The influence of sample distribution on growth model output for a highly-exploited marine fish, the Gulf Corvina (Cynoscion othonopterus)

Derek G. Bolser; Arnaud Grüss; Mark A. Lopez; Erin M. Reed; Ismael Mascareñas-Osorio; Brad E. Erisman

doi:10.7717/peerj.5582

The influence of sample distribution on growth model output for a highly-exploited marine fish, the Gulf Corvina (Cynoscion othonopterus)

Derek G. Bolser¹, Arnaud Grüss², Mark A. Lopez¹, Erin M. Reed^1,3, Ismael Mascareñas-Osorio⁴, Brad E. Erisman ¹

1Marine Science Institute, University of Texas at Austin, Port Aransas, TX, United States of America

2School of Aquatic and Fishery Sciences, University of Washington, Seattle, WA, United States of America

3Joint Institute for Marine and Atmospheric Research, University of Hawaii and NOAA Fisheries, Pacific Islands Fisheries Science Center, Honolulu, HI, United States of America

4Centro para la Biodiversidad Marina y la Conservación, La Paz, Mexico

DOI: 10.7717/peerj.5582

Published: 2018-09-17
Accepted: 2018-08-13
Received: 2018-04-13

Academic Editor: María Ángeles Esteban

Subject Areas: Aquaculture, Fisheries and Fish Science, Conservation Biology, Marine Biology, Statistics
Keywords: Gulf of California, Gulf Corvina, Growth modelling, Fish growth, Data-poor fisheries, Highly-exploited, Vulnerable

Copyright: © 2018 Bolser et al.
Licence: This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ) and either DOI or URL of the article must be cited.

Cite this article: Bolser DG, Grüss A, Lopez MA, Reed EM, Mascareñas-Osorio I, Erisman BE. 2018. The influence of sample distribution on growth model output for a highly-exploited marine fish, the Gulf Corvina (Cynoscion othonopterus) PeerJ 6:e5582 https://doi.org/10.7717/peerj.5582

The authors have chosen to make the review history of this article public.

Abstract

Estimating the growth of fishes is critical to understanding their life history and conducting fisheries assessments. It is imperative to sufficiently sample each size and age class of fishes to construct models that accurately reflect biological growth patterns, but this may be a challenging endeavor for highly-exploited species in which older fish are rare. Here, we use the Gulf Corvina (Cynoscion othonopterus), a vulnerable marine fish that has been persistently overfished for two decades, as a model species to compare the performance of several growth models. We fit the von Bertalanffy, Gompertz, logistic, Schnute, and Schnute–Richards growth models to length-at-age data by nonlinear least squares regression and used simple indicators to reveal biased data and ensure our results were biologically feasible. We then explored the consequences of selecting a biased growth model with a per-recruit model that estimated female spawning-stock-biomass-per-recruit and yield-per-recruit. Based on statistics alone, we found that the Schnute–Richards model described our data best. However, it was evident that our data were biased by a bimodal distribution of samples and underrepresentation of large, old individuals, and we found the Schnute–Richards model output to be biologically implausible. By simulating an equal distribution of samples across all age classes, we found that sample distribution distinctly influenced model output for all growth models tested. Consequently, we determined that the growth pattern of the Gulf Corvina was best described by the von Bertalanffy growth model, which was the most robust to biased data, comparable across studies, and statistically comparable to the Schnute–Richards model. Growth model selection had important consequences for assessment, as the per-recruit model employing the Schnute–Richards model fit to raw data predicted the stock to be in a much healthier state than per-recruit models employing other growth models. Our results serve as a reminder of the importance of complete sampling of all size and age classes when possible and transparent identification of biased data when complete sampling is not possible.

Introduction

Age and size data inform estimates of life history parameters that are crucial to fisheries stock assessments. In early assessments such as Beverton and Holt’s yield-per-recruit model (1957), size at age was critical for estimating reproductive output and thus the sustainability of fisheries. In today’s age-structured stock assessments, size at age is used to convert from biomass to number of fish, determine selectivity, and calculate expected length compositions (Francis, 2016). Similarly, size (i.e., length or weight) at age is used in size-structured stock assessment models to inform transitions between size bins and determine length composition (Punt, Haddon & McGarvey, 2016). Accurately representing the relationship between size and age is particularly important for vulnerable fish and in data-poor fisheries, in which life-history parameters and population structure often drive stock assessments and management decisions (Dulvy et al., 2004; Froese, 2004; Honey, Moxley & Fujita, 2010; Hordyk et al., 2016). Specifically, these types of assessments rely heavily on age–length data to confer insights on vulnerability and overfishing (Erisman et al., 2014).

