estimating disperser abundance using open population arrays · 2018. 4. 5. · draft 1 1 title:...

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Estimating disperser abundance using open population

models that incorporate data from continuous detection PIT arrays

Journal: Canadian Journal of Fisheries and Aquatic Sciences

Manuscript ID cjfas-2017-0304.R2

Manuscript Type: Article

Date Submitted by the Author: 02-Nov-2017

Complete List of Authors: Dzul, Maria; U.S. Geological Survey, Grand Canyon Monitoring and

Research Center Yackulic, Charles; USGS, Grand Canyon Monitoring and Research Center Korman, Josh; Ecometric Research Inc.

Is the invited manuscript for consideration in a Special

Issue? : N/A

Keyword: ABUNDANCE < General, MARK-RECAPTURE < General, MODELS < General, INVASIVE SPECIES < Organisms, PIT tag array

https://mc06.manuscriptcentral.com/cjfas-pubs

Canadian Journal of Fisheries and Aquatic Sciences

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Title: Estimating disperser abundance using open population models that incorporate data from 1

continuous detection PIT arrays 2

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Authors: Dzul, M.C., C.B. Yackulic, and J. Korman 4

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Maria Dzul, corresponding author 7

U.S. Geological Survey 8

Southwest Biological Science Center, Grand Canyon Monitoring and Research Center 9

2255 N. Gemini Drive 10

Flagstaff, AZ, 86001, USA 11

[email protected] 12

(928) 556-7197 (Phone) 13

(928) 556-7100 (Fax) 14

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Charles B. Yackulic 16

U.S. Geological Survey 17

Southwest Biological Science Center, Grand Canyon Monitoring and Research Center 18

2255 N. Gemini Drive 19

Flagstaff, AZ, 86001, USA 20


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Josh Korman 23

Ecometric Research Inc. 24

3560 W 22nd Ave. 25

Vancouver, BC V6S 1J3, Canada 26


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Abstract: 37

Autonomous passive integrated transponder (PIT) tag antenna systems continuously detect 38

individually marked organisms at one or more fixed points over long time periods. Estimating 39

abundance using data from autonomous antennae can be challenging, because these systems do 40

not detect unmarked individuals. Here we pair PIT antennae data from a tributary with mark-41

recapture sampling data in a mainstem river to estimate the number of fish moving from the 42

mainstem to the tributary. We then use our model to estimate abundance of non-native rainbow 43

trout Oncorhynchus mykiss that move from the Colorado River to the Little Colorado River 44

(LCR), the latter of which is important spawning and rearing habitat for federally-endangered 45

humpback chub Gila cypha. We estimate 226 rainbow trout (95% CI: 127-370) entered the LCR 46

from October 2013-April 2014. We discuss the challenges of incorporating detections from 47

autonomous PIT antenna systems into mark-recapture population models, particularly in regards 48

to using information about spatial location to estimate movement and detection probabilities. 49

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Introduction: 60

Migration and the movement of individuals between habitats is a fundamental process 61

that is central to the life history of many fish species. Often, studies are interested in evaluating 62

spatiotemporal patterns in movement that are related to species’ life history (Bahr and Shrimpton 63

2004; Lucas and Batley 1996; Yackulic et al. 2014) and behavior (Dagorn et al. 2000; Gowan 64

and Fausch 2002). Such studies typically evaluate movement using a subset of tagged 65

individuals and report movement rates in terms of individual probabilities of movement (i.e., as 66

the relative number of animals that move). While estimating movement probabilities is 67

sufficient for answering some ecological questions, there are certain applications that may 68

benefit from an estimate of the absolute number of animals that move between two habitats (i.e., 69

the flux of tagged and untagged animals). Specifically, studies that evaluate dispersal of non-70

natives (Korman et al. 2016) and studies that assess population connectivity (Fullerton et al. 71

2010) would both benefit from estimates of absolute disperser abundance, as the actual number 72

of dispersers is an important consideration when quantifying the threat posed by nonnatives or 73

the genetic exchange occurring between populations. Similarly, because fish migration can 74

impact ecosystems (Brodersen et al. 2008; Naiman et al. 2002; Wheeler et al. 2015), quantifying 75

fluxes of fish between two habitats could help scientists evaluate how fish dispersal influences 76

ecosystem processes. 77

Passive integrated transponder (PIT) tags are commonly used to evaluate fish movement. 78

PIT tags are small glass-coated electronic microchips programmed with unique alpha-numeric 79

codes (Gibbons and Andrews 2004), and thus animals implanted with PIT tags are individually 80

identifiable. Because PIT tags are powered externally (i.e., tags must be charged by an antenna 81

to electronically transmit their associated code number), fish need to be scanned to read PIT tags. 82

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Tag detection is most commonly achieved by capturing fish and manually scanning them with an 83

antenna. However, individuals can also be detected continuously over long time periods, without 84

physical recaptures, through the use of autonomous PIT tag antenna systems strategically placed 85

at one or more fixed locations (Barbour et al. 2012; Zydlewski et al. 2006). Autonomous PIT 86

antenna systems can increase detection probabilities relative to methods that rely on physical 87

capture (Adams et al. 2006; Hewitt et al. 2010). 88

Autonomous PIT antenna systems include PIT arrays, which are chains of antennae, 89

typically deployed in lotic systems and frequently spanning the entire river channel. PIT arrays 90

can be placed in strategic locations (e.g., across the channel of a tributary near its mouth) to 91

evaluate movement patterns (Roni et al. 2012; Zydlewski et al. 2006). For example, Cathcart et 92

al. (2015) used PIT array detection data to evaluate native fish movements between the San Juan 93

River and its tributary, McElmo Creek. Previous studies have used PIT array detections to 94

identify dispersers and differentiate temporary and permanent emigration from survival (Al-95

Chokhachy and Budy 2008; Horton et al. 2011). However, estimating abundance of unmarked 96

dispersers presents a challenge because autonomous PIT arrays only detect marked individuals 97

(i.e., unmarked individuals have a detection probability of zero). This problem is amplified in 98

systems where mark-recapture sampling occurs over a limited spatial scale and unmarked fish 99

from larger non-sampled areas are likely to move over the PIT tag array. In these situations, 100

determining the number of fish dispersers or migrants requires information about the spatial 101

mixing of marked and unmarked fish in the original habitat, the spatial variability of fish 102

abundances in the original habitat, and the influence of spatial location on movement probability 103

(e.g., fish that are located closer to the PIT array are more likely to move over it compared to fish 104

located farther from the array). 105

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Here we develop and apply a mark-recapture model to estimate abundance of dispersers 106

by combining information from an autonomous PIT array (located in a tributary near the mouth) 107

and traditional capture-recapture surveys (conducted at various locations in the main river). We 108

also use simulation to evaluate the bias and precision in derived disperser abundance estimates. 109

