PLEASE NOTE THIS IS A PREPRINT OF THE FINAL VERSION PUBLISH IN JOURNAL OF ECOLOGICAL MODELLING https://www.sciencedirect.com/science/article/pii/S0304380018301297 In particular, the published version contain a more thorough discussion of the SIR method.
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Characterizing the strength of density dependence in at-risk species through Bayesian model
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averaging
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Guillaume Bal1,2, Mark D. Scheuerell3 and Eric J. Ward1
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Service, National Oceanic and Atmospheric Association, Seattle, WA 98112, USA 2
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Conservation Biology Division, Northwest Fisheries Science Center, National Marine Fisheries
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Marine Institute, Rinville, Oranmore, Co. Galway, Ireland
Fish Ecology Division, Northwest Fisheries Science Center, National Marine Fisheries Service, National Oceanic and Atmospheric Association, Seattle, WA 98112, USA
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Corresponding author: G. Bal, Marine Institute, Rinville, Oranmore, Co. Galway, Ireland. (E-
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mail:
[email protected], Phone: +353 83 191 9748, Fax: +353 91 387 201)
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Developing effective conservation plans for at-risk species requires an understanding of the
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relationship between numbers of breeding adults and their subsequent offspring. In particular,
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establishing the degree to which density-dependence affects population size can be difficult due
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to errors in the data themselves, uncertainty in model parameters, and possible misspecification
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of model structure. Here we use a Bayesian framework to obtain posterior probability
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distributions for model parameters and each of four simple adult-offspring production models
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that allow for varying types of density dependence. Specifically, we analyzed 42 at-risk
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populations of anadromous Chinook salmon (Oncorhynchus tshawytscha) from the northwestern
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United States and found strong evidence that more than two-thirds of the populations exhibit
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negative density-dependent effects of adults (i.e., compensatory mortality). This was somewhat
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remarkable given large reductions in adult numbers relative to historical benchmarks, indicating
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that carrying capacity of spawning habitat has been reduced considerably. Furthermore, we also
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found non-zero probabilities of positive density-dependent effects of adults (i.e., Allee effects) in
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about one-third of the populations, which could suggest that cumulative losses of spawning
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adults over the past century has led to decreased nutrient and energy subsidies from semelparous
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carcasses, and diminished bio-physical disturbance from nest-digging activity. Importantly, our
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analysis highlights the utility of Bayesian model averaging in a conservation context wherein
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errors in choosing the best model may have more severe consequences than errors in estimating
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model parameters themselves.
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Keywords: density dependence, Bayesian model averaging, Allee effect, compensation,
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depensation, salmon.
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ABSTRACT
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INTRODUCTION One of the cornerstones of conservation biology is establishing the relationship between
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the numbers of parents and the offspring they produce. In particular, the degree to which
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organisms are affected by population density has important implications for individual fitness
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and population growth. Negative density dependence (NDD), or compensatory mortality, occurs
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when density is relatively high and any further increases in density lead to increased transmission
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of diseases or competition for resources (e.g., food, breeding locations), ultimately causing
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reductions in per capita growth rates (Hixon et al. 2002, Brook and Bradshaw 2006). Conversely,
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positive density dependence (PDD), or depensatory mortality, arises when density is relatively
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low and the addition of more individuals causes increased per capita growth rates as a result of
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cooperative foraging or defensive behaviors, increased probability of finding a mate, or
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combinations of these factors (i.e. Allee effects, Courchamp et al. 1999, Berec et al. 2007,
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Kramer et al. 2009). Understanding whether these phenomena occur and if so, to what extent, is
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necessary for determining the best options for conserving at-risk species. For example, the
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presence of NDD could indicate limited habitat availability (i.e., insufficient total area) whereas
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the existence of PDD might suggest a high degree of habitat fragmentation; rectifying those two
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types of habitat deficiencies could require rather different approaches.
