated by the Markov Regime model.

A comparison of MSY values estimated by the Markov Regime model for low (Regime 1) and high (Regime 2) productivity regimes indicate a substantial increase in potential yield across systems and the two alternative priors, however the difference in MSY between regimes is greatest for Egegik, Wood, and Igushik systems (Table 3). However, when the transition between regimes is assumed to have coincided with the 1980 shift to warm phase PDO, there is a less clear trend in MSY. The Kvichak, Ugashik and Igushik River systems are estimated to have experienced a reduction in MSY after 1980, while the Naknek, Egegik, and Wood River saw a significant increase in MSY when the spawner-recruit time series are separated at this same point in time (Table 3).

In order to define management goals that are robust to future variation in productivity regimes, we are able to use information provided by Markov Regime model to inform our evaluation of the relationship between spawning abundance and yield. The stationary distribution of the estimated transition probability matrix provides a prediction for the probability of occupying the high and low productivity regimes in the future, and is used to weight our expectation of recruitment and yield across regimes (see Fig. 3). From these yield curves the range of spawning abundances that are likely to produce greater than 90% of MSY across regimes was calculated for each system and prior choice (Table 4). Both the lower and upper bounds predicted by the Markov Regime model are lower when the mean prior on equilibrium abundance parameter (𝛽) is assumed. In general however for most Bristol Bay river systems, the lower bounds of the 90% Smsy range predicted by the Markov Regime model are higher than the lower bounds of the current escapement goal ranges, with the most striking differences observed for Egegik and Kvichak rivers.

The lower bound of the current escapement goal range for the Egegik river is 800,000 sockeye, while the Markov Regime model predicts that greater than 90% of MSY will be achieved across regimes at spawning abundances greater than 2.2 or 1.3 million sockeye, depending on whether the top 20% or mean prior was used. Both of these lower bounds predicted by the Markov Regime model are close to, or exceed, the current upper bound of the Egegik escapement goal range of 1.4 million sockeye. Similarly, the Kvichak river currently managed for a minimum escapement of 2 million sockeye and a 50% escapement rate when the inseason forecast for the Kvichak system exceeds 5 million sockeye, however the Markov Regime model estimates the lower

Top$20%$Priors Kvichak Naknek Egegik Ugashik Igushik WoodPre/Post(1980 -55% 29% 113% -63% -24% 97%Regime(1(to(Regime(2 139% 41% 245% 58% 232% 284%Mean$PriorsPre/Post(1980 -53% 40% 161% -47% -16% 126%Regime(1(to(Regime(2 71% 46% 316% 156% 231% 331%

18

bound of the escapement goal range to be 6.9 or 4.6 million sockeye for the top 20% and mean priors respectively.

In addition to differences between the current and Markov Regime model estimated lower bounds, the range of the escapement goals differs significantly. With the exception of the Kvichak and Wood river systems, the width of the range estimated by the Markov Regime model to produce 90% or greater of MSY across future regimes is substantially wider than the current escapement goal ranges. The escapement goal ranges estimated by the Markov Regime model are on average 148% or 66% larger than those currently specified in the Bristol Bay management plan, depending on whether the top 20% or mean prior from the paleo data is assumed.

In the following sections we will outline results from selected systems generated by the three spawner-recruit model types and comparisons amongst the two prior choices. Table 4. Results of Markov Regime model, in millions of sockeye. For each system and prior choice, the expectation for future maximum potential yield (MSY) and the spawning abundance required to achieve that yield (Smsy), given predictions for high and low productivity regimes and their future probability of occurrence. Also included are the ranges of spawning abundances (Smsy range), which are expected to provide greater than or equal to 90% of maximum yield (90% MSY).

