Use of a mechanistic growth model in evaluating post-restoration habitat quality for juvenile salmonids

Individual growth data are useful in assessing relative habitat quality, but this approach is less common when evaluating the efficacy of habitat restoration. Furthermore, available models describing growth are infrequently combined with computational approaches capable of handling large data sets. We apply a mechanistic model to evaluate whether selection of restored habitat can affect individual growth. We used mark-recapture to collect size and growth data on sub-yearling Chinook salmon and steelhead in restored and unrestored habitat in five sampling years (2009, 2010, 2012, 2013, 2016). Modeling strategies differed for the two species: For Chinook, we compared growth patterns of individuals recaptured in restored habitat over 15-60 d with those not recaptured regardless of initial habitat at marking. For steelhead, we had enough recaptured fish in each habitat type to use the model to directly compare habitats. The model generated spatially explicit growth parameters describing size of fish over the growing season in restored vs. unrestored habitat. Model parameters showed benefits of restoration for both species, but that varied by year and time of season, consistent with known patterns of habitat partitioning among them. The model was also supported by direct measurement of growth rates in steelhead and by known patterns of spatio-temporal partitioning of habitat between these two species. Model parameters described not only the rate of growth, but the timing of size increases, and is spatially explicit, accounting for habitat differences, making it widely applicable across taxa. The model usually supported data on density differences among habitat types in Chinook, but only in a couple of cases in steelhead. Modeling growth can thus prevent overconfidence in distributional data, which are commonly used as the metric of restoration success.

Introduction 1 model fitting has become a vital tool in conservation biology, but the models again rely 9 heavily on population numbers or densities and correlative relationships between density 10 and environmental variables [8]. 11 In many cases, however, growth can provide a more robust indication of habitat 12 quality than numbers or densities [9][10][11], especially given that growth is often correlated 13 with survival and reproduction [12,13]. This can be important in restoration ecology 14 when differences in habitat quality drive heterogeneity in growth and development 15 across individuals, leading to life-history variation and trait evolution [14]. Growth rates 16 and other life-history traits are nevertheless used only rarely to assess habitat 17 restoration efforts [15,16]. There is no shortage of models that quantify effects of habitat 18 quality on growth, ranging from calculation of suitability indices [17] to individual-based 19 behavioral models [18]. Growth models are also useful for providing descriptions of 20 growth patterns when there are multiple ecological inputs [19][20][21][22]. In cases when growth 21 data have been collected, it should therefore be possible to combine models and data to 22 provide more accurate assessments of restoration efforts. Such an approach could 23 augment, and increase confidence in, distribution and abundance studies. 24 In contrast to demographic studies, what is often missing in growth studies is the 25 use of robust statistical methods of fitting the models to data. In growth studies, 26 comparison of growth rates between habitats is usually based only on mean growth 27 rates [23]. Moreover, when mechanistic models [21,22] are fit to the data, they are 28 usually still sufficiently simple that they can be transformed to the point at which linear 29 fitting methods can be used [24]. This restriction on model complexity likely reduces 30 our ability to make inferences about growth. Modern methods of non-linear fitting 31 therefore provide an opportunity to quantify the effects of habitat restoration on growth 32 in a statistically robust fashion. With the proliferation of habitat restoration efforts in 33 species conservation, the the use of mechanistic models to describe growth data also has 34 the potential to expand the theoretical basis of restoration ecology [25]. 35 Growth and development are key features of salmonid life-cycle models, which are 36 often able to accurately predict population trajectories [4]. Studies of restoration 37 effectiveness, however, only occasionally quantify salmonid growth or survival [3,26,27], 38 requiring that related parameters in life cycle models be estimated from correlative data 39 in earlier research. The extent to which these life history traits are changed by 40 restoration is nevertheless of high importance in the use of life cycle models to predict 41 and confirm population-level responses [8]. 42 Here we present the results of a multi-year study of growth in sub-yearling 43 anadromous salmonids. We use our data to illustrate the usefulness of mechanistic 44 growth models in restoration ecology. We collected size data, as well as growth over 45 time, via fish marked and recaptured through the entire rearing season in each of five 46 years to determine whether growth of young-of-the-year Chinook and steelhead is 47 improved in restored habitat. Because these fish grow more or less continuously during 48 the growing season, we were able to fit mechanistic growth models to both types of data. 49 We used Bayesian model fitting techniques to estimate the parameters of our growth 50 functions, and we compared growth rates between individuals with different habitat 51 selection behaviors. Salmonid growth rates are rarely constant; therefore, we allowed for 52 the possibility that growth-rate parameters would change over time [28,29]. Steelhead 53 recaptures were sufficient in both habitat types, so we were able to use raw growth data 54 to determine the extent to which the model accurately reproduces growth rates in the 55 field. We then compared our estimates of growth rates with previous observations of 56 relative density [30] to determine the frequency with which increased abundance in 57 restored habitat is associated with increased growth rates. The model demonstrates 58 relatively consistently improved growth associated with restoration in Chinook, but that 59 this was the case in only two study years for steelhead. Nevertheless, key model 60 parameters describing growth rate were supported by steelhead raw growth data. Thus, 61 our model has a strong application to evaluation of salmon conservation efforts. 62 Materials and methods 63 Study System 64 Restoration of fish habitats in rivers involves a large set of approaches and can 65 sometimes target more than one species [27,31]. A common stream restoration 66 technique is the installation of engineered log jams (ELJs) in streams, which create 67 pools that serve as primary habitat for target species. In the Pacific Northwest, USA, 68 use of ELJs is a common conservation strategy for threatened and endangered 69 salmonids [2,31,32]. In the interior Pacific Northwest, ELJs are targeted toward 70 sub-yearling anadromous Chinook salmon (Oncorhynchus tschawytscha), which grow 71 and develop in streams for one year, and steelhead trout (O. mykiss) which grow and 72 develop in streams for 1-3 years, before migrating to the marine environment. Seaward 73 migration occurs at the smolt stage and, in principle, ELJs should improve growth and 74 survival to this stage. However, distribution patterns of salmonids vary 75 spatio-temporally, and differ between species [33], exacerbating the difficulty of 76 assessing restoration efficacy with distribution and abundance studies [27,30,[34][35][36].