A key feature of estimating the abundance of dispersers moving between a mainstem river and 110

its tributary is that potential dispersers moving into the tributary could come from various 111

sections of the main river which experience different levels of marking effort, which can include 112

large areas where fish were not marked. 113

After describing the general modeling approach, we apply our abundance estimation 114

method to non-native rainbow trout (Oncorhynchus mykiss or RBT) moving between the main 115

stem Colorado River and the Little Colorado River (LCR), as determined by detections on a 116

multiplexer PIT array system (MUX; Biomark Inc.) in the LCR. We fit the model to between-117

river movement over two different time intervals to obtain two estimates of disperser abundance. 118

The LCR is the main spawning habitat for endangered humpback chub in the lower Colorado 119

River (Kaeding and Zimmerman 1983), and as a consequence RBT in the LCR pose a threat to 120

the native humpback chub population because RBT prey upon and compete with humpback chub 121

(Marsh and Douglas 1997; Yackulic et al. In Review; Yard et al. 2011). Thus, an estimate of 122

RBT abundance in the LCR will help to quantify the risk to humpback chub. 123

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Methods: 125

General modelling framework 126

Our general framework assumes fish in two adjoining habitats are exposed to different types of 127

sampling: fish in one habitat are marked with PIT tags using traditional mark-recapture methods 128

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that rely on physical captures, while fish in the other habitat are exposed to PIT detections from 129

an autonomous array. In our application, fish were sampled in a large river (hereafter referred to 130

as the sampling river) and we are interested in estimating the flux into a tributary (detection 131

river) that has an autonomous array near the tributary’s confluence with the sampling river. 132

Although our application is specific to lotic ecosystems, we emphasize that our approach could 133

be modified and applied to estimate fluxes into and out of other habitats besides rivers (e.g., 134

lakes or canals). We further differentiate between two different types of movement: movement 135

within the sampling river (hereafter within-river movement) and movement from the sampling 136

river and to the detection river (hereafter between-river movement). However, because we do 137

not directly observe movement in the sampling river (but rather changes in fish location between 138

time intervals), we estimate probabilities of within-river displacements. Lastly, we use the term 139

‘dispersers’ to describe animals that make between-river movements. 140

Our approach is an extension of the multistate modelling framework (Arnason 1973; 141

Schwarz et al. 1993) and includes multiple time steps (e.g., monthly), where between-river 142

movement and within-river displacement describe the probabilities for one time step. This 143

approach requires at least three mark-recapture trips occur in the sampling river, with one trip 144

occurring (ideally) immediately before the start of the dispersal period, or immediately before 145

between-river movements commence. Additionally, PIT array detections and mark-recapture 146

efforts can occur concurrently (e.g., over the same time interval). 147

Detection probability in the sampling river is estimated using an open multistate model 148

that conditions on first capture (i.e., we only model the marked population), whereas in the 149

detection river it is estimated by treating detections from two or more antenna arrays as 150

secondary capture occasions in a robust design framework (Horton et al. 2011; Pollock 1982). 151

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This design requires at least two closely spaced PIT arrays in the detection river to allow for 152

estimation of array detection probability and, in turn, between-river movement probability. The 153

PIT arrays system should be placed in close proximity to the confluence (in order to detect 154

dispersers that spend long time periods in the detection river) but far enough from the confluence 155

to avoid detecting fish that are making short-term (e.g., diurnal) back-and-forth movements 156

between the sampling and detection rivers. The spacing of the two (or more) arrays should be 157

close enough to one another to minimize the number of fish that swim past one array but stop 158

short before the next array, but far enough from one another to minimize noise interference. In 159

general, we recommend arrays be placed 0.5 km – 2 km from the confluence, with array spacing 160

about 10-200m apart. However, in practice, the spatial arrangement of antennae will be 161

dependent on the system and focal fish species and could differ from our suggestions. 162

Based on predictions of the number of marks alive in the sampling river from the open 163

population model, and detection at the PIT array, the model estimates between-river movement 164

probability. The between-river movement probability is then multiplied by the total abundance 165

estimate in each sampling river segment to estimate the number of dispersers. The simplest 166

version of our model includes only a single site in the sampling river (or habitat) where a subset 167

of the population is vulnerable to capture in the sampling river (i.e., if the fish being studied 168

regularly moves throughout the whole sampling river and sampling occurs in a fixed portion of 169

the sampling river). We examine this approach in Appendix A, however, here we focus primarily 170

on the situation where a few discrete sites are sampled within the sampling river (or habitat). 171

An alternate approach highlighted here considers between-river movement probabilities 172

as a function of proximity to the confluence (hereafter referred to as the multi-site design). 173

Descriptions of model parameters are described in Table 1. The multi-site design requires that 174

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the sampling river be divided into segments, preferably of equal length (where at least two river 175

segments need to be sampled in this design). We use a parametric distribution to model within-176

river displacements, which allows us to compute within-river displacement probabilities from 177

sampled river segments to unsampled ones, and also reduces the number of parameters required. 178

Note also that this approach allows for multiple unobservable states in the sampling river, where 179

each unobservable state corresponds to an unsampled segment. Here within-river displacement is 180

modeled using a Cauchy distribution, but any movement model could be used (e.g., exponential). 181

Note the Cauchy distribution does not assume the sequential movement probability of an 182

exponential model (e.g., θ1,3 ≠ θ1,2 θ2,3), and that this distribution allows fish to stay within their 183

home river segment, or move one or more segments upstream or downstream (as long as this is 184

within the bounds of the sampling river segments). The Cauchy distribution has been used by 185

others to model animal movements (Korman et al. 2016; Muneepeerakul et al. 2008), and it is an 186

appropriate choice when most individuals are fairly sedentary but a small subset move far 187

distances. The Cauchy distribution includes two parameters: x0 (location or average 188

displacement), and τ (scale); and di,j represents the value of the Cauchy distribution function for 189

movement between river segments i and j. 190

191

(1) ��,� = ��

192

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The di values are then rescaled to sum to 1 using the following equation: 194

195

(2) ��,� = � ,�∑ � ,��

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where θi,j is the probability of moving from segment i to segment j and remaining in segment j, 198

and Z is the number of river segments in the sampling river. The θ values are used in a state 199

transition matrix to represent the probability of movement between sampling river segments, and 200

we refer to the θ values as the within-river movement probabilities. It is important to consider 201

that the upper and lower bounds of the sampling river can influence estimated survival 202

probabilities, because movement upstream of the upper-most segment or downstream of the 203

lower-most segment will be confounded with survival. 204

Before we model between-river movements, we must first define the confluence of the 205

detection and sampling rivers. The confluence can contain multiple river segments in the 206

sampling river, and at least one of these segments must be sampled. Next, between-river 207

movement probabilities are calculated by assuming that fish must swim through the confluence 208

segment(s) to the detection river. Specifically, to calculate between-river movement 209

probabilities, we first compute the probability of moving from segment i in the sampling river to 210

the confluence (θi,C): 211

212

(3) ��,� =∑ ��,�� 213

214

where values k and n correspond to spatial states that are considered part of the confluence. 215

Accordingly, fish must move through a confluence segment in the sampling river (i.e., ∈(k,n)) in 216

order to enter the detection river. Note if there is only one confluence segment, then k = n. 217

Now we construct the state transition matrix, which takes into account the spatial 218

arrangement of the sampling river segments relative to the confluence reach. We include Figure 219

