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Habitat Assessment and Range Updates for Two Rare Arkansas Burrowing Crayfishes: Fallicambarus harpi and Procambarus reimeri
Cody M. Rhoden, Christopher A. Taylor, and Brian K. Wagner

Southeastern Naturalist, Volume 15, Issue 3 (2016): 448–458

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Southeastern Naturalist C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 448 2016 SOUTHEASTERN NATURALIST 15(3):448–458 Habitat Assessment and Range Updates for Two Rare Arkansas Burrowing Crayfishes: Fallicambarus harpi and Procambarus reimeri Cody M. Rhoden1,*, Christopher A. Taylor1, and Brian K. Wagner2 Abstract - The Ouachita Highlands Freshwater Ecoregion harbors the 6th-highest native crayfish species density in the US and Canada. Many of these species are understudied, and the burrowing crayfishes of this region are of particular interest. We conducted field surveys in the spring of 2014 and 2015 to assess the range and habitat preferences of 2 of Arkansas’ rarest burrowing crayfishes: Fallicambarus harpi (Ouachita Burrowing Crayfish) and Procambarus reimeri (Irons Fork Burrowing Crayfish). Both crayfishes are currently of conservation concern—F. harpi is vulnerable and P. reimeri is endangered according to the American Fisheries Society. Our surveys detected new populations of both species and documented marginal and wider range expansions for F. harpi and P. reimeri, respectively. The preferred habitat for both species was characterized as wet seepage areas with an open canopy, low grasses, and abundant sedges. Our surveys support prior observations that these species are geographically constrained; however, the new populations and range expansion of P. reimeri suggest the American Fisheries Society Endangered Species Committee should reevaluate the conservation status of this species. Introduction North America harbors the highest diversity of crayfishes worldwide (Taylor et al., 2007). Within North America, 22% of the species listed as endangered or threatened in a conservation review (Taylor et al. 2007) were primary burrowing crayfishes. Primary burrowing crayfishes differ from stream-dwelling crayfishes in their life-history traits; they spend most of their life cycle underground, leaving their burrows only to forage and find a mate (Hobbs 1981). The Ouachita Highlands Freshwater Ecoregion is the 6th-most diverse freshwater ecoregion for native crayfishes in the US and Canada (Moore et al. 2013). Within the Ouachita Highlands Freshwater Ecoregion, the Ouachita Mountains Ecoregion (OME; Woods et al. 2004) of southwestern Arkansas harbors the highest diversity of primary burrowing crayfishes in the state with 6 species: Fallicambarus harpi Hobbs & Robison (Ouachita Burrowing Crayfish), F. jeanae Hobbs (Daisy Burrowing Crayfish), F. strawni (Reimer) (Saline Burrowing Crayfish), Procambarus liberorum Fitzpatrick (Osage Burrowing Crayfish), P. parasimulans Hobbs & Robison (Bismark Burrowing Crayfish), and P. reimeri Hobbs (Irons Fork Burrowing Crayfish) (Hobbs 1989). 1Illinois Natural History Survey University of Illinois Urbana-Champaign, 607 East Peabody Street, Champaign, IL 61820. 2Arkansas Game and Fish Commission, 915 East Sevier Street Benton, AR 72015. *Corresponding author - codyrhoden@gmail.com. Manuscript Editor: Bronwyn Williams Southeastern Naturalist 449 C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 The narrowly endemic nature of North American crayfishes is well documented (Morehouse and Tobler 2013, Page 1985, Simmons and Fraley 2010, Taylor et al. 2007), and Robison et al. (2008) specifically addressed the primary burrowing crayfishes in Arkansas. Endemic species are vulnerable to extirpation because they occur at such a constrained geographic scale; thus, it is important to accurately describe their habitat preferences and range to ensure the persistence of local populations through management of suitable habitat and monitoring of existing populations. Two primary burrowing crayfishes in Arkansas in need of these conservation assessments are Fallicambarus harpi and Procambarus reimeri. Little information has been published regarding the habitat preferences and range assessments of F. harpi and P. reimeri since the species were first described. Both crayfishes are endemic to the OME in southwestern Arkansas. These