The state-space model consists of two parts: (1) a process model for the underlying state dynamics, and (2) an observation model that links the data to the true underlying state. The state-space model may also be thought of as a hierarchical model where the state (abundance) evolves according to a process model for population dynamics (for example, a Beverton-Holt model) with some process error, and observations on the state (“the data”) are made conditional on the true but unobservable state.

First, to model the adult-to-juvenile transition (fig. 2), we used the Beverton-Holt model as opposed to the Ricker model to express the number of juveniles collected at Swift Dam as a density-dependent function of the number of female adults passing Swift Dam in brood year y (PacifiCorp, 2024; Ricker, 1954; Beverton and Holt, 1957):ln(RJ,y)=ln(α)+ln(EF,y)−ln1+αKEF,y+εP,J,y,(1)where

The process error term, ε_P,J,y, quantifies annual variation in productivity driven by environmental variation with a mean of zero and a standard deviation of σ_P,J. Because the productivity of juveniles directly depends on the fraction of juvenile coho salmon collected at Swift Dam, we used annual estimates of FCE (PacifiCorp, 2024), which measures fish capture relative to the number of fish that approached the floating fish collector. This provides an index of fish facility performance from year to year. We modeled the productivity parameter α as a function of the annual FCE and other covariates for coho salmon in the following manner:ln(α_y) = θ₀ + θ_ix_i,y+h,(2)where

The proportion of early spawning coho salmon was defined as those adult coho salmon that arrive at Merwin Dam and are transported during the early (September) compared to the late (October) part of the spawning migration. Note that because the stock-recruitment model was informed by just 10 years of data, we did not fit the model using all covariates on productivity simultaneously. We instead used a stepwise approach where x₂−x₅ were respectively used to estimate θ₂−θ₅ within separate models that also included the effect of FCE with θ₁x_1,y+2. Specifically, x_2,y+0 was the data vector for the proportion of hatchery-origin female spawners, x_3,y+0 was the proportion of early spawners (E_p), and x_4,y+0 was the maximum monthly winter flow indexed to brood year y. The minimum summer flows, x_5,y+2, were indexed to the yearling outmigration in year y + 2. Flow data were obtained from the U.S. Geological Survey monitoring site (site identification number 14216500) on the Muddy River located at the upper end of Swift Reservoir (U.S. Geological Survey, 2023). Importantly, when drawing inferences about the effect of FCE on productivity or SAR, we used the model that only included the effect of FCE on productivity in the model. Although the model’s framework is designed to accommodate covariates associated with oceanic effects on SAR, we refrained from adding covariates on SAR at this time because of the lack of information linking juvenile at outmigration to adult age at return to Merwin Dam.

For the juvenile-to-adult transition, we model the number of adult returns as a lognormal function of a density independent smolt-to-adult return rate (SAR):ln(R_A,y) = ln(R_J,y) + ln(SAR) + ε_P,A,y,(3)where

Given that juveniles can pass Swift Dam at ages 0–2 and adults return at ages 1–3, the initial 2 years of juvenile recruits and 3 years of adult recruits that produced returns beginning in 2013 were not linked to the two-stage stock-recruitment model. These initial state vectors were estimated as draws from common lognormal distributions with parameters ln(R_J,0), σ_RJ0 and ln(R_A,0), σ_RA0, respectively, for juveniles and adults.

Given juvenile recruits, the number of juveniles a_J passing Swift Dam in calendar year t was modeled as the following:Jt,aJ=RJ,t−aJpJ,t−aJ,aJ,(4)where

Given adult recruits, we model the number of adults returning at age a_A of sex s and outmigration strategy o as the following:Nt,aA,s,o=RA,t−aApaA,t−aA,s,o,(5)where

Outmigration strategy refers to whether juveniles were collected at Swift Dam as subyearlings, yearlings, or 2-year-old outmigrants. Because data on the age structure of returning adults for a given age at juvenile outmigration are absent, our current formulation of the two-state model does not account for juvenile age at outmigration in the adult returns, but in the future such information would allow us to estimate SAR separately for each outmigration strategy. Nonetheless, given three adult ocean-age classes (MacLellan and Gillespie, 2015; fig. 3) and three outmigration strategies, the brood-year-specific return probabilities are pA,y=paA,t−aA,s,o.

We model the brood-year composition using this structure applied to empirical data on coho salmon in the Lewis River for several reasons. First, juvenile recruitment is expressed as a function of female escapement, requiring estimation of sex structure. Second, age 0–2 juvenile proportions are of direct interest. Third, jack and adult abundances have been recorded since 2012, whereas comprehensive age-structure data have not yet been collected. Thus, using the jack and adult abundance data requires estimating the proportion of jacks in the brood-year return.