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Density-dependence has been studied extensively in fish populations because of its
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importance to both the management of healthy stocks and conservation of imperiled populations
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(Liermann and Hilborn 1997, Barrowman and Myers 2000, Barrowman et al. 2003). In classical
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fisheries management, the presence of compensatory mortality within a stock implies parental
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biomass should be harvested to the point where the surplus production of new recruits to the
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fishery is maximized relative to replacement (Hilborn and Walters 1992). Conversely, the degree
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of depensatory mortality will determine the rate at which overfished stocks will recover when
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harvest is reduced. Most conservation practitioners concentrate on the possible existence of
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positive density dependence. However, negative density dependence at relatively low densities
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can exist, implying diminished carrying capacity from factors like habitat loss/modification or
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the presence of non-native species (Achord et al. 2003), although this is often ignored in
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conservation contexts.
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Population dynamics models offer a formal means for estimating both positive and
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negative density dependence (Boyce 1992). For example, Beverton (1957) and Ricker (1954)
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models have been used to estimate the relationships between parents and offspring for decades.
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More recently, Barrowman (2000) introduced a form of piecewise regression model (known as
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the “hockey stick” model) similar to the Ricker and Beverton-Holt curves. The hockey stick
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model offers potential advantages over other models in a conservation context because it
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provides more conservative estimates of the maximum density-independent survival (i.e., slope
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at the origin) and carrying capacity (Barrowman and Myers 2000). In addition, the breakpoints in
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the hockey stick segments may provide natural reference points for management decisions.
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However, the hockey stick model does not allow for PDD (depensation).
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Regardless of model choice, three main types of uncertainties can hinder our ability to
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infer the underlying relationship between parents and their offspring. First, observation errors
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arise in the form of sampling and measurement errors. Second, model parameters are rarely
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known without error and instead must be estimated from the data. Third, uncertainty about the
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structure of the model itself affects inference about the form of the parent-offspring relationship.
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The first two concerns are often addressed through appropriate sampling designs and explicit
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consideration of both process and observation/sampling errors. However, possible
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misspecification of a particular model is typically ignored and instead the “best” model is chosen
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based on some model selection measure such as Akaike’s Information Criterion (Burnham and
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Anderson 2002). In such cases, two models with nearly identical support from the data could
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produce widely divergent predictions, especially when confronted with new data (Pascual et al.
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1997). As a guard against this very real possibility, model averaging (MA) offers a formal means
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for explicitly addressing model-selection uncertainty in problems of inference and prediction
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(Burnham and Anderson 2002, Wintle et al. 2003).
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Pacific salmon (Oncorhynchus spp.) are important to human economies and the ecology
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of coastal ecosystems across the northern Pacific Rim (Schindler et al. 2003). Fisheries for
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salmon are worth hundreds of millions of US dollars annually (NMFS 2013). Salmon also act as
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ecosystem engineers by modifying benthic habitats through nest-digging activity (Moore 2006)
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and serve as keystone species in food webs as food for fish, mammals, birds, and insects
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(Helfield and Naiman 2006). In many coastal watersheds across western Canada and the US,
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however, Pacific salmon populations have been reduced to mere fractions of their historical
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abundances due to changes in habitat, hydropower development, overharvest, and climate
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(Ruckelshaus et al. 2006). Thus, obtaining a proper understanding of whether NDD, PDD, or
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both occur among at-risk salmon populations is necessary for determining the best recovery
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options.
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Here we examined the strength of NDD and PDD among 42 populations of Chinook
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salmon (O. tshawytscha) from the northwestern USA that are currently listed as “threatened”
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under the US Endangered Species Act. To do so, we used four different piecewise-linear models
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to characterize various forms of density dependence (Fig. 1): 1) unrestricted (density-
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independent), 2) Allee effect (PDD) only, 3) carrying capacity (NDD) only, and 4) both PDD
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and NDD. Specifically, we added an additional breakpoint to Barrowman and Myers’ (2000)
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hockey stock model, which can be very useful in conservation contexts because 1) it tends to
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provide better estimates of per capita productivity at low population sizes; and 2) it offers more
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conservative estimates of mortality rates that lead to extinction. In addition, we addressed both
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parameter and model uncertainty within a unified framework through Bayesian model averaging
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(BMA), which allowed us to easily combine predictions and uncertainties across all four models.