1 to illustrate the movement parameterization in form of a diagram, and we also include a 220

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simplified form of the spatial state transition matrix for the multi-site model below, where Z=3 221

and the confluence is defined as river segment 2 (and thus θ1,C = θ1,2 ): 222

223

(4) !" =#$$$% ��,� ��,&(1 − *) ��,,(1 − *) (��,& +��,,)* 0�&,�(1 − *) �&,&(1 − *) �&,,(1 − *) * 0�,,�(1 − *) �,,&(1 − *) �,,, (�,,� +�,,&)* 0

0 0 0 0 10 0 0 0 1/

0001 224

225

Here the upper Z × Z dimensions (or 3 ×3 in the above example) of the matrix represent within-226

river displacements. Also, we add two extra states (Z+1th and Z+2th elements of the matrix) for 227

the detection river (DR): an observable detection river (ODR) state and an unobservable 228

detection river state (UDR). As an example we will focus on fish from sampling river segments 229

that are located upstream of the confluence segment (i.e., row 1 of the above matrix). The 230

probability of moving between sampling river segments located upstream of the confluence are 231

unaltered from the within-river displacement probabilities (i.e., θ), while the probabilities for 232

segments downstream of the confluence become the product of the within-river displacement 233

probabilities and the probability of not moving into the detection river (θ × (1- α)). This 234

parameterization implies that all fish that are displaced to a segment past the confluence have 235

probability α of entering the detection river, and that fish ‘decide’ to move into the detection 236

river just as they approach the confluence. 237

After swimming into the detection river, fish must first enter the ODR state, where they 238

are susceptible to PIT array detection. We assume that fish do not remain over the PIT array so 239

fish transition out of the ODR state and into the UDR state (where their detection probability is 240

zero) in the next time step. Accordingly, movement into the detection river can be interpreted as 241

permanent emigration, or alternatively, as fish transitioning permanently to a disperser state. The 242

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latter interpretation implies that dispersers retain their disperser status even if/when they move 243

back to the sampling river (note this may require altering capture histories so fish are 244

unobservable after their first PIT array detection, as was done in our applied example). 245

Including a permanent disperser state in the model enables estimation of the total number of 246

unique fish that enter the detection river over a period of time. An additional advantage of 247

having a permanent disperser state is that the survival probability in the detection river (which 248

can be difficult to estimate) does not affect the disperser abundance estimate. 249

To estimate detection probabilities, we assign additional detection categories to the ODR 250

state to use detection information from individual PIT arrays. Detection categories in the ODR 251

state for marked fish corresponded to being detected by both arrays (pant1pant2), the upstream 252

array (pant1(1-pant2)) only, the downstream array only ((1-pant1)pant2), and missed by both arrays 253

((1-pant1)(1-pant2)). Detection for sampled river segments in the sampling river are estimated as 254

trip-specific capture probabilities in months when sampling occurs. Detection probabilities for 255

reaches in the sampling river that are not sampled are fixed to zero. 256

257

Derived abundance estimates 258

Estimating the number of dispersers requires two steps: 1) calculating segment-specific 259

abundance estimates for the sampling river (or SR abundances), and 2) multiplying these 260

abundance estimates by the vector describing the probabilities of moving from any sampling 261

river segment into the detection river. Note, however, the derived disperser abundance 262

calculation described below assumes no recruitment occurs during the PIT tag array detection 263

period (apart from immigration and emigration within sampling river segments). 264

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We must first define the start and end of the PIT array detection period as t1 and t2, 265

respectively. Next, we obtain SR abundance estimates for sampled segments in the sampling 266

river at t1-1 (i.e., for the month prior to the start of the PIT tag detection period) by dividing catch 267

by estimated capture probabilities. We then use linear interpolation to obtain SR abundance 268

estimates in unsampled segments at time t1-1. The SR abundances, together with estimated 269

detection river (DR) abundances (which are assumed to be 0 at t1-1) comprise the vector of state-270

specific abundances at time t1 -1 (or 3"454). This vector is multiplied by the state transition 271

matrix (6"454) and a vector of survival rates for each state during time t (7"454) to obtain state-272

specific abundances in month t1 (3"4). 273

274

(5) 89� =89�5�[;<=9�5�>]!9�5� 275

276

This step is repeated to determine state-specific abundances at 3"4�4, 3"4�@, …3"@. 277

Finally, the estimate of disperser abundance is obtained by summing the abundance values that 278

correspond with the ODR state across all N vectors between t1 and t2. Note that disperser 279

abundance is the total number of unique fish that move into the detection river between t1 and t2, 280

and this number will likely differ from the disperser estimate in the ODR state at any given time 281

t. Accordingly, the disperser abundance estimate is more comprehensible over relatively short-282

term time scales where individuals are unlikely to make multiple between-river movements. 283

284

Important assumptions in our general framework 285

Our modeling method involves a number of assumptions. First, this model assumes all 286

PIT array detections represent upstream movements only. This upstream-only assumption does 287

limit the use of the model for all applications, but it may be appropriate when inferences are 288

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focused on estimating fluxes in one direction as in case where migrations have a strong seasonal 289

component (e.g., spawning movements) or in systems where movement over arrays is one-290

directional and permanent (e.g., over a large waterfall). We further assume that within-river 291

displacements within the same time interval represent linear movements (i.e., fish either swim 292

only upstream or only downstream within a time interval). In addition, our model makes many 293

of the same assumptions as other mark-recapture models, including the assumption that 294

movement, survival, and detection probabilities are independent processes that are equivalent for 295

all individuals. Lastly, we assume no recruitment occurs in the sampling river throughout the 296

PIT array detection period and that within-river displacement is bounded by the upper-most and 297

lower-most segments. 298

One concern specific to fish and arrays is that our approach assumes that all fish have the 299

same detection probabilities. Past studies have found that frequently a few fish have a tendency 300

to linger, swimming circles over PIT arrays, while most fish swim over the arrays just once on 301

their way upstream or downstream (Connolly et al. 2008). Lingering fish create two issues. First 302

they are more likely to be observed by both arrays in a single time period, positively biasing PIT 303

array detection probabilities. Second, they may be more likely to show up over multiple time 304

periods, suggesting between river movements and/or negatively biased detection probability 305

estimates. To address both issues, we advise only including second detections that occur within a 306

set amount of time after the first array detection of an individual (i.e., an individual can only be 307

detected by the arrays once during the study and detections on multiple arrays have to occur 308

within a short time, such as within one hour). Our modelling approach includes an unobservable 309

state in the detection river and this allows us to remove PIT array detections and sampling river 310

captures after each fish’s first PIT array detection and still estimate fluxes into the detection 311

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river. After this step, we assume all fish that pass over one array must have passed over all 312

arrays once on their way upstream (i.e., population closure between arrays). 313

314

Applied example – Non-native rainbow trout detections in the Little Colorado River 315

Field methods 316

The MUX was installed in the LCR in May 2009 to detect movement of native fishes 317

between the Colorado River (CR) and LCR. The MUX is a flat-bed array that is situated on the 318

river bottom, and its read range was estimated to be 15.1-16.1cm in 2013 (Pearson et al. 2016). 319