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The restoration project studied here was implemented in the Entiat River sub-basin 78 (49. were installed in multiple reaches of the river. We compared a single restored reach that 81 was 4.5-4.9 km from the confluence with the Columbia River, and an unrestored reach 82 that was 0.3 km upstream of the restored reach (Fig. 1). The restored reach was the 83 only one with ELJs at the beginning of the study in 2009. ELJ pools in the restored 84 reach ("restored pools"; N = 11) were compared with pools in the unrestored reach (N 85 = 11) that lacked major log and rock structures. Both types of pools occurred on the 86 stream margins where nearly all sub-yearling habitat use occurs 87 beechie2005classification. Unrestored pools were 20-40% smaller and shallower than 88 restored pools, but [30,37] showed that the unrestored pools had a similar rate of 89 increase in fish density with increasing area to the restored pools. We therefore 90 concluded that the unrestored pools served as reasonable controls for the restored pools. 91  Fish were captured using a 3 m × 1.5 m × 3 mm mesh seine. The substrate and flow 100 conditions made it impossible to pull the seine through the water, and so two field crew 101 members instead stood at the downstream end of the pool and held the seine open as 102 two other crew members, snorkeling in the water, used large hand nets to capture fish 103 individually or to coerce fish into the seine. Visibility in the Entiat River is 4-5 m, so 104 the two snorkelers could see the entire sampled area, ensuring that all visible fish were 105 captured. Captured fish were transferred to insulated, aerated buckets for enumeration, 106 marking and recording of size data (standard length, SL, in mm and mass in g).