Kvichak

The Kvichak is perhaps the most perplexing system in Bristol Bay. The overall spawner recruit curve fit using the basic Bayesian Ricker model and our priors, suggest very large stock sizes may be required to achieve MSY (see figures 6 and 7). Furthermore, it is clear from the figures 6 and 7 that the choice of prior influences our estimate of the equilibrium population size (𝛽) parameter, and therefore the estimated shape of the spawner-recruit relationship. Results from the 1980 breakpoint model for Kvichak indicate that the shape of the estimated spawner-recruit relationship is exceedingly similar for the periods before and after 1980 (figure 8), which is reflected in

System Prior MSY Smsy 90%0MSY Smsy0Range Current0RangeKvichak Top20 10.2 10.4 9.2 6.9+,+15.2 2.0+,+10.0Naknek Top20 3.5 1.8 3.1 1.1+,+2.7 0.8+,+1.4Egegik Top20 8.1 3.5 7.3 2.2+,+5.1 0.8+,+1.4Ugashik Top20 3.6 1.7 3.2 0.9+,+2.5 0.5+,+1.2Igushik Top20 0.7 0.4 0.6 0.2+,+0.6 0.2+,+0.3Wood Top20 3.3 1.4 2.9 0.9+,+2 0.7+,+1.5Kvichak Mean 8.1 7.6 7.3 4.6+,+10.6 2.0+,+10.0Naknek Mean 3.3 1.2 3 0.7+,+1.8 0.8+,+1.4Egegik Mean 6.1 2 5.5 1.3+,+3.3 0.8+,+1.4Ugashik Mean 3.3 1.3 3 0.8+,+2.1 0.5+,+1.2Igushik Mean 0.7 0.3 0.6 0.2+,+0.4 0.2+,+0.3Wood Mean 3.9 0.8 3.5 0.5+,+1.2 0.7+,+1.5

19

the very similar estimates of Smsy for both periods, however with lognormal correction the MSY estimates from the two periods are substantially different.

The Markov Regime transition model estimates periods of very different productivity. The duration of productivity regimes may be evaluated by looking at the estimated parameter values over time, as they move between states (Fig. 9). Predictions for the alpha parameter’s value across brood years indicate, that it has jumped from high to low with runs of 6-8 years in each productivity regime (Fig. 9). However, the future probabilities or occupying the high and low productivity regimes are quite close 𝑃𝑟 𝑅!"# = 0.52,𝑃𝑟 𝑅!!"! = 0.48, suggesting that future recruitment with represent a near equal mixture of the high and low productivity regimes (Fig. 10).

It is also clear from figure 10 that when the two spawner-recruit relationships for the Kvichak system are estimated by the Markov Regime transition model, are is a much more pronounced difference in shape a opposed to those estimated by the 1980 Breakpoint model (Fig. 8). This is partly due to that fact that despite a large difference in the alpha parameter of the Markov Regime model that specifies maximum productivity (max 𝑅/𝑆 = 𝑒!), the difference between equilibrium population size (𝛽) is quite similar between the high and low productivity regimes (Fig. 11).

Figure 6. Kvichak spawner-recruit curve from the basic Bayesian Ricker model with top 20% prior. Shaded regions represent the 95% and 75% credible intervals for the spawner-recruit relationship.

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ruitm

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milli

ons)

Medianalpha: 0.77beta: 20.221

DIC: 1449.9

20

Figure 7. Kvichak spawner-recruit curve from the basic Bayesian Ricker model with the mean prior. Shaded regions represent the 95% and 75% credible intervals for the spawner-recruit relationship.

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Figure 8. 1980 breakpoint model for the Kvichak River assuming the top 20% prior. Blue lines and points indicated the estimated and observed spawner-recruit relationship for the pre-1980 period, and red the post-1980 period.

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Figure 9. Changes in productivity of the Kvichak River, as reflected in estimates of the alpha parameter over time from the Markov transition model. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time.

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Figure 10. Spawner-recruit relationships for the high (green) and low (purple) productivity regimes in the Kvichak system, estimated by the Markov Regime transition model with the top 20% prior. The barplot at the right indicates the probability of future regime occupancy estimated from the transition probability matrix.