species were assessed as endangered (P. reimeri) and vulnerable (F. harpi) based on vulnerability to habitat modification or reduction and because of their restricted range (Taylor et al. 2007). The conservation categories of endangered and vulnerable are based upon determinations by the American Fisheries Society Endangered Species Committee, which followed Williams et al. (1993). These species were included in a recent petition filed by the privately funded Center for Biological Diversity (Tuscan, AZ) for protection under the Federal Endangered Species Act. To assess these conservation concerns, we developed a study with the following objectives: (1) determine the holistic range of F. harpi and P. reimeri within the OME of Arkansas, and (2) refine the description of suitable habitat for b oth species. Target species accounts Fallicambarus (Fallicambarus) harpi. The Ouachita Burrowing Crayfish was described from 2 locations in Pike County, AR, in 1985 by H.H. Hobbs Jr. and H.W. Robison (Hobbs and Robison 1985). Robison and Crump (2004) reviewed and updated the status of this endemic crayfish, and documented 12 new populations in Montgomery, Hot Spring, Garland, and Pike counties, AR. No information has been published relating to the range or habitat requirements of this crayfish since 2004. F. harpi is known from wet seepage areas with abundant sedges, such as roadside ditches and other rights of way (Robison and Crump 2004). Fallicambarus harpi most-closely resembles F. strawni and F. jeanae, but differs by possessing a free, never adnate, cephalic process on the first pleopod of the first-form male (Hobbs and Robison 1985). Procambarus (Girardiella) reimeri. The Irons Fork Burrowing Crayfish was described from 6 locations in Polk County, AR by H.H. Hobbs Jr. in 1979 (Hobbs 1979). Robison (2008) reviewed and updated the status of this Arkansas endemic and determined that the species only occurred in Polk County, in the vicinity of Mena, AR. No other information has been published relating to the range or habitat requirements of this species since 2008. This primary burrowing crayfish constructs relatively simple burrows in sandy clay soil in wet seepage areas and roadside ditches (Hobbs 1979, Robison 2008). Procambarus reimeri most closely resembles P. gracilis (Bundy) (Prairie Crayfish) and P. liberorum, but differs from Southeastern Naturalist C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 450 both by possessing a broader areola and lacking tubercles on the annulus ventralis (Hobbs 1979). Methods We conducted field surveys for these 2 primary burrowing crayfishes in southwestern Arkansas during the spring of 2014 and 2015. We sampled in the spring of both years because this is the time of peak activity for both species (Robison 2008, Robison and Crump 2004) and, thus is the time with the greatest likelihood of species detection. In 2014, we sampled historic locations listed in the Illinois Natural History Survey Crustacean Collection (Champaign, IL), National Museum of Natural History Invertebrate Zoology Collection (Washington, DC), and Arkansas Department of Natural Heritage (Little Rock, AR) records. For each species, we selected historical localities that were accessible and could be validated with geographic positioning information. These searches resulted in 20 unique and accessible locations for F. harpi and P. reimeri. We sampled 13 counties encompassing the known range of both species. From east to west, those counties were Pulaski, Saline, Perry, Garland, Hot Spring, Clark, Yell, Montgomery, Pike, Scott, Howard, Polk, and Sevier in western Arkansas. We predicted the historical and current range of both species using minimum-bounding convex polygons around all localities from historical museum records and new captures from 2015 in ArcMap using the minimum-bounding geometry toolset. In 2014, we set up three to six 50-m transects at and near each historic locality (Rhoden et al. 2016). We placed the first transect at each sampling site where burrows were present, ensuring the first transect was situated at the historic location. After we obtained a GPS location and azimuth at the 0-m mark, we placed a 1-m2- PVC quadrat over the linear transect every 10 m, for a total of six 1-m2 quadrats per 50-m transect. We completed 2–5 additional transects in adjacent habitat at the sampling