Length in assessing status of freshwater fish populations: A review
Abstract
Objective
Effective policy formulation regarding the conservation of freshwater fish necessitates an understanding of water-specific prevailing conditions and trends. Assessing fish populations in inland waters is difficult and expensive because there are many independent systems that need to be evaluated. Therefore, numerous freshwater systems are beset by insufficient data and the lack of systematic assessments of their status. To alleviate this deficiency, the objective of this study was to review length-based metrics that may have utility in evaluating the well-being of freshwater fish populations.
Methods
Length measurements can serve as proxies for a range of ecological and population dynamics attributes that are essential for the effective management of fish and associated fisheries. A review of the historical development of length measurements in fish conservation is provided, along with an examination of the potential biases that may arise from the use of lengths in practical contexts. In addition, we examine techniques that enable the spatial and temporal visualization of length data sets, as well as a range of indices and metrics that can be computed using length measurements.
Result
Building populations assessments around length may be a cost-effective strategy that allows a first cut at managing a large number of waters. Length-based assessments can signal if management intervention is necessary, if management policies are yielding the intended outcome, or if surveys beyond mere length are necessary.
Conclusion
Our review indicates that length offers a straightforward and efficient approach to evaluate the status of fish populations in inland systems. We encourage pursuing additional study and to this end propose specific areas for investigation.
INTRODUCTION
Length, defined as the straight-line distance between two endpoints, is an intuitive metric that is easily obtained and may be measured with little guidance. Next to counts, length is likely the most common information tracked by fish managers. As an illustration, our review of papers published in five prominent fisheries journals, using Scopus, indicated that in the years 2000 through 2020, fish length was mentioned in 81% of articles published in the North American Journal of Fisheries Management, 80% in Transactions of the American Fisheries Society, 73% in Fisheries Research, 72% in Fisheries Management and Ecology, and 44% in Fish and Fisheries. The ubiquitous use of fish length as a metric is predicated not only on its inherent simplicity and uncomplicated interpretation, but also for its relevance to ecological concepts, its direct connection to management activities, and its ease of communication.
Length measurements can serve as proxies for a range of ecological and population dynamics attributes that are essential for the effective management of fish and their related fisheries. For example, fish lengths indicate ecological attributes, such as life stage, trophic position, mobility, fecundity, and egg quality (Lauer et al. 2005; Hixon et al. 2014). Lengths can also provide insight into population dynamics. For example, a lack of smaller lengths can indicate recruitment deficiencies, while infrequency of larger lengths might suggest slow growth or high mortality. Length may also be considered along with covariates (e.g., weight, counts, sex, length at maturity, habitat, seasonality) to disentangle observed patterns in population dynamics and community ecology and to interpret how fish respond to environmental stressors. Length can also serve as an indicator of the status of a fishery. Length dynamics play a critical role in the application of conservation actions, such as harvest regulations (Neumann and Allen 2007). Fish lengths are readily observed and measured by fishers, and therefore they are also a critical communication link between fish managers and their constituencies. The length of fish is thus an indicator of the ecological processes of a population and can be used as a basis for population assessment and management.
Freshwaters are effectively isolated islands embedded in continental land masses (Magnuson et al. 1998). There are an estimated 100 million lakes and reservoirs >0.2 ha covering the nonglaciated areas of the planet (Verpoorter et al. 2014). Correspondingly, the surface area of running water is nearly 0.8 million square kilometers (Allen and Pavelsky 2018). Fish populations in only a few of these water bodies can be monitored, and this select group is mostly monitored to track fish species composition, abundance, and size structure. More comprehensive data pertaining to age structure, recruitment patterns, growth rates, and mortality rates required to apply statistical catch-at-age models (Methot and Wetzel 2013) are typically prohibitively expensive to obtain in freshwaters, especially in regions where many independent waters are managed (Lester et al. 2003; Trudeau et al. 2021).
We conducted a review of length metrics appropriate for assessing the status of fish populations in freshwater ecosystems. Our focus was on parameters that describe population status rather than on stock assessment models. We begin with a history of the use of lengths in fish conservation, describe methods for measuring length, and consider several biases associated with length applications. Methods for facilitating the spatial and temporal visualization of length data sets are considered. Estimates of the sample sizes required to adequately estimate population length characteristics are also reviewed. Lastly, various metrics and indices that can be calculated with length measurements are considered, focusing on those that facilitate understanding of various aspects of fish ecology, population dynamics, and vital rates. Our aim was to provide a succinct review of the use of lengths and when they may be used in place of more comprehensive surveys for the assessment of fish populations in freshwater ecosystems.
HISTORICAL BACKGROUND
According to Howe (2002), the scientific literature first employed length as a measure of fish body size in the 18th century in the works of Linnaeus (1758) and Artedi (1788–1793). Length had probably been used to indicate fish size earlier, although rarely in written documents or in standard form. The metric decimal system was first adopted in France in the late 18th century (Sant'Ambrogio and Dejours 1995). Before then, measurements were mostly unstandardized and varied regionally. In the 19th century, ichthyology books proliferated focusing largely on species discoveries and accounts, in which length played a prominent role in species descriptions (Howe 2002). The standardization developments in the late 18th and early 19th centuries might have fueled more intensive use of length in the natural sciences.
The use of lengths in the fishery literature appears to have begun sometime in the late 19th century with the works of T. W. Fulton, E. W. L. Holt, C. G. J. Petersen, and J. T. Cunningham as reported by Allen (1917), Cushing (1976), and Jackson (2007). The use of length to estimate the age of fish, and incidentally of weight in conjunction with length to estimate condition, started being applied near the beginning of the 20th century (Hecht 1916; Froese 2006; Jackson 2007). C. G. J. Petersen is thought to be the first to use modes in length-frequency distributions for aging fish (Jackson 2007), although these length-based methods were also applied by Edser (1908) and Heincke (1913). Fisheries biologists around this time were also working to establish relationships between length and weight so they could more accurately study growth and condition (Nash et al. 2006).