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Figure 11. Posterior probability distributions for the equilibrium (unfished) population size parameter (𝜷) in the Markov Regime model with top 20% prior.

0e+00

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Naknek

Results from the basic Ricker model for the Naknek system are similar to those from the Kvichak system (Fig. 6 and 7), in that the estimated value of the maximum productivity parameter (𝛼) is lower and the estimated value of the equilibrium (unfished) biomass parameter (𝛽) is higher, for the model assuming the top 20% prior on 𝛽 (Fig. 12) when compared to the mean prior (Fig. 13).

When evaluated using the regime transition model the data from the Naknek system show little in the way of regime changes, with little variation in the estimated value of the alpha parameter (Fig. 15), or beta parameter over time. In addition, the alternative spawner-recruit relationships predicted by the 1980 Beakpoint model (Fig. 14) and those estimated by the Markov Regime transition model (Fig. 16) are quite similar. Despite the similarity, the Markov Regime transition model predicts that in the future the Naknek system will spend a larger proportion of time in the slightly higher productivity regime (Fig. 16).

Figure 12. Naknek spawner-recruit curve from the basic Bayesian Ricker model with top 20% prior. Shaded regions represent the 95% and 75% credible intervals for the spawner-recruit relationship.

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DIC: 1371.1

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Figure 13. Naknek spawner-recruit curve from the basic Bayesian Ricker model with the mean prior. Shaded regions represent the 95% and 75% credible intervals for the spawner-recruit relationship.

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Figure 14. 1980 breakpoint model for the Naknek River assuming the top 20% prior. Blue lines and points indicated the estimated and observed spawner-recruit relationship for the pre-1980 period, and red the post-1980 period.

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Figure 15. Changes in productivity of the Naknek River, as reflected in estimates of the alpha parameter over time from the Markov transition model. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time.

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Figure 16. Spawner-recruit relationships for the high (green) and low (purple) productivity regimes in the Naknek system, estimated by the Markov Regime transition model with the top 20% prior. The barplot at the right indicates the probability of future regime occupancy estimated from the transition probability matrix.

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Egegik showed a dramatic increase in recruits per spawner after 1980 (Fig. 17). This increase was especially pronounced in the 1980 brood years, then declined in the 1990 brood years, and was followed by higher productivity again in the 2000 brood years.

Expectations for the spawner-recruit relationships for Egegik pre and post 1980 from the Breakpoint model (Fig. 18) show a considerably higher curve after 1980, reflecting the jump in recruits per spawner seen in figure 17, but neither data set demonstrates a clear reduction in recruitment at high spawning stock sizes and the estimate of unfished (equilibrium) population size depends strongly on the prior on β.

Figure 19 shows the two curves for the Markov transition model in green (productive regime) and purple (unproductive regime). The colored bar on the right shows the relative probability of being in the productive regime (about 65%) and the probability of the unproductive (about 35%).

Figure 20 shows the time trend in the probability of being in the productive and unproductive regimes as indicated by the median α value. If the model is 100% confident the stock was in the productive regime then the value is roughly 2.3, if the model is 100% confident the system was in the unproductive regime then is 0.9. So we see that from brood years 1963-1970 the model is confident we were in the unproductive regime, and from 1977 onwards (with the exception of 1993 and 1998) the model is confident it was in the productive regime.

Figure 17. Temporal pattern in recruits per spawner for Egegik.

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Figure 18. 1980 breakpoint model for the Egegik River assuming the top 20% prior. Blue lines and points indicated the estimated and observed spawner-recruit relationship for the pre-1980 period, and red the post-1980 period.

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Figure 19. Spawner-recruit relationships for the high (green) and low (purple) productivity regimes in the Egegik system, estimated by the Markov Regime transition model with the top 20% prior. The barplot at the right indicates the probability of future regime occupancy estimated from the transition probability matrix.

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Figure 20. Change in the alpha parameter over time from the Markov transition model for the Egegik River. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time.