site. We determined the number of transects sampled at each site based on habitat heterogeneity. We placed fewer transects in homogeneous sites; thus, sampling took less time and we were able to increase the number of sampling sites we could visit during the sampling window. We conducted supplemental sampling in the vicinity (less than 100 m) of each sampling site to verify non-detection observations from transect collections. We used our 2014 capture records to develop species-distribution models (SDMs). The SDMs were constructed with Maxent (Phillips et al. 2006), a presence-only modeling algorithm used to predict the relative occurrencerate of a focal species across a predefined landscape (Fithian and Hastie 2013). In 2015, we sampled a semi-random group of sites based on SDMs for F. harpi and for P. reimeri (Rhoden 2016). The environmental variables used for the SDM analysis consisted of canopy cover, elevation, distance to nearest waterbody, compound topographic-index value (CTI), and solar-radiation value (Table 1). We determined canopy and herbaceous cover with a spherical densiometer (Model-C; Robert E. Lemmon Forest Densiometers, Bartlesville, OK), elevation from the USGS National Elevation dataset (http://nationalmap.gov/elevation.html), and distance to nearest waterbody from the National Hydrology dataset (http://nhd.usgs.gov/). We Southeastern Naturalist 451 C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 followed Evans et al. (2010) to determine CTI and used the National Elevation dataset and ESRI spatial analyst tools to determine solar-radiation values. Our Maxent models were constructed with 30-m-resolution rasters following Peterman et al. (2013) for model construction and evaluation. We thoroughly searched each quadrat along each transect for the presence of burrows and collected the following habitat variables within each 1-m2 quadrat: percent tree-canopy cover, percent herbaceous groundcover, stem density (count of stems within a 100-cm2 quadrat placed within the upper right-hand corner of each 1-m2 quadrat), number of burrows, and the presence or absence of hydrophilic sedges (presence or absence of herbaceous plants having 3-ranked leaves, an angular stem, and a spiked fruiting body; Rhoden et al. 2016) (Table 1). We used a soil probe (AMS 2.2 cm [7/8 in] diameter open-end probe, AMS, Inc., American Falls, ID) to collect 3 evenly spaced soil samples on each transect. We analyzed the soil samples with laser diffraction on a Malvern Mastersizer 3000 (Malvern Instruments, Malvern, UK) to obtain percent composition of sand, silt, and clay for each sample. We analyzed habitat variables using generalized linear mixed models (package lme4; Bates et al. 2014, R Development Core Team 2014) to determine the fine-scale habitat preferences of both burrowing crayfish species (Table 2). The response variable in each model was the number of burrows within each 1-m2 Table 1. Habitat variables, description of each variable, source of each habitat variable, and general statistics from a study of habitat associations of 2 primary burrowing crayfishes in western Arkansas. F. harpi P. reimeri Variable Description Min/max(unit) (μ [sd]) (μ [sd]) Canopy Percent tree-canopy cover 0/100 (% cover) 22.5 (32.5) 22.5 (32.8) Herb Percent herbaceous groundcover 0/100 (% cover) 66.6 (32.4) 71.7 (31.0) Stem Number of stems per 100-cm2 0/169 (number 22.7 (19.7) 25.4 (20.7) sub-quadrat of stems) Sedge Presence of hydrophilic sedge in 1/0 (binary: yes/no) - - quadrat (binary: yes/no) Soil 1, Transformed soil variables - - Soil 2 Elevation Digital elevation model of the study 50.50/818.96 (m) 189.3 (32.8) 304.4 (59.7) site Water dist Euclidean distance to nearest 0/1740.26 (m) 176.3 (118.5) 157.2 (138.0) permanent waterbody across the study site CTI A function of slope and the upstream 2.67/27.58 (index 8.4 (1.7) 8.4 (1.5) contributing area per unit width value) orthogonal to the flow direction (Evans et al. 2010) Solar Incoming solar radiation value (watt 3542.41/64134 6062.9 (105.3) 6071.3 (42.0) h per m2) based on direct and diffuse (watt h/m2) insolation from the unobstructed sky directions Southeastern Naturalist C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 452 quadrat modeled with a Poisson error distribution and log link. To account for potential site effects, we modeled transects nested