MEASURING LENGTH
The measurement of fish length has evolved into an integral aspect of fish population assessment in freshwaters. The three most common measurements used to describe fish length are total length, standard length, and fork length (Figure 1). According to Howe (2002), it appears that the sole measurement of length utilized during the 18th century was total length. Artedi (1788–1793) provided descriptions in which total length was defined as the measurement to the end of the tail, a definition that aligns with contemporary use. The establishment of uniform length measurement techniques did not emerge in scholarly literature until the mid to late 19th century (Howe 2002). Presently, the measurement of total length is conducted using two distinct methods. The first one involves measuring from the foremost point of the longest jaw, while the fish's mouth remains closed, to the rearmost section of the tail, while the caudal rays are compressed tightly (maximum total length; Holt 1959). The second method entails measuring total length up to the tip of the tail in its “natural” posture (natural total length; Hile 1948). Standard length is most commonly defined as the measurement from the foremost point of the fish's head to the termination of the vertebral column in proximity to the caudal rays (Jennings et al. 2012). Reportedly, measurement of standard length has not been universally standardized, as there are at least eight distinct approaches for measuring standard length (Hile 1948; Howe 2002). Fork length is commonly measured as the length from the anterior tip of the longest jaw to the median point of the caudal fin (Laevastu 1965).
Each of these length measurements has its applications. However, given that it is easy to obtain for most body shapes and yields the lowest measurement error, maximum total length may be the method most often used for measuring the length of fish with standard body shapes (Carlander and Smith 1945; Hile 1948; Önsoy et al. 2011). Compared to maximum total length, standard length is not as easy to expeditiously measure in the field, as it is difficult to pinpoint the precise end of the vertebral column (Carlander and Smith 1945). Other studies have suggested that there is no one “best” length to measure on fish in terms of reducing variability and bias in measurements and that technique and consistency are more essential than the metric itself for producing reliable lengths (Kahn et al. 2004). There are a few notable exceptions in which certain techniques for measuring length may not be applicable or may yield inconsistent results with specific species (Table 1). Any method may be applicable for length-based assessment procedures so long as the same length is used throughout and an accurate record of which length was used is kept (Gulland and Rosenberg 1992).
| Exception | Solution | Source |
|---|---|---|
| Preserved specimens often lose their tail or other appendages, resulting in inconsistent length measurements | Use standard length | Thorstad et al. (2007) |
| Preserved specimens often shrink, resulting in inaccurate measurements | Depending on study objectives, formalin can be the best choice for minimizing fish shrinkage | Paradis et al. (2007) |
| The species seasonally erodes tail by fanning substrate during nesting | Use standard length | St-Hilaire et al. (2006) |
| Some species have rigid (e.g., tuna [family Scombridae]) or heterocercal (e.g., sturgeons [family Acipenseridae]) caudal fins | Use fork length | Önsoy et al. (2011) |
| Damaged or missing rostrum (e.g., Paddlefish Polyodon spathula, gar Lepisosteus spp.) | Use eye-to-fork length (e.g., anterior edge of eye to fork of tail) | Ruelle and Hudson (1977) |
| Morphological change occurs over life stages (e.g., during spawning season such as in salmons [family Salmonidae]) | Use postorbital or midorbital to fork length | Hamon and Foote (2000) |
It is a common practice to convert one length measurement system to another. Conversion enables the standardization of measurements across diverse methods of reporting and the comparison of lengths across studies. Using simple regression, lengths are converted by creating a linear equation of the form y = a + bx, where y is the form of length desired and x is the form of length available. Length–length regressions are typically linear with high coefficients of determination (r2), resulting in high precision of length conversions (Echeverria and Lenarz 1984). Length-to-length conversions for numerous fish species have been extensively published (Table 2).