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The Ugashik system also shows a pronounced increase in average productivity after 1980, when estimated using the breakpoint model (Fig. 21). However when we look at the Markov Regime transition model we see that there were periods of lower productivity after the 1980 brood year, the most pronounced of which occurred in the mid 1990s (Fig. 22). However, the long-term expectations for regime occupancy from transition probability matrix estimated in the Markov Regime transition model indicate that for the Ugashik system we should expect to spend a nearly equal proportion of time in the high and low productivity regimes (Fig. 23).

Figure 21. 1980 breakpoint model for the Ugashik River, assuming the top 20% prior. Blue lines and points indicated the estimated and observed spawner-recruit relationship for the pre-1980 period, and red the post-1980 period.

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Figure 22. Changes in productivity of the Ugashik River, as reflected in estimates of the alpha parameter over time from the Markov transition model. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time.

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Figure 23. Spawner-recruit relationships for the high (green) and low (purple) productivity regimes in the Ugashik system, estimated by the Markov Regime transition model with the top 20% prior. The barplot at the right indicates the probability of future regime occupancy estimated from the transition probability matrix.

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Igushik

The Igushik River shows no major change in spawner-recruit relationships estimated by the 1980 Breakpoint model (Fig. 24), but does appear to show transition between regimes of higher and lower productivity over time (Fig. 25). However, unlike other systems there does appear to be a three-fold difference in the maximum productivity parameter (𝛼) estimates between high and low productivity regimes (Fig. 26), which combined with a moderate difference in equilibrium (unfished) abundance parameter estimates (𝛽) for the two different regimes, allows the high productivity regime to contain one large escapement that was reasonably unproductive.

Figure 24. 1980 breakpoint model for the Igushik River, assuming the top 20% prior. Blue lines and points indicated the estimated and observed spawner-recruit relationship for the pre-1980 period, and red the post-1980 period.

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Figure 25. Changes in productivity of the Igushik River, as reflected in estimates of the alpha parameter over time from the Markov transition model. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time.

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Figure 26. Spawner-recruit relationships for the high (green) and low (purple) productivity regimes in the Igushik system, estimated by the Markov Regime transition model with the top 20% prior. The barplot at the right indicates the probability of future regime occupancy estimated from the transition probability matrix.

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The basic Bayesian Ricker mode for Wood River, assuming the top 20% prior, generally captures the spawner recruit relationship, but has difficulty in balancing the overcompensation observed in the high escapement years (Fig. 27). These years of large escapements that did not produce particularly high numbers of recruits and result in a value for the equilibrium (unfished) abundance parameter (𝛽) of just over 4 million sockeye. However, the Wood River also shows evidence for a post 1980 increase in productivity (Fig. 28), with those high escapement and low production years falling prior to the 1980 breakpoint. Similar to the 1980 Breakpoint model, the Markov Regime transition model predicts the system to be best represented by two drastically different spawner-recruit relationships (Fig. 29). From the trends in estimated values for the maximal productivity (𝛼, Fig. 30) and equilibrium abundance (𝛽, Fig. 31) parameters, it is clear that the high productivity regime is defined by brood years with a pronounced increase in both maximum production potential and rearing capacity. Furthermore, those parameters were not consistently low prior to 1980, showing a period of high production from 1976 – 1978, followed by a lower production period until the mid 1980’s.

Figure 27. Wood spawner-recruit curve from the basic Bayesian Ricker model with top 20% prior. Shaded regions represent the 95% and 75% credible intervals for the spawner-recruit relationship.

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Figure 28. 1980 breakpoint model for the Wood River assuming the top 20% prior. Blue lines and points indicated the estimated and observed spawner-recruit relationship for the pre-1980 period, and red the post-1980 period.

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Figure 29. Spawner-recruit relationships for the high (green) and low (purple) productivity regimes in the Wood system, estimated by the Markov Regime transition model with the top 20% prior. The barplot at the right indicates the probability of future regime occupancy estimated from the transition probability matrix.

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Figure 30. Change in the alpha parameter over time from the Markov transition model for the Wood River. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time.