The MUX is located 1.78 km upstream of the CR-LCR confluence, and it is comprised of two 320

arrays (each with six antennae) located approximately 100m from one another. In the winter of 321

2013-2014, the MUX detected a high number of PIT-tagged RBT (which were marked as part of 322

another effort by Korman et al. 2016) moving between the CR and LCR. Prior to these 323

detections RBT use of the LCR was believed to be minimal. We initially attempted to fit models 324

to MUX detections from October 2012-September 2014; however, movement during the early 325

portion of this period could not be estimated due to its rarity. Furthermore, PIT arrays frequently 326

malfunctioned before 2013, adding heterogeneity to MUX detection probabilities and 327

complicating model fitting. Thus, we limited our assessment of LCR movement to the winter of 328

2013-14 (i.e., October 2013-April 2014) when PIT detections were highest and PIT arrays were 329

functional. 330

Of the 36 RBT that were detected on the MUX, most were only detected one to three 331

times on the PIT array from October 2013-April 2014, but three fish had a much higher number 332

of detections (11,12, and 99), suggesting they lingered over the array. To avoid the heterogeneity 333

in detection probability these fish could create, we only included secondary detections for 334

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individuals that occurred within an hour of their first detection (see previous section). 335

Additionally, we altered capture histories so that all RBT were treated as unobserved after initial 336

detection on the MUX. This allowed us to estimate the total unique number of fish that moved 337

into the detection river over the PIT tag detection period. 338

We also assessed the directionality of array detections (i.e., whether fish that were 339

detected on both arrays were first detected on the upstream or downstream array) to determine 340

whether PIT array detections of RBT were indeed associated with upstream movements. The 341

sample size to assess this assumption was very small, as only six fish were detected on both 342

arrays within one hour of each other. However, all six fish were first detected on the 343

downstream array, thus indicating this 'upstream only' assumption is probable. Also, it is 344

worthwhile to note that preliminary data suggest upstream detection probabilities across other 345

fish species on this array to be much higher than downstream detection probabilities (Persons, 346

pers. comm.) and that other studies have observed higher detection probabilities for upstream 347

movement compared to downstream movement (Aymes and Rives 2009). 348

We combined PIT array detections with a portion of the RBT data collected in the CR. 349

Specifically, recent CR sampling trips have visited five reaches ranging from 16.3-129.6 350

kilometers downstream of Glen Canyon Dam (i.e., river kilometers or rkm) in January, April, 351

July, and September (from April 2012-September 2014 for a total of 11 sampling trips), but we 352

only included RBT from rkm 86.6-129.6 in our analyses because movement into the LCR was 353

extremely rare from more upstream reaches. We divided the CR into 22 river segments, each 354

about 2km in length (see Figure 2). Of the 22 segments in the sampling river, only 5 were 355

sampled; these included: Segments 1-3 (rkm 86.6-91.9), Segment 19 (rkm 122.0-123.6), and 356

Segment 22 (rkm 127.1-129.6). These segments were sampled with either 1 pass of 357

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electrofishing (Segments 1&3), 2 passes of electrofishing (Segments 2&19), or 5 passes of 358

electrofishing and hoopnets (Segment 22). Segments with only 1 pass of electrofishing were 359

used to estimate movement and survival, but not abundance. Upon capture, all RBT ≥ 75mm 360

fork length in river Segments 2, 19, and 22 were given a 134.2 kHz passive integrated 361

transponder (PIT) tag. Fish were sometimes also marked in Segment 1 and 3, though not 362

consistently on every trip. Since the confluence needs to include at least one sampled segment, 363

we defined the confluence as Segment 19 (where upper and lower boundaries are 2.4-0.8 km 364

upstream of the CR-LCR confluence) and Segment 20 (where upper and lower boundaries are 365

0.8 km upstream of and 1.2 km downstream of the CR-LCR confluence). The data used by our 366

model are publicly available at https://doi.org/10.5066/F7NZ86JV (Dzul et al. 2017). 367

368

Applied example – statistical analysis 369

We fit the multi-site model to RBT MUX detections in the LCR and to mark-recapture 370

data from the sampled river segments in the CR. The CR (i.e., the sampling river) included 22 371

spatial states; the location parameter of the Cauchy distribution was fixed to be zero based on 372

previous research (Korman et al. 2016), and the scale parameter (τ) was estimated by the model. 373

Note this model assumed that movement within the sampling river was temporally constant 374

throughout the course of the study. Between-river movement probability and array detection 375

probabilities were considered temporally constant from October 2013-April 2014 (the PIT array 376

detection period), and fixed to zero outside this period (i.e., prior to October 2013 and after April 377

2014). 378

For Segments 19 and 22, trip-specific and segment-specific capture probabilities were 379

estimated. Capture probabilities for segments 1-3 were also trip-specific, with the capture 380

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probabilities for Segments 1 and 3 set equal and the capture probability for Segment 2 adjusted 381

to account for one additional pass (in other words, the capture probability of Segment 2 is one 382

minus the probability of not being captured during two passes of sampling). All unsampled 383

segments were considered unobservable states and therefore their detection probabilities were 384

fixed to zero. These unobservable CR states were not differentiable from the LCR downstream 385

of the MUX (because movement into and out of the LCR below rkm 1.78 cannot be observed). 386

Based on results from Korman et al. (2016), we estimated two different monthly survival 387

rates to the CR that corresponded to above and below the LCR confluence (i.e., S1-19 and S20-22 to 388

states 1-19 and states 20-22, respectively). Additionally, summer survival (April-September) 389

was modeled as an additive offset of winter survival (October-March). We used a multinomial 390

likelihood to calculate probability of observed capture histories, which was maximized using the 391

function ‘optim’ in R (R Core Development Team 2012). Code for our applied example is 392

provided as Supplementary data. 393

The simplest sampling design would include a single, sampled segment at the confluence. 394

However, this was not the case with the LCR-CR confluence in our application. Specifically, 395

Segment 19 was located 0.8-2.4 km upstream of the CR-LCR confluence and the boundaries of 396

unsampled Segment 20 were located 0.8 km upstream of the confluence and 1.0 km downstream 397

of the confluence. Thus, we define the confluence as Segments 19 & 20 and we assume the 398

probability of LCR migration was equal for these two segments. 399

We calculated disperser abundance without recruitment. This is acceptable for our 400

applied example, as results from Korman et al. (2016) indicate that RBT recruitment in this 401

section of the CR is low (i.e., most additions to the RBT population are from immigration from 402

the tailwater located 99 km above the confluence). Furthermore, based on Korman et al. (2016), 403

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we modified the above-mentioned methods slightly by assuming RBT abundance was equal in 404

all segments below the LCR confluence (i.e., Segments 20-22 had equal abundance estimates). 405

In addition to disperser abundance estimates, we reported one additional derived parameter from 406

our multisite model: detection probability of the MUX system (i.e., pMUX or 1 − [(1 − CD�9�) ×407

(1 − CD�9&)] where pant1 and pant2 are the detection probabilities of the two antenna arrays). 408