| Study area | Number of species | Length conversion | Model type | r 2 | Source |
|---|---|---|---|---|---|
| Aegean and Marmara coast (Turkey) | 42 |
SL to TL SL to FL FL to TL |
Linear | >0.90 |
Gaygusuz et al. (2006) |
| North Atlantic (North America) | 2 | TL to FL | Linear | >0.99 | Pol et al. (2011) |
| Galveston Bay (USA) | 3 | SL to TL | Linear | >0.99 | Matlock et al. (1975) |
| Aegean Sea (Greece) | 37 |
TL to FL TL to SL FL to SL |
Linear | >0.93 | Moutopoulos and Stergiou (2002) |
| Adriatic Sea (Croatia) | 10 |
SL to TL FL to TL |
Linear | >0.90 except one | Sinovčić et al. (2004) |
| Oueme River (West Africa) | 52 | SL to TL | Linear | >0.90 | Lalèyè (2006) |
| Ganges River (Bangladesh) | 10 |
TL to SL FL to TL SL to FL |
Linear | >0.95 | Hossain et al. (2009) |
| Xingu River (Amazon basin, Brazil) | 135 | SL to TL | Linear | >0.91 | Giarrizzo et al. (2015) |
| Zarrineh River (Iran) | 6 |
FL to TL SL to FL TL to SL |
Linear | >0.98 | Radkhah and Eagderi (2015) |
| Baja California (Mexico) | 10 | SL to TL | Linear | >0.93 | Ruiz-Campos et al. (2006) |
There have been considerable advancements in the development of equipment used to measure fish length. The most common method is to place the fish on a board equipped with a built-in measuring ruler (Royce 1942; Carlander and Smith 1945; Øvredal and Totland 2002). Measuring length can be time-consuming and stressful for live fish that are waiting to be processed and released, aside from taking up field time that could be used to increase sample size. Thus, several semi- or fully automated procedures and technologies have been developed to speed up the gathering of length data and the release of fish. Automatic and electronic measuring boards, such as the FishMeter board (Øvredal and Totland 2002), can record fish length digitally, reducing human error and handling time (Chaput et al. 1992; Øvredal and Totland 2002). In addition to taking lengths, some electronic measuring boards can simultaneously record dozens of covariates (e.g., sex, weight, stomach content) and scan passive integrated transponders (i.e., PIT tags). Video and picture software have also been developed (e.g., SeaGis Photomeasure, SigmaScan, ImageJ) to measure adult and larval fish lengths and get counts (Shafry et al. 2012; Hao et al. 2016; Shafait et al. 2017; Andrialovanirina et al. 2020; Shi et al. 2020; Rasmussen et al. 2022). Some of these electronic advancements reportedly can reduce the time to measure a fish by 50–80% (Andrialovanirina et al. 2020; Rasmussen et al. 2022). These length-measuring advancements are particularly useful in culture facilities, where aquaculturists may use images of samples taken to determine growth in length and adjust feeding and medication plans (Naiberg et al. 1993; Shi et al. 2020). Though electronic measuring techniques can save time, they can be inaccurate depending on the angle of the fish to the camera, the quality of the image, and what type of references the cameras are using (Hao et al. 2016; Rasmussen et al. 2022). High-frequency multibeam sonar is a valuable technique for measuring fish lengths in environments with poor visibility or low lighting conditions, where cameras and other technologies are not suited. This method is also noninvasive, minimizing stress on the fish (Cook et al. 2019). Nevertheless, sonar-based systems exhibit lower accuracy and precision in comparison to stereo-camera techniques (Cook et al. 2019).
For some applications, such as diet analysis or archeological studies, fish lengths may have to be reconstructed from the remaining bony structures. Because bony structures are digested slowly or not at all (Garman 1982), they are frequently employed in diet studies to estimate fish prey size at time of consumption. Otoliths and cleithra are commonly used in this application (Chipps and Garvey 2007). Head, mandible, backbone, and vertebrae lengths have also been found to be good predictors of fish length (Parsons et al. 1991; Raborn et al. 2002; Isermann and Vandergoot 2005). Similarly, in archeological studies, fish length can be reconstructed with remaining bony parts, particularly vertebrae, which are often the most obvious and easily collected remains at an archeological site (Casteel 1976). In a recent archeological study in the Caribbean islands, the fossil record was used to infer overexploitation of large individuals over time (Grouard et al. 2019).
LENGTH ESTIMATION BIAS
Accurately assessing fish populations and their status is a complex task that can be compromised by various measurement biases. Measurement bias occurs when there is a systematic error in the estimation of length, resulting in inaccurate or incomplete length-frequency distributions. Bias in length estimation can stem from at least three distinct sources: (1) bias created by the spatial and temporal distribution of the fishing effort relative to the distribution of the fish, (2) bias associated with the size selectivity of gear used to capture the fish, and (3) bias introduced during the length measurement process after the capture of the fish.
The sampling effort may not align with the spatial (e.g., vertical or horizontal) or temporal distribution of fish, which could lead to captures that only represent a subset of the population length distribution if fish segregate according to size (MacLennan 1992; Pope and Willis 1996; Fischer and Quist 2014). The spatial and temporal distribution of fish may vary by length due to several factors, including ontogenetic habitat shifts, seasonal migrations, and population structure (Post et al. 1995; Makrakis et al. 2012; Jaureguizar et al. 2016). Juveniles often use separate habitats from adults and differ in the time spent foraging and time spent in refuge areas. Once reproductively mature, adults may move seasonally between breeding and feeding grounds. The spatial distribution of sampling effort may also be influenced by several factors and may not necessarily reflect the distribution of the resource, even if random sampling methods are applied. Thus, the interplay of factors that influence the spatiotemporal patterns of fish and sampling effort determines the lengths accessible to sampling. Schooling, for example, can introduce additional bias in length measurements, as schooling fish usually segregate by length (Pavlov and Kasumyan 2000).
Minimizing bias and assuring precise measurement of fish populations are influenced by the careful selection of sampling gear. The choice of gear depends on several factors, including the size and behavior of the target species, the type of habitat, and possibly the fishing regulations in place (Fischer and Quist 2014; Shoup and Ryswyk 2016). Different gear types may be more effective in specific habitats (e.g., seines in shallow water and gill nets in deep water), and constraints that may impede access to sampling areas include remoteness, travel distances, and access sites. Standardization of sampling gear and effort can increase precision in monitoring length-frequency distributions (Bonar et al. 2009) but may not remove bias. When length-frequency distributions may be biased, managers disregard length segments expected to be biased and assume the lengths retained are unbiased (e.g., excluding parts of the length distribution that may not be fully recruited to the sampling gear). Although this assumption may lead to errors, managers can still benefit from estimates obtained through an evaluation of such data, provided that precautions are implemented to reduce the chances of, or mitigate, erroneous interpretations. Selectivity curves and catchability estimates can be used in some circumstances to adjust length-frequency distributions (Breton et al. 2013; Shoup and Ryswyk 2016).