within sites as a random effect in each model. We scaled and centered all habitat variables by subtracting the variable mean from each respective value and dividing by the standard deviation of that variable. We compared candidate models with Akaike’s information criterion corrected for small sample sizes (AICc; Table 3; Akaike 1974). We examined the relative support for each model and calculated unbiased model-averaged parameter estimates from the top models (ΔAICc < 4) with the package MuMIn (Barton 2014) by means of model selection and averaging methods described by Burnham and Anderson (2002) and Luckacs et al. (2009). We hand-excavated a subset of burrows at each sampling site and along each transect to confirm the identity of the burrow occupant in each occupied quadrat. Table 2. Candidate models and hypotheses tested in the generalized linear mixed-model analysis for Fallicambarus harpi and Procambarus reimeri in Arkansas. The response variable used in each model was burrow abundance in each 1-m2 quadrat. See Table 1 for variable names. Model Variables Hypothesis Mod 1(global) Sedge + canopy + herb + stem Crayfish selection based on quadrat-level wetness + elevation + solar + water_dist characteristics, canopy cover, herbaceous + CTI + soil1 + soil2 community, erosion potential, topographic position, and soil cues Mod 2 Canopy + sedge Crayfish selection based on canopy cover and quadrat-level wetness Mod 3 Sedge + solar + water_dist Crayfish selection based on quadrat-level wetness characteristics and topographic position Mod 4 Sedge + CTI Crayfish selection based on quadrat-level wetness characteristics and topographic position Mod 5 Water_dist + CTI Crayfish selection based on topographic position Mod 6 Solar + CTI Crayfish selection based on topographic position Mod 7 Elevation + water_dist Crayfish selection based on topographic position Mod 8 Sedge + stem Crayfish selection based on herbaceous community Mod 9 Canopy + herb Crayfish selection based on canopy and herbaceous community Mod 10 Herb + soil1 + soil2 Crayfish selection based on herbaceous community and soil cues Mod 11 Canopy + soil1 + soil2 + sedge Crayfish selection based on canopy cover, soil cues, and quadrat-level wetness Mod 12 Canopy + solar + soil1 + soil2 Crayfish selection based on canopy cover, solarradiation potential, and soil cues Mod 13 Canopy + sedge + stem Crayfish selection based on canopy cover, quadratlevel wetness, and erosion potential Mod 14 Soil1 + soil2 Crayfish selection based on soil cues Mod 15 Canopy Crayfish selection based on canopy cover Mod 16 Solar Crayfish selection based on solar-radiation potential Southeastern Naturalist 453 C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 Hand excavation consisted of using a hand shovel to slowly dig around the burrow entrance to follow the main burrow-tunnel, feeling for crayfish as we worked our way through the burrow complex. We chose this method because of its high success-rate and the limited amount of time required at each burrow location (Ridge et al. 2008). We collected, preserved in 70% ethanol, and deposited in the Illinois Natural History Survey Crustacean Collection voucher specimens of all crayfish species detected from all occupied sites with burrows present. Results During the 2014 and 2015 field surveys for F. harpi and P. reimeri, we sampled 1392 quadrats across 232 transects at 180 sites in western Arkansas (Fig. 1). All of the historical localities sampled were less than 50 m from primary, secondary, and tertiary roadways. In 2014, we sampled adjacent habitat less than 100 m from the historiccollection sites and out of the right of way in an attempt to survey different habitat types spatially available to individuals at each site. In 2015, all of the sites sampled were in the rights of way of primary, secondary, and tertiary roadways. We found F. harpi in 79 quadrats across 16 sites and P. reimeri in 90 quadrats across 24 sites. During the field sampling we encountered 14 other crayfish species: Cambarus ludovicianus Faxon (Painted Devil Crayfish), F. fodiens (Cottle) (Digger Crayfish), F. jeanae, F. jeanae x F. strawni, F. strawni, P. (Girardiella) sp., P. acutus (Girard) (White River Crayfish) P. curdi Reimer (Red River Burrowing Crayfish), P. liberorum, P. parasimulans, P. regalis Hobbs & Robison (Regal Burrowing Crayfish), P. simulans (Faxon) (Southern Plains Crayfish), P. tenuis, and P. tulanei Penn (Giant Bearded Crayfish). Fallicambarus harpi Of the 91 sites sampled for F. harpi, we found the species at all historical sites (n = 11) and 6% of selected sites sampled in 2015 (n = 5) (Fig. 2). The longest- Table 3. Model name, number of model parameters (K), Akaike’s information criterion adjusted for small sample size (AICc), difference in AICc (ΔAICc), Akaike weights (wi), and log liklihood (LL) for the top habitat models (ΔAICc < 4) from a suite of variables modeled with a generalized linear mixed-model analysis for 2 primary burrowing crayfish species, Fallicambarus harpi and Procambarus reimeri in Arkansas. See Tables 1 and 2 for a description of each model and the variables included. Species/model K AICc ΔAICc wi LL Fallicambarus harpi Mod 12 7 469.67 0.00 0.35 -227.80 Mod 3 6 470.32 0.65 0.25 -229.10 Mod 16 4 471.28 1.61 0.15 -231.60 Mod 6 5 471.66 1.99 0.13 -230.80 Mod 1 13 473.36 3.70 0.05 -223.40 Procambarus reimeri Mod 2 5 543.56 0.00 0.62 -266.74 Mod 13 6 545.59 2.03 0.22 -266.73 Mod 11 7 546.59 3.03 0.14 -266.21 Southeastern Naturalist C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 454 persisting population of F. harpi was first discovered in 1982. We captured F. harpi along 1 transect with P. parasimulans present and 1 transect with P. simulans; both transects were removed from the dataset prior to the final statistical analysis. We calculated a historical range of 147 km2 and a current range of 265 km2 for F. harpi. The modeling analysis showed that the number of burrows in a quadrat was negatively associated with canopy cover (Table 4). Burrows were generally present in quadrats with little to no canopy cover (n = 79, μ = 4.8%, σ = 18.7). The presence of hydrophilic sedges and amount of solar radiation were positively associated with the number of burrows in a quadrat (Table 4). Sedges were present in 76% of the quadrats with burrows present (n = 60) and 40% of quadrats where burrows were absent (n = 247). Procambarus reimeri Of the 89 sites sampled for P. reimeri, we found the species at all but one historical site sampled (n = 8) and 20% of the selected sites sampled in 2015 (n = 16) (Fig. 2). The oldest resampled site continuing to harbor populations of P. reimeri was first discovered in 1973. We captured Procambarus reimeri along 1 transect where P. tenuis was present; this transect was removed from the dataset prior to the final statistical analysis. We calculated a historical range of 80 km2 and a current Figure 1. Sampling locations for 2 species of primary burrowing crayfish, Fallicambarus harpi and Procambarus reimeri, in western Arkansas in the spring of 2014 and 2015. Southeastern Naturalist 455 C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 Figure 2. Historical and current capture locations of Fallicambarus harpi (right) and Procambarus reimeri (left) in western Arkansas. Table 4. Unbiased model-averaged parameter estimates of the top models (Table 3) for 2 primary burrowing crayfish species (Fallicambarus harpi and Procambarus reimeri) in Arkansas. See Table 2 for a description of the variables included. Sedge1 = presence of sedge in quadrat, CL = confidence limits. Variable Model-averaged estimate (SE) 95% CL Fallicambarus harpi Canopy -0.84 (0.38) -1.59, -0.10 Sedge1 0.43 (0.21) 0.03, 0.83 Solar 2.24 (0.68) 0.91, 3.57 Stem 0.07 (0.14) -0.21, 0.35 Herb -0.05 (0.20) -0.44, 0.34 Soil1 0.06 (0.50) -0.93, 1.04 Soil2 -0.48 (0.36) -1.19, 0.24 Water_dist -0.37 (0.52) -1.39, 0.64 CTI 0.12 (0.10) -0.07, 0.30 Elevation -0.72 (0.70) -2.10, 0.65 Intercept -6.27 (1.05) -8.33, -4.21 Procambarus reimeri Canopy -0.83 (0.26) -1.34, -0.32 Sedge1 2.24 (0.47) 1.33, 3.15 Stem 0.01 (0.12) -0.23, 0.25 Soil1 -0.01 (0.55) -1.01, 1.08 Soil2 0.51 (0.61) -0.68, 1.70 Intercept -5.54 (0.85) -7.21, -3.88 Southeastern Naturalist C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 456 range of 1467 km2 for P. reimeri. Modeling revealed that the number of burrows in a quadrat was negatively associated with canopy cover (Table 4). Burrows were generally present in quadrats with little to no canopy cover (n = 90, μ = 4.1%, σ = 12.8). The presence of hydrophilic sedges was positively associated with the number of burrows in a quadrat (Table 4). Sedges were present in 94% of the quadrats with burrows present (n = 