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Figure 31. Change in the beta parameter over time from the Markov transition model for the Wood River. Shaded regions represent the 95% and 75% credible intervals for the parameter value over time. Discussion

In this report we present three alternative methods for modeling the stock-recruitment relationships for the Kvichak, Naknek, Egegik, Ugashik, Igushik, and Wood River systems of Bristol Bay, Alaska. The Bayesian approach to parameter estimation in this instance was superior maximum likelihood methods because it allowed samples to be drawn from posterior probability distributions that describe uncertainty in model parameters, and permitted the incorporation of prior information for uninformed parameters. The first method fits a simple Bayesian Ricker model to the available data, but incorporates an informative prior on the equilibrium unfished abundance parameter (𝛽) based upon reconstruction of sockeye abundance prior to modern abundance enumeration methods, from paleolimnological sampling of marine derived nitrogen isotopes in nursery lake sediment cores. Two alternative priors were utilized to evaluate the sensitivity of model predictions for management reference points (i.e., MSY and Smsy) to prior information, one which used the calculated mean and variance of the reconstructed salmon abundance trend (1750 – 1890) and a less informative prior which was defined by the mean of the top 20% of reconstructed abundances and a CV=0.5. The

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second method allowed for a shift in system productivity occurring in brood year 1980 so as to coincide with the shift in the Pacific Decadal Oscillation from cold to warm phase, and estimating separate Ricker model parameters for the two time periods. The third method assumes that each river system has shifted between two productivity states historically, treating the transition between these productivity states as a 1st order Markov process whereby the productivity in a specific brood year is conditioned on the estimated productivity in the previous year.

Data for each river system was modeled using each of these three methods and the two alternative priors. Each of the modeling methods implicitly assumes something about the state of the system. The simple Bayesian Ricker model assumes that the spawner recruit relationship has been constant over time, and that differences between observed recruitment and that predicted by the spawner-recruit model are independent and identically distributed random process deviates. The Breakpoint Ricker model assumes that each system experienced a shift in production during the period of time evaluated, and that that shift occurred in 1980. Conversely, while the Markov Regime transition model also assumes that the spawner-recruit data is best approximated by two production relationships, it makes no assumption about the number of shifts that has occurred, nor when those shifts took place. In this ways the Breakpoint Ricker model is a more restrictive subset of the Markov Regime transition model.

As we move from the basic Ricker to the Breakpoint model, to the Markov Regime model, the flexibility of the spawner-recruit model in fitting to the available data increases. One method for evaluating the fit of each model to these data is to look at the estimated standard deviation in the lognormal error distribution. For both the Egegik and Wood River systems, the 1980 Breakpoint model was estimated to have a lower standard deviation in the distribution of residual error in the model fit, indicating that it was successfully able to explain more of the variation in recruitment over time independent of the prior assumed for 𝛽. Instead of assuming one shift between production regimes has occurred and fixing the date of that shift occurred in 1980, the flexibility of the Markov Regime model allows it to explain more of the variability in the observed spawner-recruit data, leading to lower estimates of residual error variance for all systems but the Naknek River, and the Ugashik River when the top 20% prior was assumed. This is not altogether surprising given the lack of difference in Markov Regime transition model parameters over time in the Naknek River (see Fig. 15) When the variance in the error distribution is compared between the 1980 Breakpoint model and the Markov Regime model, estimated variance parameters for all systems and prior choices were lower for the Markov Regime model.