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Mild anaesthetization using MS-222 (<0.1 g·l -1 ) exposure for 2-3 minutes made it 108 possible to measure, weigh and apply an identifying mark to each fish. We avoided PIT 109 tagging because it would have caused too much handling stress [39] to allow for robust 110 mark and recapture, especially given the small size (< 50 mm SL) of some fish at the 111 start of field sampling each year. We therefore marked fish with a subcutaneous per year from handling/marking procedures. Following handling, fish were held in an 116 aerated bucket for at least 10 min, or until they displayed a full righting response and 117 normal activity, and were then released into the pool in which they were captured.

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After 24 hrs field crews returned to the same pools as part of a separate short-term 119 behavioral study (Polivka in review) and captured all fish using the same methods.

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Recaptured fish were released to the pool and newly captured fish on that day were 121 given marks as described above. The pools were sampled again 10-14 days later, with 122 size data recorded on recaptured fish and marks applied to newly captured fish that had 123 immigrated into the pool during the sampling interval.

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To analyze our data, we constructed mechanistic growth models that described fish 126 growth over each growing season, and we fit these models to data on fish size. Size data 127 included observations both of individuals marked and recaptured, and of individuals 128 that were marked but never recaptured. We then compared best-fit growth models in 129 restored and unrestored habitats. In these species, individual growth data are typically 130 collected via mark-recapture studies, but the high mobility of Chinook [40] often limits 131 their recapture rates, especially in unrestored habitats [38], and fitting growth models 132 allowed us to nevertheless use these data. To do this, we combined the repeated size 133 measurements on the recaptured individuals with individuals captured only once, 134 regardless of the habitat of initial capture. The repeated measurements of recaptured 135 fish ("recaptures") required use of a random effect in the model specification (see 136 below), whereas the individuals captured only once ("others") only required fitting the 137 model to the sizes observed during the course of the season. This analysis for Chinook 138 is more conservative, because at least some of the "others" occupied restored habitat for 139 some portion of the interval of time between sampling events. If there is a benefit to 140 restoration some portion of the "others" will therefore be more comparable to We constructed our growth model to describe the growth of young-of-the-year Chinook 147 salmon and steelhead in streams during the growing season. Size-over-time models in 148 fisheries, e.g., [19] usually describe growth over the lifetime of the organism. One 149 important consideration was that inspection of our data showed that there were 150 inflection points; therefore, we used a generalized version of a logistic growth 151 model [20][21][22]. This model allows for accelerating growth in the early stages of the 152 growing season, followed by an inflection point, after which growth decelerates, leading 153 to a maximum size at the end of the first season. The use of a size asymptote is 154 consistent with patterns evident in our data, and in size data reported from several 155 years of downstream migration of smolts in the spring [41]. Our model is: Here, l t is the length at time t, Y is the lower bound on size, effectively a minimum, 157 and a is the rate of increase in the size curve. L ∞ is the maximum size that a fish can 158 reach during the growing season, so that Y ≤ l t ≤ L ∞ for all time t. The inflection 159 point of the curve occurs att =α a . In these cases, fish are close to the lower bound on the curve Y when time is less than zero, which is to say, before sampling begins. In all panels Y = Y ∞ = 35.
To avoid confusion with other models such as the Von Bertalanffy model, we 167 re-parameterized the model according to Y ∞ = L ∞ − Y , so that Y ∞ is the total 168 amount by which a fish increases in size during the growing season. The model is then: 169 When non-linear models are fit to data, a standard approach is to transform the model into a linear form to accommodate linear least-squares fitting routines and their assumptions, such as normally distributed residuals. For our data, however, it turned out that the residuals are normal even without transformation of the model, and the variance in the residuals for the untransformed model was roughly constant. We therefore did not transform the model. Instead we simply added residual variation to the non-linear model, according to: Here i indicates individual fish i. We then assume that the residual it for fish i at time 170 t follows a normal distribution with mean zero, and variance σ 2 residual , which we 171 estimated from the data. We thus have i,t ∼ N (0, σ 2 residual ), so that σ 2 residual is the 172 residual variation in length. Thus, Y and Y ∞ describe growth bounds averaged over 173 individuals in the population. 174 We then extended the model to allow for fixed treatment effects, and random year 175 and individual effects. First, to include habitat treatment effects, restored vs. 176 unrestored, we allowed the four model parameters (Y , Y ∞ ,α, and a) to vary by habitat 177 type, as fixed effects. Second, to account for the statistical properties of our sampling design, we included random effects of year, to allow for annual variability in the 179 parameters. Third, in the case of individuals that were recaptured during a season, we 180 accounted for repeated measurements by including random effects of individual. Given 181 these considerations, the model becomes: Here i is again the individual, h is the habitat type, restored or unrestored, y is the 183 sampling year, and t is time within a growing season. The 's represent the different 184 types of random effects, as follows.