Vectors of brood-year-specific marginal and conditional probabilities were modeled hierarchically as draws from a Dirichlet distribution (Fleischman and others, 2013; Scheuerell and others, 2021):Py, jack/not jack|ajDirichletπjack/not jack|aj,τjack/not jack|aj Py, aA|jack,ajDirichletπaA|jack,aj,τaA|jack,ajPy,aA,s=M,F|not jack,ajDirichletπaA,s=M,F|not jack,aj,τaA,s=M,F|not jack,aj,(6)whereπ

is the vector of mean proportions in each age, sex, and age at ocean entry category and is an inverse variance parameter.

Given age-specific juvenile emigration and adult returns from brood year y in calendar year t, the total number of juveniles passing Swift Dam in calendar year t is the sum of the abundance-at-age over all ages:Jt=∑aJJt,aJ.(7)Note that the current model for coho salmon at Swift Dam accounts for brood stock removal but does not account for juvenile age at outmigration or harvest rates; however, we show here how these estimates can be incorporated into the current framework of the model for future applications. For example, the total number of returning adults in year t is the sum of the abundances across all age, sex, and outmigration classes:

Returns of subclasses in year t are calculated similarly by summing over the appropriate indices. For example, the number of female returns in year t isNF,t=∑o∑aANt,aA,s=F,o.(9)In addition, the age, sex, and outmigration structure in the adult returns each calendar year isqaA,t,s,o=NaA,t,s,oNt,(10)where

Escapement (E_t) of spawners past Swift Dam in calendar year t includes hatchery-origin adults released upstream from Swift Dam to spawn in the wild (H_t) plus naturally produced spawners (N_t). The model can accommodate those naturally produced spawners that survive harvest (C_t) or those taken for hatchery brood stock (B_t):

Where the annual harvest rate below Swift Dam can be a composite of ocean harvest that is assumed constant across ages and sexes of returns each year, such that catch is:C_t = U_tN_t,(12)where

Escapement of female spawners, which dictates juvenile recruits, is shown in equation 13:E_F,t = N_F,t − q_F,tC_F,t − q_F,tB_t + f_F,tH_t,(13)where

Observations to inform the parameters of the state model include the following:

All parameters and unknown states were estimated in a Bayesian framework using JAGS software (Plummer, 2009) as implemented through the ‘runjags’ package of the R statistical programming platform (R Core Team, 2023). JAGS is a Bayesian estimation software package that implements Markov chain Monte Carlo (MCMC) sampling using a Gibbs or Metropolis-Hastings sampler. Prior distributions for most parameters were set according to Fleischman and others (2013). However, a preliminary model using a weakly informed prior distribution indicated that the data were insufficient to estimate capacity. Therefore, we used information from a meta-analysis on coho salmon populations throughout the Pacific Northwest region to inform the prior distribution on capacity (Korman and Tompkins, 2014). We compare posterior distributions when using the weakly informed versus the informed prior distribution. We used the lognormal distribution parameters provided by Korman and Tompkins (2014) with a mean of 7.46 and standard deviation (SD) equal to 0.67 as the informed prior distribution on capacity. For the weakly informed prior, we used a truncated normal distribution with a μ of 10,000 and an SD of 1,000,000. Comparison of posterior distributions resulting from different prior distributions should help provide a range in capacity estimates for the estimated habitat upstream from Swift Dam. Importantly, the Korman and Tompkins (2014) estimates of capacity were provided on a per-kilometer basis. Therefore, we used 186.9 km of habitat upstream from Swift Reservoir and an additional 14.5 km for the reservoir, resulting in a total of 201.4 km of habitat for salmon. This assumes similar habitat carrying capacity between the reservoir and riverine habitats upstream from Swift Dam.

We ran 3 independent MCMC chains each for 80,000 iterations, discarding the first 30,000 to ensure each chain had converged to its stationary stable distribution. We then thinned the final 50,000 iterations to 1 in 50 to reduce autocorrelation, yielding a final sample of 1,000 draws from each chain. Convergence of each parameter was checked visually to ensure mixing of the chains, and quantitatively by ensuring that the Rubin-Gelman statistic (R^) was less than 1.1 (Gelman and others, 2004).