We believe the Markov Regime model represents the best tool for evaluating stock-recruitment relationships and defining biological escapement goals for most Bristol Bay sockeye populations for several reasons. First, almost all stocks show evidence for changes in productivity in response to periodic regime change. The Kvichak, Ugashik and Igushik show strong changes that are not well explained by a one time shift around 1980, and interestingly all show a jump to the low productivity regime in the 1990s. Only Naknek really appears to be “best” explained by a single Ricker curve (Fig. 15). Egegik has a strong change from low to high productivity beginning in brood year 1975,

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with two single year switches (Fig. 20). Wood River showed a bump in in productivity in the mid 1970s, and then a gradual increase in the last few decades (Fig. 30 and 31). Second, by allowing the data to define the timing and frequency in shifts between regimes, the Markov Regime transition model is better able to capture the individual production dynamics of the system as opposed to an arbitrary breakpoint imposed by the analyst based on assumed shifts in the frequency and timing of changes in production. Third, by estimating elements of the transition probability matrix defining the 1st order Markov process governing transition between productivity states (regimes), we are able to estimate the future probability for a system occupying the high or low production states. These future predictions can then be used to define BEG’s that will maximize yield across the different production regimes likely to be observed in the future (weighted in proportion to their probability of occurrence). These BEGs can be expected to produce higher yield given uncertain production patterns in the future.

We note that for the Igushik, Kvichak, and Wood River systems the probability of occupying the high productivity state (regime) nearly equal to that of the low productivity state. Conversely, for the Egegik and Naknek systems, the probability of occupying the high productivity state in the future greatly exceeds the probability for the low productivity state. These differences manifest themselves in how the estimated BEGs for the Naknek and Egegik systems rely more heavily on spawning stock sizes which maximize yield in the higher production regime.

Predictions for the Markov Regime transition model for the timing of regime transitions illustrate several interesting things. First, very few of the systems with the exception of the Egegik River, have experienced a single shift in productivity consistent with the Breakpoint models assumptions. The Kvichak, Igushik, and Ugashik systems have seen cyclic patterns of production, historically consisting of between 2 and 5 transitions between production regimes, which would not be well represented by a one-time shift coinciding with the PDO transition. Despite a rather consistent trend toward higher production in the Wood River following the 1980’s, the 1980 Breakpoint model would have severely underestimated production during the period 1974 – 1977 (Fig. 30 and 31). If we were to choose a single model for all stocks the Markov Regime transition model would seem to be preferred, but a case could be made for using a single curve for the Naknek, and perhaps a fixed breakpoint for Egegik in brood year 1975.

When using informative prior information for parameters, which are poorly defined by the available data, it is important to evaluate their influence on model predictions. The mean and top 20% priors specified for unfished (equilibrium) abundance parameter (𝛽) influence the estimated value of several model parameters. When the less informative top 20% prior is specified, estimates of both MSY and Smsy are higher independent of the model type. Across systems, the estimates of Smsy are more strongly influenced by prior choice, as opposed to those for MSY. Overall our preference would be to use the 20% prior as it is a broader and overall less informative prior, and therefore estimates of Smsy and MSY are more directly influenced by the available data.

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Limitations

Our analysis and selection of BEGs is predicated on the usual assumption that the Smsy will be achieved in each year, with no allowance for implementation error, and that our models are in fact correct. If BEGs are not achieved in reality, using them as a management goal will not maximize average yield. Implementation error is a reality that cannot be ignored, and while many would argue that once you include implementation error in an analysis you are looking at something other than BEGs it must be recognized that BEGs are not likely to be consistently achieved in reality. We know from experience that the actual escapement is not independent of return sizes, and in years of large returns escapements end up being higher than in years of low returns.

Our results are based on an assumption that historical patterns in production will mirror those in the future. In reality however, Western Alaska has seen dramatic changes in climate in the last few decades and almost all scientific analysis suggest the warming trend will continue. Thus it seems unrealistic to expect that production relationships seen in the past will continue in the future. This could take the form of the probability of different regimes occurring would change, or that the regimes themselves might change.

Finally the objectives of salmon management are more complex than biological yield alone. There are economic and social objectives, such as the economic viability of the fishery and allocation among users. Ideally, these other objectives should be incorporated in an analysis of escapement goals.

Future work

The most obvious future work should be directed at correcting for the limitations identified above. Simulations that incorporate the management process will provide a better estimate of the likely consequences of different escapement goals on yield and ultimately, on economic and social objectives known to exist.