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The random effects of year on each of the four model parameters, 1...4 hy , vary between 186 the two habitats, such that, for example, 1 variation in initial size Y measured across habitats h and years y. For the random effect 188 of recaptured individuals, 5 individual h is the variance in 189 fish length among recaptured individuals from a given habitat type. This random effect 190 of individual affects only the overall length, rather than the rate of growth. Among 191 other things, this means that we can account for the possibility of observing fish that 192 start off larger than average being more likely to continue being larger than average 193 throughout the growth period pfister2002genesis. To be conservative, we began by 194 assuming that no Chinook in the unrestored pools were recaptured, and so the random 195 effects of individual Chinook are only relevant for recaptures in the restored habitats.

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Finally, the residual variation is allowed to differ between the two habitats, where . This is particularly important for the Chinook data set, because 198 some of the variation among the recaptured individuals in restored habitats can be 199 explained by the random individual effects. In the unrestored habitats in contrast, no 200 individuals were recaptured, and therefore more variation was left unexplained. Because 201 of this, it turned out to be the case that σ 2 residual unrestored > σ 2 residual restored .

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In general, we used vague priors for our parameters, but to aid HMC performance, 208 we centered the prior distributions of Y and Y ∞ on realistic values, based on previous 209 work with these two species. Also, we constrainedα and a to fall within realistic ranges, 210 namely 0-10 and 0-1, respectively, again based on previous work. Sensitivity analysis 211 showed that these priors did not strongly influence our posterior inferences, and that 212 the posterior was clearly dominated by the likelihood, rather than by the prior 213 distributions. For each model, we ran three HMC chains for 9000 iterations, using the 214 first 2000 iterations as a warm up. We then thinned by 7 iterations to produce a total 215 of 1000 samples per chain. We evaluated HMC chain convergence based on Gelman and 216 Rubin's potential scale reduction factor,R [45]. We visually inspected chain mixing 217 using traceplots, running mean plots, and marginal posterior density plots [45], and we 218 saw no evidence of pathological MCMC behavior. Although we thinned our chains to 219 avoid auto-correlation, our effective sample size was close to the total number of 220 iterations, and thinning was likely unnecessary [46].

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To compare parameter estimates between the restored and unrestored habitats, we 222 calculated differences between paired parameter values for the restored and unrestored 223 habitats for each sample in the model's joint posterior. For example, to calculate 224 differences in the shape parameter between restored habitats r and unrestored habitats 225 January 25, 2020 6/18 u, we calculatedα r −α u for each of the 3000 samples in the posterior (our 3 Markov 226 chains each produced 1000 samples). We then calculated the 95% credible interval (CI) 227 of that set of differences. This approach allowed for the possibility of correlations in 228 parameters across the samples in the posterior, and is therefore a more robust method 229 than calculating parameter differences by making independent draws of each parameter 230 from the posterior [47].