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3.1
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MATERIAL AND METHODS Allee Hockey Stick model Barrowman and Myers’ (2000) hockey stick (HS) model assumes the number of
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offspring increases linearly up to an asymptote, beyond which it is independent of parental
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abundance. For a given population, the model is described by two linear functions,
𝑂𝑡 = {
𝛽𝑃𝑡 𝑖𝑓 𝑃𝑡 < 𝑃𝑐 } 𝛽𝑃𝑐 𝑖𝑓 𝑃𝑡 > 𝑃𝑐
(1)
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where is the density-independent per-capita reproductive rate, 𝑃𝑡 is the number of parents
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breeding in year t, and 𝑃𝑐 is the threshold parental abundance above which compensatory
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density-dependent effects dominate and restrain the number of offspring. As 𝑃𝑐 approaches
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infinity, the degree of density-dependence goes to zero and there is no limit to population size.
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To allow for Allee effects at relatively low adult abundance, we modified the original HS model (Barrowman and Myers 2000) to allow for a third linear segment. This new “Allee
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Hockey Stick” (AHS) model allows growth rates to become depressed when adult abundance is
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less than some threshold, 𝑃𝑑 𝛽1 𝑃𝑡 𝑂𝑡 = {𝛽1 𝑃𝑑 +𝛽2 (𝑃𝑡 − 𝑃𝑑 ) 𝛽1 𝑃𝑑 +𝛽2 (𝑃𝑐 − 𝑃𝑑 )
𝑖𝑓 𝑃𝑡 < 𝑃𝑑 𝑖𝑓 𝑃𝑑 < 𝑃𝑡 < 𝑃𝑐 𝑖𝑓 𝑃𝑡 ≥ 𝑃𝑐
(2)
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where 0 < 𝛽1 < 1 < 𝛽2, and 𝑃𝑑 and 𝑃𝑐 are the breakpoint thresholds respectively. By fixing
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parameters at zero or infinity, the AHS model is capable of producing a range of dynamics,
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including three less complex models nested within Equation 2 (Fig. 1): unrestricted growth (i.e.,
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𝛽1 = 0, 𝑃𝑑 and 𝑃𝑐 = ∞), original HS with carrying capacity only (i.e., 𝛽1 = 0, 𝑃𝑑 = 0), and
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Allee effects only (i.e., 𝑃𝑐 = ∞).
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3.2
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Parameter estimation and model weighting Though Bayesian estimation could be done in a 2 step procedure, we illustrate here how
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to conduct both estimation and model selection in a single framework, using the Sampling –
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Importance – Resampling (SIR) algorithm (Rubin 1988). This approach aims to approximate the
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posterior distribution by using an importance function to resample parameter draws from a
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proposal distribution; we adopted the same version of the algorithm as described and
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implemented by Moore (2008). We assigned equal probabilities a priori to each of the 4 models,
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and calculated the posterior marginal likelihood as the mean likelihood across all parameter
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vectors from the SIR algorithm (Carlin and Louis 2000, Gelman et al. 2004). The SIR algorithm
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can be run independently for each of the proposed models, and the marginal likelihoods can then
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be normalized so that ∑𝑖 𝑃(𝑚𝑖 |𝑥) = 1. These normalized marginal likelihoods are interpreted as
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conditional probabilities of models on observed data. Compared to other Bayesian estimation
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approaches, such as MCMC, the SIR algorithm offers several advantages. The main is that
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estimating the mean marginal likelihood mean is more straightforward than other MCMC
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approaches, such as WinBUGS (Lunn et al. 2000) or JAGS software (Plummer 2003). The SIR
148
algorithm also does not require a burn-in period or assessment of MCMC diagnostics for
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convergence. All priors used in this study were weakly informative uniform priors and shared
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among models whenever possible (Table I). Lower and upper boundaries were based on
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Barrowman’s (2000) work on the congener O. kisutch.
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3.3
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Model Application For management purposes, Pacific salmon species are grouped into evolutionarily
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significant units (ESUs), defined as a group of salmon that (1) is reproductively isolated from
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other conspecific populations, and (2) represents an important component in the evolutionary
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legacy of the species (Waples 1991). We estimated the strength of density dependence among 42
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Chinook salmon populations within 3 distinct ESUs from Washington, Idaho, and Oregon in the
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northwestern USA (Fig. 2). Chinook salmon within the Puget Sound ESU are “ocean-type”,
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which means juveniles spend less than 1 year in fresh water before migrating to the ocean, but
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salmon from the Snake River and Upper Columbia ESUs are “stream-type” and spend 1 full year
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in fresh water before migrating to sea (Taylor 1990). This offers us an opportunity to compare
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whether the type or strength of density dependence varies between the contrasting life histories.