The occurrence of error and bias in length measurements made by people is widely acknowledged, although the extent of these is seldom known. It is generally accepted that the extent of these errors when made by professionals is small (Ferguson et al. 1984; Gutreuter and Krzoska 1994) and that they rarely have a significant effect on the validity of applications under most circumstances (but see Phelps et al. 2013). However, managers sometimes must rely on measurements made by fishers, such as in surveys of catch-and-release fisheries. Results from studies that have examined fishers' measurement error indicate that fisher-reported lengths are less precise because of digit bias (e.g., rounding to the nearest whole number), varied with species, and increased with fish size and with the presence of length limit regulations that for various reasons influence attentiveness to measurement (Ferguson et al. 1984; Page et al. 2004; Bunch et al. 2013; Matlock 2014).
Another source of bias may occur in the analysis of length-frequency data (Pope and Willis 1996; Heery and Berkson 2009; Fischer and Quist 2014). The interpretation of a standing length frequency assumes some degree of stability or uniformity over time. There may be serious bias in the accuracy of the metrics when there are departures from equilibrium conditions (i.e., steady state). In general, annual or seasonal variability in recruitment, natural mortality, and fishing effort, and to a lesser extent growth, are all sources of bias that may introduce errors into the interpretation of a standing length-frequency distribution (Beamesderfer and Rieman 1988). Variability in recruitment violates the steady state assumption but is not particularly troublesome if the variability is random with no systematic trend. In contrast, temporal variability in mortality, whether induced by natural or by fishing processes, violates the assumption of steady state. Such variability may be prompted by an evolving shift in fisher behavior (e.g., the growing prevalence of catch-and-release practices in recreational fishing; Sass and Shaw 2020) or by systematic transformation in fishing effort induced by fishing regulations. Bias can have significant implications for the management and conservation of fish populations. Thus, reliance on length as a tool for fish management may require intervallic monitoring to double-check results and the adoption of traditional age-based methods in situations when there is unsettling uncertainty and high stakes in management outcomes.
SIMULATING LENGTH-FREQUENCY DISTRIBUTIONS
Simulations of length-frequency distributions can be employed alongside empirical length data to facilitate modeling exercises and enable comparisons with empirical distributions. Simulated length-frequency distributions constructed with specified rates of growth, mortality, and recruitment can remove estimation biases and generate reference populations. There are various ways to simulate length-frequency distributions (Hampton and Majkowski 1987; Jones 1987), all of which are relatively straightforward, as only a few basic population parameters are needed. A growth curve (e.g., von Bertalanffy model) can be used to estimate mean length of a cohort over time. The distribution of lengths around the mean length can be assigned based on an assumed deviation (e.g., standard deviation, coefficient of variation [100 × standard deviation/mean]) and a statistical distribution (e.g., normal, lognormal). A constant or variable annual mortality rate can be applied to individuals over years to randomly remove them from the distribution and prevent advancement into the next year. When the remaining individuals are combined across all years, such distribution of lengths creates a reference length-frequency distribution that has the specified starting population parameters. Simulated reference length-frequency distributions can serve various purposes, including (1) identifying reference points, (2) calculating indexes such as spawning potential ratio, and (3) estimating sample sizes needed to adequately characterize length metrics.
Identifying reference points
Length reference points are useful for setting goals and evaluating management actions. One of these indicators commonly used in North America is relative stock density (RSD), defined as the ratio of two counts: Pn/N, where N represents the count of fish that exceeds a smaller benchmark length, and n the count of fish that exceed a longer benchmark length (Gabelhouse 1984; Table 3). A simulation approach may be used to generate an idealized length-frequency distribution, which can serve as a basis for deriving a RSD reference point. Reynolds and Babb (1978) constructed such simulated length-frequency distribution for Largemouth Bass Micropterus nigricans populations utilizing the average length over multiple ages, variability around length at age i, and an annual interval mortality of 30% and 50% representing low and high levels, respectively. Then they used these two simulated length frequencies to compute an RSD. Based on species-specific benchmarks (200 mm TL characterized as stock size and 300 mm TL characterized as quality size; Gabelhouse Jr 1984), the authors estimated that target RSD should be 43–60%, representing the high-mortality and low-mortality length-frequency distributions, respectively. It is not difficult to extend this simulation exercise to find reference points for other species or other length metrics, such as those in Table 3.