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Note that, in the case of Chinook, the u group consisted of all uncategorized fish, 232 meaning transient individuals that were not recaptured, regardless of habitat of original 233 capture. Accordingly, when we tested for interactions between habitat and year for 234 Chinook, we took into account the random year effects. For example, in testing for 235 differences in the shape parameter between years for Chinook, we calculated 236 (α r + 1 r,y ) − (α u + 1 u,y ) for each posterior draw, where 1 r,y and 1 u,y are the random 237 year effects in year y. 238 We considered the parameter estimates for the two habitats to be meaningfully 239 different if the 95% CIs of their differences did not overlap zero. For most parameters, 240 however, the fraction of differences that was above or below zero was reasonably 241 consistent across years, even if the 95% CI overlapped zero. Because this consistency 242 provides at least modest additional support for some of our arguments, we report the 243 fraction of differences that were above or below zero in each case.

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Our best-fit models suggested that fish almost always exceeded the minimum size Y 245 before sampling began. To estimate fish size at the beginning of sampling, we therefore 246 used the best-fit models to back calculate median fish length at time t = 0 when 247 observations began. This also allowed us to confirm that there were no differences in 248 fish size among habitat types at the start of the season. Such initial differences can lead 249 to growth advantages that are independent of environmental factors such as habitat 250 quality [48].

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Because we recaptured steelhead in both restored and unrestored pools, we were able to 253 directly compare observed growth rates for recaptured steelhead between habitat types, 254 in addition to comparing our estimates of growth that came from fitting models to the 255 data. In carrying out analysis of measured growth rates, a key consideration is that, in 256 many organisms, individual growth rates are strongly size dependent [48][49][50]. Because 257 we encountered a wide range of initial sizes (40-75 mm SL), it was important to allow 258 for the possibility of size-dependent growth when comparing steelhead growth rates 259 between habitats. 260 We generated a predicted growth vs. size regression for steelhead for each year, by 261 carrying out a log-linear regression of change in size (mm · day -1 = 262 SL recaptured −SL marked days ) on initial standard length, for each recaptured individual. To 263 compare growth in restored vs. unrestored habitat in these regressions, we kept track of 264 the habitat in which each individual was recaptured. We then used the residual for each 265 point as an indicator of growth rate relative to the population average for that size, 266 where the average for the size was calculated from the regression line of growth on size. 267 A positive residual indicates that an individual has a higher than average growth rate, 268 adjusted for size, whereas a negative residual indicates that an individual has a lower 269 than average growth rate, again adjusted for size (Fig. 3). In comparing size-dependent 270 growth rates between habitats, we compared the mean residual from restored habitat to 271 the mean residual from unrestored habitat for each year of the study, using two-sample 272 t-tests. Because growth rate in mm · day -1 corresponds to the a term in Equations 1-4, 273 we compared differences among habitats in mean residuals with differences among 274 habitats in the a parameter. Use of residual values from a size vs. growth rate regression to compare growth in different habitats. The arrow shows the distance from the points to the line. Mean residual valuesR are calculated for all individuals in each habitat, and ifR r >R u , then we conclude that growth is higher in restored than in unrestored habitats.

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From a management perspective, our goal was to determine whether observations of 277 density differences between habitats are supported by growth differences. We therefore 278 compared our model output first to the fish density data from [30]  length records from Chinook. In the restored habitats, we recaptured 238 Chinook, for 285 a total of 481 length records, with five fish that were recaptured more than once. The 286 best-fit growth curves in Fig. 4 show that Chinook recaptured in restored habitats were 287 generally larger early in the season compared to unrecaptured fish. This visual  In 2009 and 2010, however, values of the growth rate a were larger for unrecaptured 297 fish than for recaptured fish. This observation, in combination with the largerα values 298 for unrecaptured fish describing small early-season size, indicates that unrecaptured fish 299 experienced a rapid increase in size later in the season (Fig. 4) that compensated for the 300 early season disadvantage. The difference in those two years was indicated in >99% of 301 differences in sample values drawn from the posterior distributions of a (Fig. 5).

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Although the 95% CI for the differences in a for all years together overlapped zero, they 303 were below zero in 91.1% of the draws, consistent with the trend observed in 2009 and 304 2010.