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We used the total number of spawning adults in a given year as our estimate of parent
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abundance. Because salmon from the same cohort mature at different ages, our estimates of
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offspring are then the sum of subsequent adults that were born in a given year, but that vary in
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age and return over sequential years (e.g., the offspring of Snake River stream-type adults that
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spawned in 2000 return as 3-6 years later in 2003-2006). All data used here were compiled by
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the National Marine Fisheries Service and are available upon request. To ensure time series are
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long enough to potentially detect positive density dependence (Brook and Bradshaw 2006,
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Gregory et al. 2010), we restricted our analysis to only include populations with at least 15 years
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of data. This cutoff yielded 42 time series (Table 2).
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4
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RESULTS For both life history types, we found the most support from the data for a model with
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only NDD (i.e., the median posterior model probabilities were 0.72 and 0.56 for ocean- and
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stream-type Chinook salmon, respectively; Fig. 3). We found very little evidence for models
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containing Allee effects (PDD) only, but models that included both PDD and NDD had median
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probabilities of being the best across ESUs of 0.10 and 0.075 for ocean- and stream-type salmon,
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respectively.
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Among populations within a specific life history type, however, the relative support for
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the different models varied considerably (Fig. 4). Although there were a few instances where
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model probabilities were evenly split among the four model forms (e.g., 4, 15), most populations
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showed strong support in favor of one particular model. For ocean-type Chinook salmon, most
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populations showed strong indications of NDD, but a few populations had very little evidence
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for either form of density dependence (e.g., 7, 18, 20); we found similar results for stream-type
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Chinook. Notably, the evidence for models with both PDD and NDD was reasonably strong in a
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few populations (e.g., 25, 26).
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For each population, the model averaged relationship differed slightly from the best
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model. For example, the best models selected for the Duwamish, Skykomish, and Imnaha rivers
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(i.e., populations 4, 15, 25 in Table 2) were the AHS model, linear model, and HS model,
190
respectively. Applying BMA to estimate the shape of this curve may yield a result similar to the
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single “best” model, but can also result in more conservative results (Fig. 5). In the case of the
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Duwamish River, a simple model selection would select a linear relationship whereas the shape
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coming from BMA exhibits Allee effects when the population size falls under 865 fish. In the
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Imnaha River, the estimated carrying capacity from the best model is about 50% higher than the
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model-averaged estimate. For the Skokomish River, however, there is very little difference
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between the best and MA models.
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5
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DISCUSSION Most management decisions for harvested, threatened and/or endangered populations are
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based on estimates from a single population model, but the estimated relationship between
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population size and per capita growth is driven by the choice of models used in estimation
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(Wintle et al. 2003). Although prediction intervals can be constructed based on parameter
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uncertainty, ignoring model uncertainty creates overconfidence in the model prediction and may
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obfuscate decision-making (Hoeting et al. 1999, Wintle et al. 2003, Ellison 2004). In examining
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support for processes such as positive or negative density dependence, most previous approaches
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have simply identified whether or not data provide evidence for density dependence being
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present. Model averaging improves on this by allowing a range of forms of density dependence
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to be evaluated simultaneously.
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Our results reveal two important messages regarding the role of density-dependence in
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the population dynamics of Chinook salmon in the Pacific Northwest. First, although we found
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that linear relationships between offspring and parents with no density dependence were
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probable, especially within the Snake River ESU (median model posterior probability = 0.27),
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we found much stronger support for models with NDD (median model posterior probability =
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0.56) even though parental abundances across all populations have been greatly reduced from
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historical numbers and hence we expected less evidence for NDD. This suggests that future
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restoration efforts ought to focus on increasing overall carrying capacity. Unfortunately we lack
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much of the age-specific data for many of these populations to assess where in the life-cycle
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capacity is most limiting.