| Method | Definition | Applications |
|---|---|---|
| Frequency | ||
| Length frequency | The count or percentage of fish in length intervals |
The conventional method for characterizing length-frequency distributions. The capacity to visually compare a large number of populations or samples may be limited. Interval widths can influence visual impact. |
| Relative length frequency | Comparison between observed and expected (i.e., standard) length-frequency distribution | If a suitable benchmark is available, it could be an efficient way to identify deviations from the norm or of spotting abnormalities. |
| Cumulative distribution | The sum of the frequency in a length interval and all preceding intervals | Presents an alternative visualization method to length frequency, one that can handle more distributions in a single graph panel. The resultant curve may be easily fitted with nonlinear models to compare distributions or uncover variables that are connected with the shape of the distributions. |
| Length ordination | Uses similarity coefficients to ordinate length frequencies | Can handle more populations or samples than the methods above, and the ordination of lengths can be examined in the context of outside variables to explore what abiotic or biotic factors influence the distribution of lengths. |
| Single value | ||
| L i | Length of the ith fish | Measured as per described in this article. Multiple Li are needed for length analyses (see section on sample sizes for length metrics). |
| L c | Length at first catch | The smallest fish collected with little bias. The Lc varies with collection method and is frequently employed to demarcate the subset of the length distribution under examination. While the value may be treated as fixed, it can be coupled with a probability distribution. |
| L c,f | Length at first catch in a fishery | The smallest fish caught in the fishery. Could be coupled with a probability distribution if based on fisher preferences. Alternatively, could be a fixed value if there is a minimum length limit in place. |
| L mean | Mean of Li ≥ Lc | Reflects the average length of fish in a population considering Lc. The Lmean is sensitive to growth, mortality, and recruitment. It decreases with mortality and recruitment and increases with growth. It can be misleading if influenced by one or few exceptionally large lengths in the sample. |
| L mean,f | Mean of Li ≥ Lc,f |
Reflects the average length of fish harvested in a fishery. It is influenced by the same elements as Lmean. It is also anticipated to function as a metric for fishing effort, as its level decreases in response to harvests that targets the largest specimens. |
| L X% | Li that is larger than x percentile (e.g., 25th, 75th, 95th percentile) of all recorded lengths ≥ Lc | Can indicate mortality, growth, or recruitment depending on x. For example, if L95% is low, this could indicate high mortality, low growth, or high recruitment. |
| L max | The highest recorded Li in a sample | It could be taken as an index of fish growth, or potentially low mortality rates, within the population. Nevertheless, it is a tenuous estimate as it relies on a single fish. |
| L max5% | Mean length of largest 5% Li | Like Lmax but avoids the weakness of relying on a single fish. Because it is derived from the right-hand tail of a length distribution, it is less affected by fluctuations in recruitment than Lmean (Miethe et al. 2019). |
| L max30 | Mean length of largest 30 Li | Like Lmax5% but based on an absolute fish count rather than a relative one. Thus, unlike Lmax5%, this estimate is not affected by size of the collection. |
| L ∞ | Asymptotic mean length in population estimated with fish ≥ Lc | Estimated as a/(1 – b) in a linear regression y = a − bx, where x is the lower limit of each length interval in a length-frequency distribution and y is the mean length of all fish larger than x (Wetherall 1986). The L∞ is another estimate of the mean maximum size potentially achieved by a population. It is a mean because the least-squares regression fits a and b in a way that minimizes the sum of the squared vertical distances between the line and the x–y points. |
| L mat | Length at which nearly 50% of females become sexually mature | Although there may be differences in this value across different populations, Lmat is difficult to estimate and therefore it is common to use a generic value obtained after a review of existing literature. |
| Compound value | ||
| L i /L ∞ | Relative length expressed as length of the ith fish relative to the asymptotic mean length in the population | Suitable for comparing across populations and possibly across species. |
| L mean /L ∞ | Ratio of mean Li ≥ Lc in population to asymptotic length | Values approximating 1 indicate large fish in population, possibly low mortality. Values nearing 0.66 are generally desirable (Jensen 1996). |
| L max5% /L ∞ | Ratio of mean length of top 5% largest fish in population to asymptotic length | Values near 1 indicate large fish in population, and possibly low mortality. Values > 0.8 are generally desirable (Fitzgerald et al. 2018). Lmax30/L∞ may be more appropriate if comparing samples with different Lc. |
| L mean /L mat | Ratio of mean Li ≥ Lc in population to length at maturity | Values >1 are desirable (Fitzgerald et al. 2018). However, values ~1 are acceptable when using fishery data under the tenet that all fish should spawn at least once (Myers and Mertz 1998; Froese 2004). |
| Ratio of the length interval above the mean length to the length interval between mean length and first catch | Values closer to zero indicate large fish in population, possibly low mortality. This ratio corresponds to Z/K, where Z is the instantaneous total mortality and K is the growth coefficient in a growth curve such as von Bertalanffy's (Beverton and Holt 1956). This ratio varies across taxa with values around 1.5–2.0 being most common (Prince et al. 2015). Any Z/K ratios >2 may be in a critical zone where mortality could be too high relative to growth, and further scrutiny is warranted. A species with a high Z/K has a low probability of reaching large size. | |
|
P n/N |
The ratio between two counts, N and n, where N represents the count of fish that exceed a smaller benchmark length, and n indicates the count of fish that exceed a longer benchmark length | Include various stock density indices (e.g., relative stock density; Anderson and Neuman 1996). In general, the benchmark lengths used for these metrics vary by species. Target values vary by species and policy and are available in a large body of North American literature. |
Spawning potential ratio
In marine fisheries exploited fish stocks are often managed with specific consideration given to a ratio called the spawning potential ratio (SPR), but this ratio has also been applied in freshwaters (e.g., Heller et al. 2022). The SPR is the ratio of eggs produced per recruit over a lifespan when the population is fished, relative to the eggs produced when the population is unfished. Fishing shortens fish lifespans, thus reducing their potential for producing eggs. This ratio helps determine whether a fishery may harvest more or fewer fish, and it therefore drives management in some situations. A population with a ratio close to 1 is relatively unfished, ratios of 0.2–0.5 are considered marginal, ratios <0.2 are considered overfished, although these values depend on species life history traits, such as fecundity, natural mortality, and longevity (Mace and Sissenwine 1993). Commonly, the ratio is computed based on fish age schedules (Goodyear 1993; Miranda and Killgore 2013). However, the ratio can also be estimated based on length data, provided there is a fecundity-length equation available to convert simulated length frequencies to egg counts. This approach was refined into a length-based SPR by Hordyk et al. (2015), including an R package which is accessible at https://cran.r-project.org/web/packages/LBSPR/index.html. An alternative approach is to substitute egg counts for the biomass of fish contributing to the spawn (Goodyear 1993). Thus, SPR can be expressed as the ratio of the fished to unfished spawning stock biomass per recruit. A weight–length equation may be used to convert the length-frequency distributions into biomass.