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All else equal, smallerα values lead to earlier inflection pointsα/a. To test whether 306 the smaller values ofα for Chinook recaptured in restored habitat in all years combined, 307 and in 2009, 2010, and 2013, led to earlier inflection points for those fish, we drew pairs 308 of values of the shape parameterα and the growth parameter a for recaptured and 309 unrecaptured Chinook. We then calculated differences in inflection pointsα/a for 310 recaptured and unrecaptured fish in each year and for all years combined, as we did for 311 the other parameters.

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This procedure showed that inflection points for recaptured fish did indeed occur 313 earlier than for recaptured fish in 2009, with 99.8% of sampled differences being less 314 than zero. In 2013, there was a similar trend, with 94.6% of sampled differences less 315 than zero. Differences in the other years, and for the combined data, were not 316 meaningful, likely because independent variation in the two parameters obscured 317 differences in the inflection points. We therefore conclude that there is at least modest 318 evidence that restoration is associated with larger Chinook size earlier in the season as 319 indicated by the shape parameterα, but that variation in the two parameters prevents 320 a consistently earlier inflection point.

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We recorded the lengths of 2,871 steelhead across the five sampling years. In the 323 restored habitats, we recaptured 306 individuals for a total of 689 observations, whereas 324 in the unrestored habitats, we recaptured 179 individuals for a total of 413 observations. 325 The number of steelhead recaptured from unrestored habitats was thus sufficient to fit 326 the model directly to recaptured individuals in each habitat type, in contrast with 327 Chinook. In 2013, however, there were no recaptures of steelhead in unrestored habitat 328 for recapture intervals of longer than 24h. We therefore excluded the 2013 data from 329 our analyses.

330
For all years combined, there were no differences in growth parameters in steelhead 331 recaptured in restored versus unrestored habitats (Fig. 6), but there were differences in 332 some parameters in 2009 and 2010 (Fig. 7). In those two years, both the shape 333 parameterα and the growth rate parameter a were higher for fish in restored habitat in 334 99-100% of samples from the posterior distributions. Over all years, almost 90% of 335 differences inα were above zero, and almost 93% of differences in a were above zero, The higher values of a for fish in restored habitat mean that growth rates were 338 higher for steelhead in those two years, but the higher values ofα mean that fish in 339 restored habitat reached larger size later in the season and caught up to the fish in 340 unrestored habitats that were larger earlier. The lack of differences in total growth Y ∞ 341 supports this conclusion. The inflection point was later for steelhead in restored 342 habitats in 2010 only (99.3% of sampled differences above zero), again suggesting that 343 the inflection point is not a specific indicator of growth differences among habitats. 344 Nevertheless, mid-season occupancy of restored pools was associated with rapid growth. 345 Fig 7. The difference in estimates of all model parameters ± the 95% credible interval between the two capture types of young-of-the-year steelhead (recaptures in restored habitat -recaptures in unrestored habitat). Year-specific estimates incorporate the random effect of year, and combined differences are based on the average parameter values among years. The number in parentheses represents the fraction of posterior draws for which the difference was above or below zero, depending on trend of the data. For all panels except the lower right, this represents the fraction of posterior draws above zero. ∆ Length at time = 0 was substituted for model parameter Y . The analysis omits 2013 owing to lack of long-term steelhead recaptures in unrestored habitats.

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Residuals from the log-linear regression of growth (mm · day -1 ) on initial size were more 347 positive, on average, for individuals marked and recaptured in restored habitat than in 348 unrestored habitat, indicating higher growth in unrestored habitat. This trend, however, 349 held for 2009 and 2010 but not 2012 or 2016 (Fig. 8). Inspection of the regression plots 350 for 2009 and 2010 indicated that the largest differences in residuals between habitats in 351 those two years were among individuals 55-60 mm in length. Separate residual analyses 352 on individuals < 60 mm, and individuals > 60mm, produced significant differences in 353 residuals only in individuals < 60 mm (not shown).