We obtained confidence intervals for the derived parameters using a parametric bootstrap 409

from a multivariate normal distribution that is based on mean parameter estimates and the 410

variance-covariance matrix generated by the model. This was implemented using the package 411

MASS (Venables and Ripley 2002) to simulate from a multivariate normal distribution, and we 412

provide R code for a simplified version of the derived abundance equation and CI calculation in 413

Supplementary data. Specifically, the variance-covariance matrix of the each model was used to 414

simulate a multivariate normal distribution of model parameters, and derived parameters were 415

then calculated for each simulation. We incorporated variability in sampling river abundances 416

using a Horvitz-Thompson estimator, which accounts for both variability in capture probability 417

and catch (McDonald 2005). We conducted this simulation 10000 times to obtain a distribution 418

of derived parameters. Furthermore, we tested the sensitivity of the derived disperser abundance 419

estimates (NLCR) to parameters used in the above calculation. To account for parameter 420

correlation in the sensitivity analysis, we first increased and decreased each parameter used in 421

the derived abundance calculation by two times its standard error, then we adjusted estimates of 422

other parameters using a conditional multivariate normal distribution from the R package 423

condMVNorm (Varadhan 2015) and calculated NLCR from adjusted parameter values. 424

We also evaluated bias of the multi-site model, by first simulating 300 data sets from our 425

RBT model, then fitting models and calculating disperser abundance estimates from catch and 426

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parameter estimates. We computed relative bias from these simulations by subtracting the 427

known number of dispersers from the estimated values and divided by the true value. 428

429

Results 430

A summary of catch from the CR (sampling river) is provided in Table 2 and the 431

summary of PIT array detections in the LCR (detection river) is shown in Table 3. The model 432

estimated that the MUX detection probability from October 2013-April 2014 was 59.8% (95% 433

CI: 37.9-80.0%), and that the monthly between-river movement probability (i.e., movement from 434

the CR into the LCR) was 1.0% (95% CI: 0.6-1.7%; Table 4). The 1.0% monthly movement rate 435

indicates that less than 10% of the rainbow trout located at the confluence moved 1.8km into the 436

LCR over the seven month period (Oct 2013- Apr 2014). This translates to an estimate of 226 437

RBT (95% CI: 127-370). Importantly, this may be a conservative (i.e., low) estimate of the 438

number of RBT dispersers because RBT that swim from the confluence and into the lower 439

reaches of the LCR (i.e., below rkm 1.78 km) cannot be detected by the MUX. While our model 440

did estimate lower winter survival and lower survival below the LCR confluence, we do not 441

interpret these findings because sampling river survival is confounded with movement outside of 442

the sampling river boundaries (i.e., upstream of the upper-most segment and downstream of the 443

lower-most segment) due to the spatial truncation in our movement model. 444

The sensitivity analysis demonstrated that LCR abundance estimates were highly 445

sensitive to the between-river movement probability, slightly sensitive to the September 2013 446

capture probability, and not sensitive to estimates of winter survival or within-river displacement 447

probabilities (Figure 3). Importantly, low precision in the between-river movement probability 448

had a large effect on the disperser abundance estimate in part because the between-river 449

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movement parameter was low. For instance, between-river movement probability estimates of 450

1.5% and 1.0% would result in disperser abundance estimates of 314 and 226, respectively. In 451

contrast, between-river movement probabilities of 5.5% and 5.0% would result in disperser 452

abundance estimates of 1061 and 975, respectively, which are more similar when viewed on a 453

relative scale. 454

Simulations indicated that derived abundance estimates had slightly negative relative bias 455

in the mean (-2.5%) and median (-4.1%) disperser abundance estimate. Furthermore, the 95% 456

confidence interval for relative bias was fairly wide (-27.0% to 33.8%) and included large 457

positive outliers (Figure 4). This suggests that, despite low bias in median and mean disperser 458

abundance, large outliers can result when between-river movement probabilities are low. 459

460

Discussion 461

Although autonomous PIT antenna systems are useful tools for studying animal 462

movements, it is not entirely clear how to best incorporate these remote PIT detections into 463

mark-recapture models. In particular, problems arise due to the continuous stream of detections 464

and because array detection probabilities are subject to individual heterogeneity (e.g., due to 465

certain fish lingering over the array compared to fish that just swim over arrays once). 466

Nonetheless, many studies have successfully incorporated remote detections into mark-recapture 467

models to quantify movement probabilities using a variety of approaches (Al-Chokhachy and 468

Budy 2008; Buchanan and Skalski 2010; Horton et al. 2011). Despite the burgeoning literature 469

on PIT array data and mark-recapture models, one of the remaining limitations of PIT array data 470

is the inability of this technology to detect unmarked fish, and this limitation has hindered the 471

use of PIT array data for abundance estimation. Thus, although numerous other studies have 472

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evaluated movement over PIT arrays, few studies quantify the number (i.e., abundance) of fish 473

that move over PIT arrays or other similar remote tag detection systems (Dudgeon et al. 2015). 474

This modeling approach accounts for spatial heterogeneity in between-river movement 475

probabilities by dividing the sampling river into discrete segments and modeling each segment as 476

a separate state. This approach requires the sampling river is divided up into discrete segments, 477

and the size of these segments is admittedly subjective. An alternative approach would be using 478

spatial mark-recapture models (Royle et al. 2013), where models estimate activity centers for 479

individual animals as well as abundance or density. While the spatial mark-recapture approach 480

would allow for finer resolution in assigning movement probabilities (i.e., individual-level as 481

opposed to segment-level), it has the disadvantage of being difficult to fit for species with low 482

capture probabilities, as their center of activity is difficult to approximate. Furthermore, spatial 483

mark-recapture would not help with estimation of abundance because unmarked individuals still 484

have zero capture probabilities on autonomous PIT technologies. Nonetheless, spatial-mark 485

recapture models may hold promise for movement studies of fishes (Raabe et al. 2013), 486

particularly if long-range movements could be incorporated into their model structure. 487

Combining continuous detection data with capture-recapture sampling presents multiple 488

challenges. For example, many mark-recapture models are designed for discrete sampling 489

occasions, and consequently it can be difficult to determine how to best incorporate continuous 490

detection data from PIT arrays into a mark-recapture model. In the current study, we discretized 491

continuous MUX detection data by binning detections across months. While binning continuous 492

PIT detections presents one option for mark-recapture modeling (Pearson et al. 2015), some 493

studies have instead opted for a Barker model for PIT array data because the Barker model 494

couples discrete mark-recapture data with continuous resight data (Al-Chokhachy and Budy 495

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2008; Barker 1997). Although the Barker model typically produces less biased estimates of 496

survival compared to a Cormack-Jolly-Seber (CJS) model for continuous resight data (Barbour et 497

al. 2013), the Barker model was intended for systems where resightings encompass a broad 498

spatial scale relative to the area sampled during mark-recapture studies. When resight data are 499

localized, however, Barker and CJS models exhibit similar performance (Conner et al. 2015). 500