Sample sizes for length metrics
Knowledge of sample size is important because taking too few or too many samples squanders effort and other resources. Miranda (2007) used bootstrapping from simulated length-frequency distributions considered reference populations to provide insight into the sample sizes needed to estimate length frequencies and other length metrics. Several reference populations were established to represent species with varying maximum lengths and mortality patterns, as well as to account for the size of the bins used to generate the length-frequency distribution. The reference populations were subjected to random resampling using bootstrapping techniques. By systematically increasing the sample sizes, it was possible to determine when the samples ceased to deviate significantly from the reference length-frequency distribution. In general, simpler length metrics, smaller species, smaller populations, and populations with higher mortality required fewer length samples (Miranda 2007).
SUMMARIZING LENGTH FREQUENCIES
Various methods have been devised to summarize, characterize, and report length measurements collected on fish. To facilitate organization, we classify these metrics into three distinct categories: frequency distributions, single-value metrics, and compound-value metrics (Table 3). Lengths may be reported for the entire sample or for segments of a length-frequency distribution thought to be adequately represented by the collection. In the sections that follow, we present a range of metrics and provide illustrative examples using simulated and empirical data. We recognize that a single-value or a compound-value metric distilled to only one number is likely insufficient to represent all the components that it is meant to capture. Their goal is to synthesize complex information into a value that is easy to convey and understand and that indicates whether management actions are achieving the intended outcome.
Frequency distributions
Frequency distributions allow visualization of patterns in the length structure of a population. They are usually organized into a histogram to reflect patterns in size structure (Figure 2A). The patterns can be interpreted to reflect recruitment, growth, and mortality, although it may be difficult to sort out which one. For example, a population with a high frequency of small fish can reflect high recruitment or slow growth, or an extended length-frequency distribution could reflect low mortality or fast growth. Consequently, local ecological knowledge or additional metrics are required for an accurate interpretation of the length-frequency distribution. Frequency histograms give a rich picture of the size structure. Nevertheless, the abundance of information they offer might complicate the process of comparing multiple histograms.
Relative length frequency contrasts an observed size structure against a regional or an idealized reference size structure that is species-specific (Bonar 2002). Choosing the reference is important so that the contrast is relevant to the goal of the comparison. Establishing reference length frequencies can be controversial, making them rarely available. Consequently, this very practical application may not have received the appropriate level of attention in the main literature.
Cumulative frequency distributions provide an alternative visual for length-frequency data. At each length interval, the frequency in all preceding intervals is accumulated resulting in a progressively rising curve that is quasi-S-shaped, and its changes in slope may vividly describe the length characteristics of a population. However, working with more than a few overlapping cumulative distributions makes frequency distribution visualization challenging (Figure 2B). Additionally, in cumulative distributions it can be difficult to assess the contribution of different length-groups. Cumulative curves are amenable to analyses with nonlinear models to estimate descriptive curve parameters and to statistically compare curves spatially or temporally (Flather 1996). Curves may be considered along with covariates to test if the parameters that represent the shape of the cumulative curve are influenced by factors such as environmental characteristics or fishing effort. For instance, a logistic regression curve may be used to represent a cumulative frequency distribution, as demonstrated by Barlow (1990). Furthermore, a multiple logistic regression model can be employed to determine if certain environmental conditions have a substantial impact on the shape of the logistic curves, as explored by Ohlmacher and Davis (2003).
An ordination (Figure 2C) offers still another point of view by arranging samples with high similarity close together in a multiaxis plane and placing samples with low similarity far apart (Miranda 2024). The vectors represent the correlations between axes scores and size-groups and are such that their length and direction reflect the degree of emphasis of a given length at each sample or location. While effectively communicating information regarding variations in size structure, ordinations can aid in the representation of a substantial number of collections. This approach frequently provides improved visualization capabilities and allows for the representation of complex interactions between size structure and covariates. The correlations between the axes scores in a length ordination and external covariates may be depicted as vectors that are superimposed in the ordination along with the vectors that represent the length-groups. Ordinations can also be constrained by the covariates to get a better understanding of what aspects of the size structure are best correlated to outside covariates (see Figure 3 in Miranda 2024).
Single-value metrics
Singular metrics serve as concise summaries of the length-frequency attributes of a fish population, encapsulating them with a single numerical value (Table 3). For instance, the mean length Lmean denotes the arithmetic average of the lengths within a given collection, or the mean length in a fishery Lmean,f, each with different length at first catch (i.e., Lc or Lc,f). Lengths at percentiles (e.g., 25th, 75th, and 95th) can complement central tendency metrics. Metrics that describe the upper bounds of length-frequency distributions can indicate aspects like reproductive status, mortality, and anthropogenic effects (Stergiou and Karachle 2006; Edelist et al. 2014). Alternative metrics to describe the upper bounds may include mean length of the largest 5% of the sample, mean length of the largest 30 fish in the sample, or simply maximum length (Stergiou and Karachle 2006; Miethe et al. 2019). Because they are derived from the right-hand tail of a length distribution, these metrics are less affected by recruitment variability and may more precisely reflect changes in size structure (but see comments in Table 3). Moreover, a mean asymptotic length L∞ can be estimated from the length-frequency distribution to predict a mean maximum length in the population (Wetherall 1986). Similarly, measures that describe the lower boundaries of a length-frequency distribution (e.g., L25%) might be helpful in delineating other pertinent specificities like recruitment or gear selectivity. These metrics often play a significant role in the characterization of populations and the understanding of population dynamics.