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In each year of our study, Chinook salmon were more abundant in restored habitat than 356 in unrestored habitat whereas parameters from the growth model only showed a growth 357 benefit in three of the five years (2009, 2010 and 2013; Table 1). For all years combined, 358 however, density and growth were concordant for Chinook. For steelhead, 2009 was the 359 only year in which there was concordance between observed fish abundance and growth 360 patterns. In 2010, when there was no significant difference in density among habitats, 361 there was nevertheless more rapid growth in restored pools, as indicated above by the a 362 term in the model and by the positive mean residuals from the growth vs. size curve 363 ( Fig. 8 and Table 1).

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Mechanistic ecological models play an important role in restoration ecology [51], 366 because their description of observed data increases the chances that species 367 conservation efforts will be successful. Perhaps the best known set of tools comes from 368 population viability analysis. Population viability analysis has been widely used in the 369 needs to be restored [52]. Species distribution analysis is a related approach that is used 371 to identify the habitat characteristics that are important for conservation [53,54]. Like 372 population viability analysis, species distribution analysis is used to predict how species 373 respond to management actions or environmental change [55]. Predator-prey theory has 374 been used to construct more complex models, again to analyze the outcomes of 375 conservation actions [56].

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All these methods, however, assume that there are correlations between abundance 377 and habitat characteristics, but the support for such correlations is often mixed [57].

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Growth data in contrast are infrequently used in restoration ecology. Surveys of relative 379 abundance cannot easily identify the processes that determine ecological differences 380 between habitat types [9,58,59]. Data sets on growth thus provide opportunities to 381 expand the use of mechanistic models in describing specific effects of habitat restoration 382 in conservation efforts. By comparing growth between restored and unrestored habitat, 383 we have shown that restoration enhances growth rather than simply redistributing 384 organisms among habitats or increasing numbers. Compared to observations of 385 abundance, fitting mechanistic growth models to data can thus provide stronger 386 evidence in favor of restoration.

387
Growth is an important life history trait because of its correlation with fitness in 388 many species. Growth models can therefore help identify the potential fitness 389 advantages of a given habitat type. Accordingly, although we designed our analysis to 390 focus on a specific study system, our over-arching goal is to to provide an adaptable 391 tool for a wide range of studies. Moreover, by using state-of-the-art Bayesian methods, 392 we have shown how growth models can be used to make statistically robust inferences in 393 restoration ecology. The development of Bayesian analytical methods has similarly restoration ecology [60][61][62].

398
Although our model describes a specific life history trait for our study species, in the 399 form of growth during early development, we constructed our model by adapting models 400 that describe growth over the lifetime of an organism. Our model could thus be 401 expanded to describe longer segments of a species' life history. Moreover, our model 402 offers two advantages over previous models. First, our model includes parameters 403 describing not only the rate of growth, represented by a, but also the timing of size 404 increases, represented by the shape parameter,α. Second, the model allows for spatial 405 variation, in that its parameters and residuals were allowed to vary by habitat type, 406 thus allowing for direct description of habitat differences in the data.

407
Our estimates of the model parameters for sub-yearling Chinook salmon and 408 steelhead demonstrate that habitat restoration can make growth conditions more 409 favorable. Our estimates of the shape parameterα showed that Chinook reached large 410 size earlier in three of the five study years, and across all years combined, as a result of 411 remaining in restored pools. In two of the study years, the growth rate parameter, a, 412 was higher among individuals that moved among habitat types, but this occurred later 413 in the season. The interpretation of model parameters therefore can be specific to the 414 study system.