Alternative escapement goals should also be evaluated across a wide range of climate-impact scenarios so that policies that are robust to climate induced uncertainty can be identified.

Identifying the trade-offs between various objectives should be included as part of any analysis. References

Baker, T.T., L.F. Fair, R.A. Clark, and J.J. Hasbrouck. 2006. Review of salmon escapement goals in Bristol Bay, Alaska 2006. Anchorage.

Burkenroad, M.D. 1946. Fluctuations in abundance of marine animals. Science 103(2684): 684-686.

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Burkenroad, M.D. 1953. Theory and practice of marine fishery management. Journal du Conseil 18(3): 300-310.

Clark, J.H., McGregor, A., Mecum, R.D., Krasnowski, P., and Carroll, A.M. 2006. The

commercial salmon fishery in Alaska. Alaska Fishery Research Bulletin 12(1): 1-146. Cunningham, C.J., Hilborn, R., Seeb, J., Smith, M., and Branch, T.A. 2012. Reconstruction of

Bristol Bay sockeye salmon returns using age and genetic composition of catch. University of Washington, Seattle, WA.

Fair, L.F., Brazil, C.E., Zhang, X., Clark, R.A., and Erickson, J.W. 2012. Review of salmon

escaopement goals in Bristol Bay, Alaska, 2012. Fisheries Manuscript Series. Alaska Department of Fish and Game, Anchorage, Alaska.

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. 2004. Bayesian data analysis. Chapman &

Hall/CRC, Boca Raton, Fla. Gelman, A., and Shalizi, C.R. 2012. Philosophy and the practice of Bayesian statistics. The

British journal of mathematical and statistical psychology. Gilbert, D.J. 1997. Towards a new recruitment paradigm for fish stocks. Canadian Journal of

Fisheries and Aquatic Sciences 54(4): 969-977. Hare, S.R., Mantua, N.J., and Francis, R.C. 1999. Inverse production regimes: Alaska and West

Coast Pacific salmon. Fisheries 24(1): 6-14. Hilborn, R. 1985. Simplified calculation of optimum spawning stock size from Ricker stock

recruitment curve. Canadian Journal of Fisheries and Aquatic Sciences 42(11): 1833-1834. Mantua, N.J., and Hare, S.R. 2002. The Pacific decadal oscillation. J Oceanogr 58(1): 35-44. Ricker, W.E. 1954. Stock and recruitment. Journal Fisheries Research Board of Canada(11): 556-

623. Rogers, L.A., Schindler, D.E., Lisi, P.J., Holtgrieve, G.W., Leavitt, P.R., Bunting, L., Finney,

B.P., Selbie, D.T., Chen, G., Gregory-Eaves, I., Lisac, M.J., and Walsh, P.B. 2013. Centennial-scale fluctuations and regional complexity characterize Pacific salmon population dynamics over the past five centuries. Proceedings of the National Academy of Sciences of the United States of America 110(5): 1750-1755.

Schindler, D.E., Leavitt, P.R., Brock, C.S., Johnson, S.P., and Quay, P.D. 2005. Marine-derived

nutrients, commercial fisheries, and production of salmon and lake algae in Alaska. Ecology 86(12): 3225-3231.

Schindler, D.E., Leavitt, P.R., Johnson, S.P., and Brock, C.S. 2006. A 500-year context for the

recent surge in sockeye salmon (Oncorhynchus nerka) abundance in the Alagnak River, Alaska. Canadian Journal of Fisheries and Aquatic Sciences 63(7): 1439-1444.

Vert-pre, K.A., Amoroso, R.O., Jensen, O.P., and Hilborn, R. 2013. Frequency and intensity of

productivity regime shifts in marine fish stocks. Proceedings of the National Academy of Sciences of the United States of America 110(5): 1779-1784.

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An#Evaluation#of#Biological# Escapement*Goalsfor ... 22, 2015 Suggested citation: Cunningham, C.J., D. Schindler, R. Hilborn. 2015. An evaluation of biological escapement goals for