Compound-value metrics
Compound metrics combine two or more single-value metrics to produce a new metric that represents population status. Relative length, the ratio of fish length to asymptotic length (i.e., Li/L∞) standardizes length between 0 and 1 to allow comparing populations, and possibly different species, using a common length scale. The ratio of mean length of fish relative to asymptotic length (i.e., Lmean/L∞), or the ratio of mean length of longest 5% in a sample relative to asymptotic length (i.e., Lmax5%/L∞), can suggest both natural and fishing mortality in a population, as ratios closer to 1 indicate presence of larger (i.e., older) fish (Table 3). Alternatively, values below 1 for the ratio of mean length of fish relative to mean length at maturity (i.e., Lmean/Lmat, Lmean,f/Lmat) can indicate recruitment overfishing. The ratio of the length interval above the mean length to the length interval between mean length and first catch, , reflects instantaneous mortality adjusted for instantaneous growth rate (i.e., Z/K; Beverton and Holt 1957). Various ratios of fish counts classified according to broad but meaningful length-groups have been applied. For example, RSD represents the percentage of stock-length fish that are also of quality length in recreational fisheries (Anderson and Neuman 1996; Guy et al. 2007). Stock and quality lengths, and other benchmark lengths included in these and related metrics, vary by species and policy and are available in a large body of the North American fishery management literature.
EXAMPLES OF LENGTH ANALYSES
Length-frequency methods
We illustrate the use of frequency methods with a length data set representing fishes in oxbow lakes. Fish were collected from 13 lakes in the original floodplain of the lower Mississippi River by boat electrofishing (collection methods described by Dembkowski and Miranda 2014). All fish were counted, measured to the nearest 1 mm TL, and binned into 10-cm length-groups, disregarding any taxonomic distinction to focus on fish community size structure, which is of conservation importance in these systems. Length-frequency distributions, cumulative distributions, and ordination were employed to organize the bins for visualization of the data set.
All three of these methods provide insightful and complementary descriptions of the fish size structures in oxbow lakes (Figure 2). The ordination (Figure 2C) separates lakes with a higher representation of intermediate size fish to the left and a higher representation of small fish to the right. Lakes with a high representation of small fish include Horseshoe, Pleasant, Robinson, South Swan, Long Brake, and Yazoo Cutoff. The high representation of small fish is illustrated by the distribution of lengths in the histograms (Figure 2A), by the vectors in the ordination, and by the large y-intercepts in the cumulative frequency distribution (Figure 2B). Within this group of six lakes, South Swan and Long Brake also included some very large fish along with the small fish and were positioned at the lower right-hand end of the ordination, whereas Horseshoe, Pleasant, Robinson, and Yazoo Cutoff tended to include a slightly higher representation of midsize fish and were positioned at the upper right-hand end, and both were represented by the position of the length vectors. In the cumulative length frequency, these patterns were reflected by flatter lines that had high intercepts and reached asymptote later (South Swan and Long Brake) and steeper lines that had lower intercepts and reached asymptote sooner (Horseshoe, Pleasant, Robinson, and Yazoo Cutoff). The ordination also illustrates a notable increase in fish of intermediate size towards the left side in lakes such as Mossy, Blue, Wasp, and Townsend. Moreover, lakes located in the lower left quadrant of the ordination, such as Townsend, Patterson, and Hampton, place particular emphasis on the presence of both intermediate and large fish. The observed patterns were manifested in the cumulative length frequency through the presence of steep S-shaped curves, which exhibited a tendency to approach an asymptote at a relatively bigger size. The diversity of length distributions in these oxbow lakes is associated with lake features, and while outside the scope of this paper, they are explored by Miranda (2024).
Single-value and compound metrics
We illustrate the use of single-value and compound-value length metrics with simulated length-frequency distributions derived from assumed growth curves and basic population parameters. The original von Bertalanffy growth model is , where Lt is the total length (mm) at age t, L∞ is the asymptotic length, L0 = is the average total length at hatch, and K is the growth coefficient indicating how fast length approaches the L∞. We assumed a Largemouth Bass population with L∞ = 580 mm, K = 0.2, L0 = 4 mm, and t = 1–15 years. These values of L∞ and K correspond to the median for Largemouth Bass populations in the United States and Canada (Beamesderfer and North 1995), and L0 is the length at hatch (Miranda and Bettoli 2019). The simulation started with 10,000 fish at t = 1 distributed normally around Lt with a coefficient of variability of 10%. Each year, 80% or 30% of the length distribution of each age-group was randomly selected to survive until the following year (i.e., interval annual mortality, A, was either 20% or 70%). The lengths of fish surviving in each year-class were combined across age-groups to create the reference length-frequency distributions. Assuming Lc = 20 cm for this hypothetical population, the resulting reference length distributions were used to estimate some of the single-value and compound-value metrics listed in Table 3.
The simulated length-frequency distributions demonstrate the impact of low and high mortality rates on single-value and compound-value metrics (Figure 3). The expectation was that the single-value and compound-value metrics would reflect the trend in the mortality disparity. As annual mortality increased from 20% to 70%, the Lmean values decreased from 38 to 25 cm, Lmax5% from 58 to 39 cm, and Lmax from 70 to 53 cm. Furthermore, Lmean/L∞ decreased from 0.63 to 0.40 with only the higher value within desirable range, Lmax5%/L∞ was above the 0.8 target threshold for the low-mortality population but below for the high-mortality population, Lmean/Lmat decreased from 1.48 to 1.08 and at the higher mortality approximated the precautionary threshold of 1, and Z/K was below the threshold of 2 for the low-mortality population but beyond the threshold for the high-mortality population. The resulting RSD values of 0.15 at high mortality and 0.68 at low mortality are in line with the 0.4–0.6 RSD at 30% and 50% annual mortality suggested by Reynolds and Babb (1978) for Largemouth Bass in Midwestern ponds.