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Second, the existence of PDD among these Chinook salmon populations is a possibility,
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which agrees with previous studies of other salmon species (Liermann and Hilborn 1997,
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Barrowman et al. 2003). Although we cannot identify the exact cause of PDD, some reasonable
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hypotheses exist. First, salmon gain about 90% of their adult biomass in the ocean, and therefore
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act as net importers of marine-derived nutrients and energy to freshwater ecosystems (Schindler
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et al. 2003). Low adult abundance could therefore reduce survival during earlier juvenile life
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stages. Second, nest-digging activity by female salmon mobilizes fine sediment (Moore et al.
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2004) and decreases the probability of stream-bed scour and excavation of buried salmon eggs or
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embryos (Montgomery et al. 1996). Thus, losses of adult salmon may reach a point whereby the
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ecosystem cannot support the juvenile production it once did (Achord et al. 2003, Scheuerell et
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al. 2005).
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For salmon in particular, absence of observed Allee effects could be linked to high
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straying rates and colonization capacities (Makhrov et al. 2005, Correa and Gross 2008, Launey
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et al. 2010, Westley et al. 2013). Alternatively, it could indicate that availability of juvenile
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habitat is more important than direct mortality from predators (Mogensen and Post 2012). For
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Puget Sound populations, in particular, much of the lowland areas have been converted from
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forest to agriculture, which has greatly decreased the carrying capacity for juvenile Chinook
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salmon (Scheuerell et al. 2006). Furthermore, increasing evidence points to NDD among adult
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life stages during their ocean residency owing to competition with hatchery fish, (Connors et al.
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2012, Ruggerone et al. 2012), which shifts the focus away from habitat actions and more toward
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hatchery reforms (Buhle et al. 2009).
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The varying degrees of support for different forms of density dependence among the
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many populations provide some important insights for conservation. Decisions to list Pacific
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salmon as threatened or endangered under the Endangered Species Act are made collectively at
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the relatively large ESU level, but recovery plans are developed and implemented at much
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smaller scales relating to specific populations. Here we have used posterior model probabilities
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to assess the degree to which PDD, NDD, or a combination thereof exists for each population
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within an ESU. Managers could potentially use this information to further direct location-
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specific actions that might decrease the intensity of PDD or NDD.
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Our analysis highlights the utility of model averaging for estimating relationships among
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parents and their offspring. Model averaged results were more conservative than the best model
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alone, and therefore applying model averaging may be particularly important for populations of
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conservation concern. The use of the HS and AHS models was largely chosen based on the taxa
251
in our analysis; other model sets may be developed for other species (and unlike our analysis,
252
there is no requirement that all candidate models be nested). While there are a number of ways to
253
implement Bayesian model averaging, the SIR approach is best for relatively simple models like
254
those we used here. Increased model complexity will decrease the speed of the SIR approach
255
exponentially, however. Using other estimation routines such as Markov Chain Monte Carlo
256
(MCMC), the model-averaged procedure could also be constructed hierarchically across
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populations, to estimate common strengths of density dependence across populations.
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6
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This research was performed while Dr. Guillaume Bal was a post-doctoral research associate
260
sponsored by the National Research Council. We thank Brian Burke and Kevin See for helpful
261
comments on an earlier draft of the manuscript.
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ACKNOWLEDGMENTS
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7
TABLES
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Table I. Uniform distributions used as Bayesian priors for the parameters in the four competing
265
models described by equation 2 in the methods. Pmax and Omax are the maximum numbers of
266
observed parents and offspring, respectively, for a given population.
Parameter
Linear
β1 β2
Allee Unif(0, 1)
Unif(1, 25)
Pd
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Allee Hockey Stick Unif(0, 1)
Unif(1, 25)
Unif(1, 100)
Unif(1, 100)
Unif(0, Pmax / 3)
Unif(0, Pmax / 3)
Unif(0, Pmax / 3)
Pc CV
Hockey Stick
Unif(Omax / 5, Omax) Unif(Omax / 5, Omax) Unif(0, 3)
Unif(0, 3)
Unif(0, 3)
Unif(0, 3)
Table 2. Summary of parent and offspring data for Chinook salmon used in the study. PS, S, and UC refer, respectively, to Puget Sound, Snake, and Upper Columbia Evolutionary Significant Units (ESU).