To implement the two-stage lifecycle model, we first had to estimate the abundance of juveniles collected at Swift Dam. Of the total fish collected, a varying fraction of fish is sampled to determine size and age composition. Although 100 percent of the fish are sampled on most days, on infrequent days during high fish passage, sample rates are lowered for logistical reasons. Over the time series, daily sample rates were as low as 0.1, but 87 percent of the time, sample rates=1. We used the daily sample rates in year t (r_d,t) and the counts of juvenile salmon in the sample tank at Swift Dam (c_d,t), to estimate the age-specific (a) total abundance of juveniles collected (J_d,t) at the dam.

Estimating age-specific abundance requires classifying juveniles by age (0, 1, and 2). Catch-at-age was determined by PacifiCorp personnel, and we assumed no classification error in age assignment (fig. 4). Daily counts of age-0 fish were considered a complete census, without classification error, owing to their small size at date. Thus, estimation of the total number of age-0 fish (J_0d,t) involved simply inflating sample-tank counts by the sample rate, if counts (c_0d,t) were binomially distributed:c_0d,t ~ binomial(r_d,t,J_0d,t),(19)and for age-1,c_1d,t ~ binomial(r_d,t,J_1d,t),(20)and age-2 fish

However, protocols for age assignment of older fish varied across years. In years prior to 2017, age-1 or age-2 was assigned based on size thresholds for subsampled groups (g) whose lengths were measured. The subsampled groups of the collected fish were pooled over 1–6-month periods depending on the year. Because age assignments were based on a measured subsample, we accounted for this sampling error by first estimating age fractions from the subsample and then applying these fractions to the total catch of fish in the sample tank. First, the total number of age-1 and age-2 fish subsampled for a group (s_g,t) and the number of age-1 fish (n_1g,t) were used to estimate the fraction of age-1 fish (p_1g,t):

Given that day d,t fell within the period encompassed by group g,t the daily abundance of age-1 juvenile coho salmon was calculated using the following:J_1d,t = p_1g,t × J_d,t.(23)By subtraction, the abundance of age-2 juvenile coho salmon was calculated using the following:

Thus, we assume a constant age fraction over each subsample period. The total annual abundance for each age at outmigration (o) was then obtained by adding the J_o,d,t over the total number of sample and fish collection days in calendar year t (D_t):

The total escapement of coho salmon upstream from Swift Dam was variable but increased slightly over time with the lowest escapement recorded in 2015 (3,754 fish) and the highest escapement in 2020 (9,486 fish; fig. 5). The annual escapement of hatchery-origin fish averaged 5,417 (SD=1,486) with the highest number and fraction of hatchery fish escaping in 2014, whereas natural-origin escapement increased steadily over the time series (fig. 6). In 2013, just 28 naturally produced adult coho salmon were transported upstream from Swift Dam and by 2020, 4,863 natural-origin adults were transported. The fraction of hatchery-origin female spawners (pHOS) generally decreased over time, but pHOS was highly variable, averaging 0.75 (SD=0.182) over the time series. Values of pHOS ranged from a high of 0.99 in 2013 to a low of 0.48 in 2017 and 2020 (fig. 7). The fraction of early spawning coho salmon transported upstream from Swift Dam (Ep) was 1.0 during 2013 and 2014, sharply dropping to 0.08 in 2015, and then showing an increasing trend with Ep=0.8 in 2023 (fig. 8).

The number of natural-origin juvenile coho salmon collected at the Swift FSC (from hatchery and natural-origin spawners) showed an increasing trend over time (fig. 9). Over the annual time series, the average number of juvenile coho salmon collected at the Swift FSC was 50,896 fish (SD±30,771) and ranged from 9,201 fish (SD±46) in 2014 to 100,137 fish (SD±766) in 2019.

Calculation of the proportion of juveniles transported by age at outmigration showed that over the entire time series, age-1 juveniles always represented a majority (greater than 0.55) of the fish collected at the Swift FSC each year (fig. 10). However, in some years either age-0 or age-2 fish represented a sizable proportion of the total number of fish transported. For instance, in 2016, age-2 fish represented 30 percent of all fish collected. Similarly, age-0 fish represented 36 percent of all collected juveniles in 2017 and 30 percent in 2021. Thus, even though age-1 fish were predominant in each year, in some years either very young or older fish represented a sizeable proportion of the juveniles collected.

Fundamentally, the number of juveniles collected at the Swift FSC is determined by the number of spawning females upstream from Swift Dam and the FCE of juveniles at the Swift FSC. Juvenile coho salmon FCE at Swift Dam averaged 0.375 (SD=0.196) and ranged from a low of 0.06 in 2013 to a high of 0.64 in 2019. The time series of FCE suggests an increasing trend in FCE over time (fig. 11). More recent years (2019–23) had higher FCE (mean=0.536, SD=0.129) than years before 2019 (mean=0.242, SD=0.127; fig. 11). These changes in FCE correspond to improvements made to the Swift FSC since it was commissioned in late 2012 (PacifiCorp, 2024). Because the number of spawning adults and juveniles has increased, the number of female spawners and the effect of FCE must be accounted for when estimating lifecycle productivity.