Thus, because resight data in the current study were highly localized (i.e., rainbow trout could 501

only be remotely detected at one spatial location), we decided to use a CJS multistate model 502

because it provided a flexible model framework for our application. 503

Prior to learning of RBT MUX detections in the current study, RBT were considered rare 504

visitors to the LCR. Furthermore, RBT rarely enter hoop nets (the main sampling gear used in 505

the LCR) and no sampling trips visit the LCR from November to March. Therefore, the large 506

influx of RBT that occurred during the winter of 2013-14 would likely have remained 507

undiscovered without the MUX. Compared to other tagging technologies, PIT arrays may be 508

particularly useful for detecting rare movements, whose timing is difficult to predict a priori. 509

Advantages of PIT array systems over other types of electronic tags (e.g., sonic and GPS tags) 510

include their relative small size, low cost, longevity (i.e., the tag is typically good for the lifetime 511

of the organism), and reliability (Cooke et al. 2013; Smyth and Nebel 2013). These features 512

make PIT arrays good options for augmenting mark-recapture data when detection or movement 513

probabilities are low, as in these situations having a large number of marked fish can help with 514

identifying dispersal patterns. However, PIT arrays do also have some disadvantages over other 515

tags (e.g., relatively short read range of PIT tags, noise interference, difficult or impossible to 516

deploy in large river systems), and the advantages and disadvantages must be evaluated on a 517

case-by-case basis. 518

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Our applied example illustrates some of the challenges in quantifying rare movements, as 519

the low number (i.e., 36) of unique RBT detected on the MUX resulted in low precision in 520

movement and detection parameters. Improving precision in the between-river movement 521

parameter could be accomplished by maximizing the mark rate (i.e., the proportion of fish that 522

have marks) in the sampling river. Other options would include improving precision in the mark 523

rate estimate (and thereby population size in the sampling river), or incorporating environmental 524

covariates effects on movement in order to reduce uncertainty in movement estimates. Of 525

course, maximizing detection on the PIT array would also help reduce uncertainty in the 526

between-river movement parameter, though if PIT array detection is already relatively high this 527

would likely be less effective than increasing the mark rate in the sampling river. Additionally, 528

mark-recapture data from the detection river (if such data exist) could be incorporated into the 529

model by adding additional detection states in the detection river. Supplemental mark-recapture 530

sampling in the detection river (or alternatively, information about downstream detections) 531

would allow for estimation of detection river survival, thus permitting evaluation of potential 532

tradeoffs in survival and movement. This would also allow for modelers to estimate the number 533

of dispersers in the detection river at any period in time. 534

In our applied example, the estimate of 226 RBT (95% CI: 127-370) dispersers accounts 535

for both marked and unmarked fish that originate from sampled and unsampled CR river 536

segments. The degree to which 226 RBT in the LCR pose a threat to endangered humpback 537

chub is debatable. Yard et al. (2011) evaluated the effects of RBT predation on juvenile 538

humpback chub in the CR, and observed that RBT diet consisted mainly of drifting invertebrates 539

and only a small proportion of fish. This would suggest that RBT are a major threat to 540

humpback chub only when RBT population density is high. A more recent study by Yackulic et 541

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al. (In Review) suggests that RBT display a moderate negative effect on juvenile humpback chub 542

survival. If the number of RBT in the LCR continues to increase over time, additional research 543

assessing the RBT diet composition in the LCR may help determine the risk that RBT pose to 544

endangered humpback chub. 545

Fish movements can be attributed to spawning behavior (Kaeding and Zimmerman 1983; 546

Lucas and Batley 1996; McAda and Kaeding 1991), environmental variables such as flow 547

and(or) temperature (Albanese et al. 2004), and(or) food availability (Osmundson et al. 1998). 548

In particular, differences in water temperatures in the CR and LCR may influence RBT 549

movement, as water temperatures in the CR are typically colder and more seasonally stable than 550

the LCR. Thus, from November to April, mean monthly water temperatures in the LCR (~13-18 551

°C) are closer to the 17.2°C thermal optima value (which is contingent on food availability) for 552

RBT (Hokanson et al. 1977) compared to CR temperatures (~8-12 °C). However, from June to 553

September, LCR water temperatures typically exceed 20 °C while CR water temperatures remain 554

much cooler (11-14 °C). This suggests the LCR may be a harsh environment for RBT during 555

summer months, as previous studies have documented marked decreases in RBT survival and 556

growth occurring between 21-23 °C (Ebersole et al. 2001; Ojolick et al. 1995). 557

Although other studies have used PIT array data or other remote tagging technologies to 558

inform abundance estimates, this is the first study (to our knowledge) to use PIT array detections 559

to directly estimate disperser abundance (i.e., abundance of both marked and unmarked 560

dispersers) when mark-recapture sampling and PIT array detections occur in different rivers, and 561

the number of unmarked fish passing over PIT arrays is unknown. Previously, Pearson et al. 562

(2015) presented abundance estimates for humpback chub in the Little Colorado River (LCR) 563

during periods when PIT array detections and mark-recapture sampling occurred at the same 564

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time; however, the current paper differs from Pearson et al. (2015) because MUX detections and 565

mark-recapture sampling occur in two different rivers. Similarly, Dudgeon et al. (2015) also 566

combined mark-recapture sampling with acoustic receiver detections to improve precision of 567

survival estimates of broadnose sevengill sharks (Notorhynchus cepedianus) in Norfolk Bay, 568

Tasmania. Using Monte Carlo simulation, these survival estimates were then integrated into a 569

Jolly-Seber model to estimate abundance. Fell et al. (2013) estimated abundance from PIT array 570

data, but the study used supplemental information in the form of fish resistivity counter data to 571

estimate the number of unmarked fish that swam over PIT arrays. 572

The modeling approaches described in the current paper provide a useful framework for 573

estimating abundance by combining mark-recapture data with autonomous PIT technologies. 574

Accordingly, the current study helps illustrate that autonomous PIT detection technologies 575

represent a powerful tool that, when paired with conventional sampling trips, can be used to 576

improve population models. In order for remote PIT technologies to reach their full potential, 577

however, studies must develop innovative statistical methods to integrate continuous PIT 578

detection data into mark-recapture models (Cooke et al. 2013). We hope that, by providing a 579

methodological framework to estimate the number of migrants or dispersers in a population, the 580

current study will advance the use of remote PIT arrays for answering ecological questions and 581

informing management. 582

583

584

585

586

587

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492-502. 721

722

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Table 1. List of parameters and mathematical values. State-specific parameters include both 723

sampling and detection rivers. 724

725

726

Parameter Type Description

di,j scalar Density of displacement from segment i to j over one time interval in sampling river

θi,j scalar Probability of displacement from segment i to j over one time interval in sampling river

τ scalar Scale parameter for Cauchy distribution used to describe within-river displacement

x0 scalar Location parameter for Cauchy distribution used to describe within-river displacement

C scalar or vector Index for which river segment(s) are considered part of the confluence

α scalar Probability of moving from confluence into the detection river

pant1 scalar Probability of detection for array 1

pant2 scalar Probability of detection for array 2

pMUX scalar Probability of MUX detection (i.e., probability of being detected on at least one array)