85) and present in 44% of quadrats where burrows were absent (n = 269). Discussion Our study assessed the range and habitat of 2 primary burrowing crayfish species in southwestern Arkansas. We recorded F. harpi at all of the historical sites sampled and P. reimeri at all but one historical site. Although we did not visit all historical locations, historical sites sampled encompassed the (then) known range of both species. Our sampling scheme resulted in most (91% and 93% for F. harpi and P. reimeri, respectively) of the sampling sites to be near or in the rights of way of primary, secondary, and tertiary roadways. Although we conducted most of our sampling in these highly altered habitats, our field surveys revealed new populations of both species. Both species showed common patterns of habitat preference, which correspond with early descriptions of the habitats preferred by both species (Hobbs 1979, Hobbs and Robison 1985). Based on our findings, we recommend that F. harpi retain its conservation-status category of vulnerable and that scientists re-evaluate the conservation category of endangered for P. reimeri. Our samples documented a marginal range expansion for F. harpi of ~118 km2 and 5 new populations, including 1 found in a county (Clark) not previously documented within the species’ range (Fig. 2). Our records document a greater range expansion for P. reimeri of ~1387 km2, including 16 new populations. We discovered 1 of these populations in a county (Montgomery) previously not considered in the range of P. reimeri (Fig. 2). Despite our findings, the documented ranges for F. harpi and P. reimeri are relatively small compared to most of the other primary burrowing crayfishes in the OME (e.g., F. jeanae, F. strawni, P. liberorum, and P. parasimulans; Taylor et al. 2007). Although new populations of both species may be discovered on the periphery of the range reported here, we believe the updated range extent is accurate based on the breadth of sites sampled across the OME (Fig. 1). We captured P. reimeri in Arkansas less than 1 km from the Oklahoma state line (Fig. 2), a discovery which led us to suspect that P. reimeri may occur in Oklahoma; however, more field surveys are needed to assess this possibility . Our modeling and field observations revealed both F. harpi and P. reimeri showed common patterns of habitat preference. These burrowing crayfishes were most abundant in treeless, wet seepage areas with an abundance of low grasses and sedges. The soil composition at 92% and 94% of quadrats occupied by F. harpi and P. reimeri, respectively, was loam and silt loam (Soil Survey Division Staff 1993). The burrows of F. harpi and P. reimeri that we excavated in the field were complex, 0.5–1-m deep, and connected to groundwater. Our results highlight the importance of these wet, open-canopy habitats as a preferred environment f or both species. Southeastern Naturalist 457 C.M. Rhoden, C.A. Taylor, and B.K. Wagner 2016 Vol. 15, No. 3 Our success in finding F. harpi at all of the historical sites sampled and 5 new populations indicates that this crayfish is geographically constrained, but continuing to persist. Its total range is estimated to be 265 km2. These results suggest F. harpi should remain categorized as vulnerable due to its restricted range (Taylor et al. 2007). Our documentation of P. reimeri at all but one historical site and 16 new populations, one of which was >50 km away from the previously known range, indicates this crayfish is somewhat geographically constrained, but is more widespread than originally thought. We estimate its total range at ~1467 km2. This finding suggests that a re-evaluation of the conservation status given by Taylor et al. (2007) is warranted for P. reimeri. Acknowledgments We thank Josh Seagraves, Andrea Daniel, Allison Fowler, Benjamin Thesing, and Justin Stroman for permitting and field assistance, with special appreciation to Sarah Tomke and Dan Wylie for field assistance. This work was funded with a State Wildlife Grant from the Arkansas Game and Fish Commission. Thanks to Mike Dreslik, Robert Schooley, and Bill Peterman for help with initial study design and analysis, and Henry Robison for his extensive collections of these species that formed the basis for our study. Literature Cited Akaike, H. 1974. 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