One factor that could also affect the productivity and survival of juvenile coho salmon upstream from Swift Dam is the minimum summer flows during outmigration. Lower summer flows could result in warmer temperatures and physiological stress, higher predator activity, reduced habitat area and thus, lower fish production at Swift Dam. On average, the minimum mean monthly summer flow for our index site on the Muddy River was 149 cubic feet per second (ft³/s; SD=29.4) and ranged from 111 ft³/s in 2015 to 209 ft³/s in 2017 (fig. 12).

Posterior distributions represent the uncertainty about the model parameters (table 1; figs. 13 and 14). Annual FCE was found to have a strong positive effect on productivity with a median of 1.962 (95-percent credible interval [CI]=0.247–4.079). The fraction of the posterior distribution less than zero was 0.008, indicating that FCE at the Swift FSC had a positive effect on the number of recruits per spawner that contribute to the population. The median of the posterior for mean productivity across years when FCE=1 was α_FCE=1=64 naturally produced juvenile recruits per female spawner (95-percent CI=10.37–245.79; table 1.1; fig. 14). Assuming a mean fecundity of 2,600 eggs (Beacham, 1982) per female, this equates to a mean egg-to-juvenile outmigrant survival of about 2.4 percent at low spawner density.

Evaluations of individual effects on productivity over and above the effects of FCE indicated that minimum monthly summer flow had a negative effect on annual productivity (tables 1, 1.1, and 1.2; fig. 13). About 95 percent of the posterior distribution for θ₅were less than 0, indicating statistical support for an effect of minimum summer flow on productivity. Also, the fraction of hatchery spawners tended to have a negative effect on productivity, with 80 percent of the posterior distribution having θ₂ less than 0. In contrast, a larger fraction of early spawning coho salmon was associated with generally higher juvenile productivity, with 17.5 percent of the posterior distribution having θ₃ less than 0. The effect of maximum monthly winter flows on juvenile productivity was not statistically supported with a posterior that encompassed zero, with 34 percent of the posterior less than zero.

Using population level estimates of capacity from Korman and Tompkins (2014) to inform the prior distribution for capacity (K) of Lewis River coho salmon resulted in a well-defined posterior distribution with a median capacity of 297,901 (95-percent CI=84,584–812,488) maximum juveniles produced (table 1; figs. 14 and 15). Uncertainty about the estimate of capacity is demonstrated by how the choice in prior distribution can influence the estimate of capacity. For example, when a weakly informed prior using a truncated normal distribution where K ~ normal(μ=10,000, SD=1,000,000), median capacity was 645,779 (95-percent CI=58,373–1,903,420) juveniles, and the posterior distribution was much more protracted compared to the posterior distribution informed by Korman and Tompkins (2014; fig. 14), K ~ lognormal(μ=7.34, SD=0.67). Nonetheless, the posterior distributions were similar to their respective prior distributions, supporting the conclusion that the data provided little information towards the estimation of capacity in the lifecycle model.

The curves from the Beverton-Holt stock-recruitment model illustrate how recruitment of naturally produced juveniles varies as a function of the number of female spawners (fig. 16). The data used to inform the stock-recruitment model encompassed a relatively narrow range in the number of spawners and juveniles, from a low of 1,074 to about 4,993 female spawners. So, our estimated stock-recruitment model parameters provide inference over a narrow range of observed population levels as supported by the uncertainty in our estimates of capacity (fig. 15).

For the juvenile-to-adult transition, the median of the posterior for the mean SAR across all years was 0.045 (95-percent CI=0.001–0.13; table 1). Thus, in the absence of accounting for harvest, a median of 4.5 percent of juvenile outmigrants returned to the Lewis River. Owing to the hierarchical structure of the state-space model, annual variation in productivity and SAR can be estimated as random effects drawn from the lognormal process error (σ_P,J and σ_P,A; table 1.1; figs. 16 and 17). SAR and productivity vary considerably over time and generally increase after brood year 2015. The higher productivity since brood year 2015 coincides with increasing FCE since 2015 (fig. 11). The number of juveniles collected at Swift Dam given the number of female spawners showed the increase in productivity of natural juveniles upstream from Swift Dam because of higher juvenile fish collection performance at the Swift FSC (figs. 16 and 18).