Nt vector Abundances of fish in each state at time t

St vector Survival probabilities of fish in each state at time t

Mt matrix State transition probabilities of fish in each state at time t

K scalar Index corresponding to start of confluence segments

N scalar Index corresponding to end of confluence segments

Z scalar Number of spatial states in the sampling river

t1 scalar Index for first month of PIT array detection period

t2 scalar Index for last month of PIT array detection period

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Table 2. Number of individual rainbow trout (Oncorhynchus mykiss) marked during sampling 727

trips. Rainbow trout were captured in three reaches in the Colorado River: Segments 1-3 728

(located 86.6-91.9 km downstream of Glen Canyon Dam (GCD)), Segment 19 (122.0-123.6 km 729

downstream of GCD), and Segment 22 (127.1-129.6 km downstream of GCD). Fish were 730

marked using passive integrated transponder (PIT) tags. 731

732

Trip Segment(s)

PIT tag

marks

PIT tag

recaptures

April 2012 1-3 554 0 19 308 0 22 138 0

July 2012 1-3 775 72 19 276 65 22 173 27

September 2012 1-3 617 222 19 214 91 22 193 43

January 2013 1-3 794 194 19 266 83 22 270 65

April 2013 1-3 733 155 19 325 95 22 83 53

July 2013 1-3 719 166 19 336 98 22 228 54

September 2013 1-3 492 251 19 320 161 22 293 68

January 2014 1-3 984 216 19 529 124 22 358 60

April 2014 1-3 977 184 19 311 82 22 132 66

July 2014 1-3 752 183 19 293 138 22 308 88

September 2014 1-3 877 347 19 496 206 22 371 87

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Table 3. Detection history of passive integration transponder (PIT) tag array detections of unique 733

rainbow trout, Oncorhynchus mykiss, that swam from Colorado River into the Little Colorado 734

River from October 2013 to April 2014. 735

736

737

738

Month Year Detection Frequency

Oct 2013 upstream array only 2

Nov 2013 both arrays 5

Nov 2013 downstream array only 6

Nov 2013 upstream array only 7

Dec 2013 both arrays 1

Dec 2013 downstream array only 2

Dec 2013 upstream array only 4

Feb 2014 downstream array only 1

Feb 2014 upstream array only 3

Mar 2014 upstream array only 4

Apr 2014 upstream array only 1

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Table 4. Estimated and derived parameter estimates from a population model describing non-739

native rainbow trout (Oncorhynchus mykiss) in the Colorado and Little Colorado Rivers (CR and 740

LCR, resp.) 741

742

743

Parameter Description Mean

95%

confidence

interval

pMUX Conditional detection probability of PIT array 0.598 0.379-0.800

pSep Sep 2013 capture probability in river segment 19 0.283 0.244-0.325

NLCR LCR RBT abundance 226 127-370

pant1 Conditional detection probability of upstream array 0.458 0.254-0.679

pant2 Conditional detection probability of downstream array 0.195 0.156-0.241

τ Cauchy scale parameter 0.195 0.185-0.206

S1-19, winter Monthly survival probability- segments 1-19 - Oct to Apr 0.910 0.894-0.923

S20-22,winter Monthly survival probability- segments 20-22 - Oct to Apr 0.815 0.786-0.841

S1-19, summer Monthly survival probability - segments 1-19 - Apr to Oct 0.967 0.952-0.979

S20-22,summer Monthly survival probability - segments 20-22 - Apr to Oct 0.928 0.895-0.954

α Probability of movement from CR confluence to LCR 0.010 0.006-0.017

744

745

746

747

748

749

750

751

752

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Figure 1. Conceptual diagram illustrating the within-river and between-river movement 753

probabilities estimated by the multi-site model. Here, the sampling river has six segments, and 754

the PIT array is located in a smaller tributary (hereafter detection river). The top row shows the 755

probability of within-river displacement from segment i to segment j and the probability of 756

between-river movement from segment i into the detection river. These values were calculated 757

from a Cauchy distribution with location = 0 and scale = 0.5, and the between-river movement 758

probability was set to 5%. The bottom row illustrates the mathematical expressions used to 759

calculate movement probabilities, with θi,j being the within-river displacement probability from 760

segment i to j, and the between-river movement probability equal to α. 761

762

Figure 2. Map of rainbow trout (Oncorhynchus mykiss) sampling locations in the lower Colorado 763

River (from 86.6 to 129.6 kilometers downstream of Glen Canyon Dam). The map divides the 764

Colorado River into 22 river segments, with each reach measuring roughly 2 km in length, and 765

the river flows downstream to the south. River segments include both sampled (numbered using 766

large, black font) and unsampled (numbered using small, gray font). Additionally, the map 767

illustrates the location of a remote PIT tag multiplexer array system (i.e., MUX) that detected 36 768

unique rainbow trout between October 2013 and April 2014. 769

770

Figure 3. Sensitivity of rainbow trout (Oncorhynchus mykiss) abundance estimates in the Little 771

Colorado River (NLCR) to model parameters: winter survival of river segments 1-19 (S1-19), 772

winter survival of river segments 20-22 (S20-22), the within-river displacement scale parameter of 773

the Cauchy distribution (τ), September 2013 capture probability (pSep), and the between-river 774

movement probability (α). Error bars represent the range of NLCR values that are calculated by 775

increasing and decreasing each parameter value by two times its standard error. 776

Figure 4. Relative bias of disperser abundance estimates from a multi-state mark-recapture 777

model quantifying the number of migrant fish moving between two different rivers. Bias 778

estimates were obtained by simulating 300 datasets using parameter values from the model 779

describing movement of non-native rainbow trout Oncorhyncus mykiss between the Colorado 780

River and Little Colorado River. Relative bias was calculated as the true value minus the 781

estimated value divided by the true value of migrant abundance. 782

783

784

785

786

787

788

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Figure 1. Conceptual diagram illustrating the within-river and between-river movement probabilities estimated by the multi-site model. Here, the sampling river has six segments, and the PIT array is located

in a smaller tributary (hereafter detection river). The top row shows the probability of within-river

displacement from segment i to segment j and the probability of between-river movement from segment i into the detection river. These values were calculated from a Cauchy distribution with location = 0 and scale

= 0.5, and the between-river movement probability was set to 5%. The bottom row illustrates the mathematical expressions used to calculate movement probabilities, with θi,j being the within-river

displacement probability from segment i to j, and the between-river movement probability equal to α.

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Figure 2. Map of rainbow trout (Oncorhynchus mykiss) sampling locations in the lower Colorado River (from 86.6 to 129.6 kilometers downstream of Glen Canyon Dam). The map divides the Colorado River into 22 river segments, with each reach measuring roughly 2 km in length, and the river flows downstream to the

south. River segments include both sampled (numbered using large, black font) and unsampled (numbered using small, gray font). Additionally, the map illustrates the location of a remote PIT tag multiplexer array

system (i.e., MUX) that detected 36 unique rainbow trout between October 2013 and April 2014.

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Figure 3. Sensitivity of rainbow trout (Oncorhynchus mykiss) abundance estimates in the Little Colorado River (NLCR) to model parameters: winter survival of river segments 1-19 (S1-19), winter survival of river

segments 20-22 (S20-22), the within-river displacement scale parameter of the Cauchy distribution (τ),

September 2013 capture probability (pSep), and the between-river movement probability (α). Error bars represent the range of NLCR values that are calculated by increasing and decreasing each parameter value

by two times its standard error.