415
After growth to the parr stage (late August/early September), Chinook begin 416 migrating downstream to overwintering habitat [40,63]. As Chinook migrate 417 downstream, slower growing, later hatching, or previously more transient Chinook can 418 immigrate into the pools being vacated by earlier occupants. This can lead to 419 compensatory growth because of release from competition [64]. Chinook previously at a 420 growth disadvantage may then increase their growth once their competitors leave, and 421 may ultimately grow faster than their competitors once did [65]. This effect also likely 422 depends on the overall population size and distributional differences between habitat 423 types. The largest difference in Chinook abundance between habitats was in 2009 and 424 2010 [30]. Thus, fish in those two years were more likely to have been originally 425 captured in restored pools. By mid-season, individuals captured in restored pools could 426 have occupied a pool for up to 10-14 days, the typical interval between sampling events, 427 even if not recaptured later. This could explain the higher growth rate a in 428 unrecaptured Chinook initially captured during mid-season of 2009 and 2010 (Fig. 5). 429 As an explanation for observed habitat differences in steelhead growth in 2009 and 430 2010, competitive release is also consistent with estimated growth parameters. In this 431 study system, steelhead are less abundant overall and overlap with Chinook, 432 particularly in restored pools [30]. When steelhead are less abundant than other 433 salmonids, they usually respond by adopting a more generalist habitat use pattern [66]. 434 Indeed, steelhead recapture rates in unrestored pools indicate higher residency time 435 there compared with Chinook [38], despite the fact that they are usually relatively more 436 abundant in restored pools [30,67]. If early season growth by steelhead in restored pools 437 is slowed by competition with Chinook, then steelhead might do better in unrestored 438 pools, especially in early season.

439
Moreover, the competitive pressure exerted by Chinook is probably more severe 440 when Chinook are at high density. Further evidence that competition affects steelhead 441 growth then comes from the observation that theα parameter for steelhead was larger 442 in restored habitat in 2009 and 2010, when Chinook densities were highest. In those two 443 years, theα parameter indicated that steelhead attained larger size earlier in unrestored 444 habitat. The opportunity to grow rapidly in restored habitat with less competition 445 becomes available mid-season when Chinook begin downstream migration. Because our 446 best-fit models also showed that transient Chinook showed higher growth during this 447 time, our model-fitting approach has apparently revealed the effects of both intra-and 448 interspecific competition.

449
Direct measurements of growth in in steelhead supported the inferences we made by 450 model-fitting. Although direct measurements can in some cases provide more obvious 451 answers, our model-fitting approach has important advantages. First, our model-fitting 452 approach is capable of not only comparing habitats directly, but also of using time series 453 of size to compare recaptured and non-recaptured individuals (as in the case of 454 Chinook). Thus it also integrates over the effects of environmental variables, which are 455 difficult to include as individual correlates in other approaches, across the whole rearing 456 season. Second, model-fitting allowed us to test for effects of different habitats on 457 Chinook, even though the Chinook recapture rate in unrestored habitat type was nearly 458 zero [38]. Third, our approach demonstrated that subtle differences in growth exist 459 between early-and mid-season for both species.  Indeed, exploratory multiple regression analyses attempting to predict growth data with 467 variables such as temperature and current velocity yielded no discernible pattern [38].

468
Our work has implications for the use of life cycle models to predict population-level 469 responses of salmonids to restoration [4,68], but there are also substantial uncertainties. 470 First, although we found that there was higher growth in restored habitat, and although 471 changes in habitat capacity are allowed for in life cycle models [4], such changes have 472 not yet been detected in nearby sub-basins [8]. Second, positive growth in restored 473 habitat is evidence of capacity increase, but whether this is important for the entire life 474 cycle remains unclear. Salmonid growth at one life stage is sometimes a strong predictor 475 of growth at a subsequent stage [69], but not always [70]. More broadly, observations of 476 increased growth at a few sites of course does not mean that growth would be increased 477 over larger areas.

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Model fitting nevertheless allowed us to show that growth can be reliably quantified 479 as a biological response to habitat quality differences, a rare accomplishment in fish 480 restoration. Model fitting also provided at least partial confirmation that differences in 481 growth between habitats are concordant with differences in relative density between 482 habitats. It is also important to note that, in 2010, higher steelhead growth was not 483 associated with higher abundance, emphasizing that there is more to restoration than 484 abundance. Finally, the model strongly suggests that phenology and competition may 485 be modulated by habitat restoration. Growth models may thus provide a tool of general 486 usefulness in restoration ecology.