CONCLUSIONS
A comprehensive understanding of existing conditions and trends is crucial for effective policy formulation in the conservation of freshwater fish. However, monitoring fish populations in inland ecosystems is challenging and costly because of the substantial number of waters that may require assessment. As a result, many freshwater systems suffer from inadequate data and lack systematic assessments of their status. A significant obstacle in fish conservation remains the need to create cost-efficient methods for assessing the status of small-scale freshwater systems. Our aim was to review length metrics potentially valuable in assessing the status of freshwater fish populations. To this end, we combed through the historical, contemporary, and latest literature on freshwater and salt water in pursuit of length-related concepts, formulations, procedures, and innovations pertinent to the assessment of freshwater populations. We omitted some size-structured models that are used in marine fisheries (Punt et al. 2013; Chong et al. 2020) but that are rarely or never used in freshwater. With this review, we hope to spark additional research activity into one of the most widely used concepts in freshwater fish management, yet arguably neglected in terms of research and development.
Our review suggests that length frequencies offer a straightforward and efficient approach to evaluate the status of a population. Working solely with lengths does not necessitate any data on fish age, recruitment, growth, mortality, or population size. Depending on length as a measure can exclude the necessity of relying on fish relative abundance or relative weight (i.e., condition) estimates. The spatial and temporal variability of catches within the same system and within the same day or week can compromise relative abundance (Pope and Willis 1996; Fischer and Quist 2014), while problems with accurately measuring fish weight in the field and with finicky weight–length standard equations can compromise relative weight (Froese 2006). Furthermore, achieving a standardized effort for most collection gears can be challenging because they are susceptible to operator and ambient conditions biases. Thus, reliable estimates of length may be easier to obtain and track than estimates of population densities, and building population assessment strategy around length may be a cost-effective strategy with only minimal data requirements.
Inevitably, a metric based solely on length cannot do justice to all the elements that affect it. Its purpose is to condense and synthesize highly complex information into a simplified form that is easily comprehensible and communicable. It serves to indicate if management intervention is necessary, if management policies are yielding the intended outcome, or if surveys that are more comprehensive are necessary. This strategy, while not professing great accuracy, allows a first cut at managing a large number of waters.
Although length-based methods are the cornerstone of many population assessments, the collection and analysis of accurate length distributions pose challenges. Some length-based approaches rely on the assumption of temporal stability or uniformity (i.e., steady state), and so accuracy of the findings may be compromised when there are deviations from equilibrium circumstances. Various factors affect length distributions, such as size-selective sampling, fluctuations in recruitment, interannual variability in growth and mortality, and variations in sampling effort from year to year, all of which contribute bias that could lead to inaccuracies in length-based estimations (Isaac 1990; Somerton and Kobayashi 1991; Schwamborn 2018). To mitigate the possible ambiguities inherent in length-based approaches, we strongly advise complementing them with age-based methods in situations where making the correct judgment is of utmost importance. Using more than one approach for fish population assessment incorporates parallel or competing lines of scientific evidence and alleviates unease associated with difficult choices (Kraak et al. 2010).
Given the importance of length in freshwater fish conservation and its widespread use in the fisheries literature, it is surprising that there is not a larger emphasis on research in this area. Considering this limitation, we encourage pursuing additional study and propose several specific areas for investigation. An essential question is what level of accuracy and precision is required from length data to successfully tackle decision making in typical conservation scenarios. Such investigations would provide guidance regarding the situations in which lengths alone are enough or inadequate. Another important component is the processing of length data, whether as frequencies, single-value metrics, or compound-value metrics. Some of the aspects that require elaboration include reliability when there are departures relative to the ideal equilibrium conditions, biases, sample sizes for estimation considering population sizes, target and reference values, and special limitations linked to species characteristics. Previous examinations have revealed that certain length-based approaches can be unreliable (Isaac 1990; Somerton and Kobayashi 1991; Schwamborn 2018). Exploring the integration of correlated data might be a promising avenue for investigation. The integration of length data with angler survey data, environmental data, or a minimalist aging program has the potential to improve and expand the analysis and interpretation of length measures.
Lastly, fisheries management has historically been rooted on a yield paradigm that was established for commercial marine fisheries and considers populations in relation to recruitment, growth, and mortality (Beverton and Holt 1957; Pauly 1998). These demographic variables are incorporated into models that prioritize maximum yield as the goal and fishing effort as the main driver for change. The paradigm dictates that recruitment, growth, and mortality need to be managed to attain sustainable harvests on steady state populations to avoid the recurrence of classic scenarios such as recruitment or growth overfishing (Beverton and Holt 1957). With this mindset, the paradigm also dictates which length metrics are relevant. The paradigm does not consider the diversity of environmental drivers, the large number of broadly dispersed and partially isolated inland waters, and the diversity of human needs other than fishery yield. Given the diversity of inland waters, an alternative paradigm may encourage goals such as protecting natural resources, ecosystem functionality, promotion of recreation participation, and implementation of transparent management. Each of these goals might have some relevance in some inland waters. Existing length metrics may be applicable, although employed in different contexts, or alternative innovative metrics may emerge once mindsets are liberated from the traditional yield paradigm.
ACKNOWLEDGMENTS
Helpful and constructive reviews were provided by Dan Dembkowski, Nate Smith, and Steve Tyszko. The Mississippi Cooperative Fish and Wildlife Research Unit is cosponsored by the U.S. Geological Survey, Mississippi Department of Wildlife, Fisheries, and Parks, Mississippi State University, U.S. Fish and Wildlife Service, and Wildlife Management Institute. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.
CONFLICT OF INTEREST STATEMENT
The authors declare that there are no competing interests.
ETHICS STATEMENT
There were no ethical guidelines applicable to this study.