Index
ESU
River name
Time period
Parent range
Offspring range
1
PS
Cascade R.
1981-2005
83-625
92-2458
2
PS
Cedar Creek
1965-2005
126-1896
8-9099
3
PS
Dungeness R.
1986-2005
43-955
47-1486
4
PS
Duwamish R.
1968-2005
2014-11551
657-52907
5
PS
Elwha R.
1986-2005
164-5309
418-5485
6
PS
Lower Sauk R.
1952-2005
112-3896
321-22021
7
PS
Lower Skagit R.
1952-2005
409-9263
577-62260
8
PS
North Fork Nooksack R.
1984-2005
10-7473
13-5337
9
PS
North Fork Stillaguamish R.
1974-2005
330-1849
353-89489
10
PS
Puyallup R.
1968-2005
527-5387
74-23354
11
PS
Sammamish R.
1983-2005
34-550
20-1708
12
PS
South Fork Nooksack R.
1984-2005
103-625
100-1364
13
PS
South Fork Stillaguamish R.
1974-2005
73-391
15-14265
14
PS
Skokomish R.
1968-2005
189-3184
39-16734
15
PS
Skykomish R.
1965-2005
1681-7703
110-110621
16
PS
Snoqualmie R.
1965-2005
324-3603
149-24714
17
PS
Suiattle R.
1952-2005
167-1804
87-10407
18
PS
Upper Sauk R.
1952-2005
108-3345
57-27343
19
PS
Upper Skagit R.
1952-2005
3586-20040
6146-114870
20
PS
White R.
1965-2005
7-2131
9-7779
21
S
Big Creek
1957-2002
5-1858
6-3349
22
S
Camas Creek
1964-1998
3-554
1-1611
23
S
Catherine Creek
1955-2002
28-3161
4-6515
24
S
Grand Ronde R. (Upper Mainstream)
1956-2002
3-1028
2-1991
25
S
Imnaha R.
1954-2002
258-6267
197-5361
26
S
Lemhi River
1957-2002
10-3357
13-7633
27
S
Lostine River
1959-2002
37-1585
35-3372
28
S
Marsh Creek
1957-2002
13-1845
4-4461
29
S
Minam River
1954-2002
54-4104
26-4068
30
S
Pahsimeroi River
1989-2002
27-306
64-1187
31
S
Secesh River
1957-2002
48-1395
21-1581
32
S
Snake River (Lower Mainstem)
1957-2002
11-4888
24-2444
33
S
Snake River (Upper Mainstem)
1962-2002
18-3554
41-5267
34
S
South Fork Salmon River (East Fork)
1958-2002
23-1257
17-1444
35
S
South Fork Salmon River (Mainstem)
1958-1999
224-5029
138-6763
36
S
Tucannon River
1979-2002
11-897
11-16050
37
S
Wenaha River
1964-2002
48-2682
46-5838
38
S
Salmon River Lower Mainstem
1963-2000
11-432
52-636
39
S
Salmon River Upper Mainstem
1964-2000
18-2047
137-2545
40
UC
Entiat River
1962-1998
18-714
35-1111
41
UC
Methow River
1962-1998
43-2813
141-3894
42
UC
Wenatchee River
1960-2000
58-3523
166-6126
8
FIGURES
Figure 1: Shape of the four alternative population production models described in the Methods. Thin grey line is the 1:1 replacement line.
Figure 2. Map of the study area. Grey shading indicates ocean-type populations within the Puget Sound ESU; black shading represents stream-type populations from the Upper Columbia and Snake River ESUs.
Figure 3. Boxplots of models probabilities for the two different life-history types of Chinook salmon. Whiskers and hinges indicate the 95% and 50% credible intervals, respectively, around the median.
Figure 4 Stacked barplots of the posterior probability for each of the four competing models: linear (L), linear with Allee effect (A), hockey stick showing compensation (HS), hockey stick with depensation plus Allee Effect (AHS)
Figure 5. Plot of the stock recruitment data (in thousands) together with the posterior median of predicted values coming from model the Bayesian model model averaing and the most credible model only.
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LITERATURE CITED
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