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Figure 4. Relative bias of disperser abundance estimates from a multi-state mark-recapture model quantifying the number of migrant fish moving between two different rivers. Bias estimates were obtained by simulating 300 datasets using parameter values from the model describing movement of non-native

rainbow trout Oncorhyncus mykiss between the Colorado River and Little Colorado River. Relative bias was calculated as the true value minus the estimated value divided by the true value of migrant abundance.

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Appendix A.

Single-site open population model

Here we present the single site approach, which is best for systems where fish

distributions are highly localized near the confluence of sampling and detection rivers. The

assumptions of this approach are similar to that of the multi-site model referred to in the main

paper. The single site model requires multiple visits to one sampling site in the sampling river

that is centered on the confluence of sampling and detection rivers (hereafter confluence site).

This model uses a non-parametric approach to model within-river displacements of the

confluence site superpopulation. We define this superpopulation as the group of fish that could

potentially be present in the confluence site. In other words, the superpopulation is comprised of

individuals that are permanent residents of the confluence site as well as fish that may move

between the confluence site and other reaches in the sampling river. We assign two spatial states

to the sampling river: 1) an observable sampling river state (OSR) for fish in the confluence site

and therefore susceptible to capture, and 2) an unobservable sampling river state (USR) for fish

that move from the confluence site into other unsampled segments of the sampling river.

Similarly, we assigned two spatial states for the detection river : 1) an observable detection river

state (ODR) to represent dispersers that are susceptible to PIT array detection, and 2) a detection

river unobservable state (UDR) for fish that had moved upstream of the PIT array. We include

an example state transition matrix (M) below, where rows and columns represent states at times t

and t+1, respectively, and values in italics represent transition probabilities. The indexing of

rows/columns is as follows: OSR (1), USR (2), ODR (3), and UDR (4).

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(A1) �� = �(1 − ��,�� − ��,��) ��,�� ,�� 0��,�� (1 − ��,��) 0 00 0 0 10 0 0 1�

In other words, movement parameters are constrained so that fish in the OSR state can

either remain in their current state, transition to the USR state, or transition to the ODR state.

Fish in the USR state can either remain in the USR state or transition into the OSR state. Similar

to the multi-site model, fish in the ODR state must transition into the UDR the following month

and remain in the UDR state permanently. Thus, out of 16 potential movement parameters (not

including transitions to and from the dead state), 3 are estimated, 2 are calculated by subtraction

(i.e., transition probabilities from each state must sum to 1), 9 are fixed at 0, and 2 are fixed at 1.

Importantly, including unobservable states in the sampling river accounts for temporary

emigration, which helps with survival estimation. For a description of detection probability

parameterization, refer to the main paper.

Determining the number of fish in the unobservable sampling river state

For the single-site model, the number of fish in the USR state before the PIT array

detection period (hereafter referred to as ��where t1-1 is the time period before the start of the

PIT array detection period) will influence the disperser abundance estimate. One option is fixing

��; however, this value is rarely known in practice. Thus, we present one method that can be

used to estimate ��, given no recruitment occurs during the open PIT array detection period

(apart from immigration and emigration). This method assumes that abundances are known in

all spatial states (except for the USR state) at time t1-1. If the study occurs at the onset of

migration, it may be reasonable to assume 0 fish in the two detection river states (i.e., NODR,t1-1 =

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NUDR,t1-1 = 0), and the estimate of fish in the OSR state can be obtained by dividing catch by

capture probability from mark-recapture sampling. Furthermore, this method requires an

estimate of the number of fish in the OSR state at some future time (tf), which can also be

obtained by dividing catch by capture probability at some later mark-recapture event within the

MUX detection period. First, we define q as the elements of the matrix A:

(A2) � = ∏ [�(��)]�� !��

Where Si and Mi are the survival vector for each state at time i and the state transition matrix at

time i, respectively. Thus, the matrix A represents the probability of transitioning between states

during multiple time steps (specifically, between t1 and tf). Then, the estimate of η is:

(A3) �̂�� = #$%&,' �($%&,$%&#$%&,'� ()%&,$%&

This value of �̂�� can then be used in the *�� vector of the derived abundance equation

described in the main body of the paper. Note that, unlike in the multi-site abundance equation,

SR abundance is not interpolated across sites in the single-site model.

Simulation

We simulated 300 data sets to evaluate the single site model using parameter values from

the multi-site model for RBT in the LCR. Capture histories from the multi-site simulations were

altered to only keep fish that were captured in the two confluence segments. We fit the single-

site model to these simulated data and calculated disperser abundance under three scenarios: 1)

USR zero (where we assumed zero fish in USR state in the first month), and 2) USR known

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(where the number of USR fish at the beginning of the detection period was known based on

simulations), 3) USR-estimated (where we used methods described above to estimate the number

of fish in the USR state at the start of the MUX detection period). We fit the single site model to

our data for all three scenarios, calculated derived abundance estimates from catch data and

model parameters, and computed relative bias as the difference between the true disperser

abundance value and the estimated value, divided by the true value.

Simulations illustrated that bias of our modeling approach was dependent on the

sampling scenario. Results of simulations from the single-site model indicated the USR zero

scenario displayed strong negative relative bias (mean = -24.0%, median = -25.4%), and that bias

decreased under the USR known scenario (mean = 6.0%, median = -0.6%; Figure A1).

Compared to the USR zero scenario, bias was slightly reduced under the USR estimated scenario

(mean = -19.7%, median = -22.1%). Collectively, simulation results suggest that the unknown

number of fish in the USR state may produce substantial bias in disperser abundance estimates,

and that this represents one limitation of the single-site approach. Further investigation is

required to determine how these estimators vary with differing detection probabilities,

abundances, and movement probabilities.

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Figure A1. Bias in disperser abundance estimates from simulations of a single site population

model designed to combine autonomous PIT detections with mark-recapture sampling in two

different rivers. Relative bias was calculated by first taking the difference between true and

estimated disperser abundance, then dividing this difference by the true disperser abundance

value. Abundance calculations included three scenarios: assuming zero fish in unobservable

state in the sampling river at the onset of the study (USR zero), 2) assuming known number of

fish in unobservable state in the sampling river at the onset of the study (USR known), and 3)

estimating the number of fish in the USR state (USR estimated). Distributions are based on 300

simulations.

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Figure A1. Bias in disperser abundance estimates from simulations of a single site population model designed to combine autonomous PIT detections with mark-recapture sampling in two different

rivers. Relative bias was calculated by first taking the difference between true and estimated disperser

abundance, then dividing this difference by the true disperser abundance value. Abundance calculations included three scenarios: assuming zero fish in unobservable state in the sampling river at the onset of the study (USR zero), 2) assuming known number of fish in unobservable state in the sampling river at the

onset of the study (USR known), and 3) estimating the number of fish in the USR state (USR estimated). Distributions are based on 300 simulations.

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estimating disperser abundance using open population arrays · 2018. 4. 5. · draft 1 1 title:...

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