When modelling the relationship between age and size for the purposes of assessment, and for any purpose, each age and size class must be sufficiently represented to generate growth parameter estimates that reflect biological growth (Cailliet et al., 1986; Cailliet & Tanaka, 1990; Francis & Francis, 1992; Cailliet & Goldman, 2004). It is important to make the distinction between this type of sampling and sampling to reflect population structure, which should not be the goal of age and growth studies as this reflects bias due to the relative scarcity of large and old individuals. Sufficiently representing each size and age class may be especially difficult in highly-exploited species, as exploitation alters the population structure of fishes by preferentially selecting for large and old fish individuals (Mason, 1998; Berkeley et al., 2004). The ramifications for failing to acknowledge selection are clear, as length-selective fishing mortality distorts growth curves (Walker et al., 1998). Further, the lack of representation of large and old individuals could result in underestimation of lifespan and longevity, which makes fishery management measures less effective (Campana, 2001; Cailliet & Andrews, 2008; Goldman et al., 2012). Large and old fish drive estimates of the maximum average length parameter L_∞, and without them, L_∞ is underestimated and the growth rate (typically denoted by K) is overestimated. The underestimation of L_∞ and the overestimation of K lead to the assumptions of a shorter generation time and less mortality, and thus more resiliency to high levels of fishing pressure (Campana, 2001; Goldman et al., 2012; Harry, 2017). The L_∞ term is particularly important when growth models are incorporated into stock assessment (Wells et al., 2013). This problem may also occur in growth modelling for vulnerable fish or in data-poor fisheries, where lack of representation of each age and size class due to sampling constraints or the scarcity of individuals may similarly affect parameter estimates. Fishery dependent data are often the only data available for growth modelling, which may be acceptable only as long as the inherent biases and limitations are acknowledged.

Several models have been developed to quantify the relationship between age and size, with body length being the most common metric of size. Typically, asymptotic growth models are used to quantify this relationship. These models describe fast growth in the earliest years of life and slower growth in later years. Despite some criticism (Roff, 1980; Czarnołe‘ski & Kozłowski, 1998), the most widely used is the von Bertalanffy growth model (Chen, Jackson & Harvey, 1992; Kimura, 2008). Rooted in bioenergetics, this model is intended to give a biologically relevant representation of how catabolic and anabolic processes work within a fish to change growth over the lifespan of fishes (Von Bertalanffy, 1938; Pauly, 2010). Over the years, there have been many re-parameterizations of the von Bertalanffy model with incorporation of growth-influencing factors and applications to a variety of situations (Gallucci II & Quinn, 1979; Ratkowsky, 1986; Helser & Lai, 2004; Kimura, 2008; Brunel & Dickey-Collas, 2010; Van Poorten & Walters, 2016), but the original parametrization is still the most commonly used (Lorenzen, 2016). Other asymptotic growth models are commonly used in fisheries, such as the Gompertz growth model (Gompertz, 1825) and the logistic growth model (Ricker, 1975).

In recent years, fish growth models have moved from a foundation in bioenergetics to being more statistically driven (Van Poorten & Walters, 2016). These models are inherently more flexible, allowing them to capture subtleties in growth patterns that may not be captured by the more inflexible growth models. The Schnute model (Schnute, 1981), for example, has four curve families that the model may assume based on which types of data the model is fit to and what other functions are incorporated into the framework. Another flexible growth model, the Schnute–Richards model (Schnute & Richards, 1990), can describe biphasic growth among several other forms. By design, the Schnute–Richards model may be equivalent to the other growth models discussed above when the proper values are specified for its dimensionless parameters. Fish growth is inherently plastic and fish do not all grow the same way (Weatherley, 1990; Lorenzen, 2016), so a flexible growth model may be advantageous in certain situations. However, these flexible models may also be more sensitive to sampling biases in data, potentially producing growth patterns that reflect the size-frequency distribution of fish collected rather than the biological growth pattern of the species.

The Gulf Corvina (Cynoscion othonopterus) is an ideal species to examine the performance of multiple growth models in a highly-exploited marine fish. Endemic to the northern Gulf of California, Mexico (Robertson & Allen, 2008), it is currently listed as vulnerable under the International Union for the Conservation of Nature (IUCN) Redlist (Chao et al., 2010). Gulf Corvina have experienced persistent overfishing on their spawning aggregations for the past two decades, which have resulted in growing concern for the fishery’s stability and longevity (Erisman et al., 2012; Ruelas-Peña, Valdez-Muñoz & Aragón-Noriega, 2013; Erisman et al., 2014; Ortiz et al., 2016). The life history of this species has been well documented and provides an ideal foundation for future analysis of individual and population growth (Román-Rodríguez, 2000; Gherard et al., 2013). With a documented maximum size of 1,013 mm total length (TL) and a documented maximum age of 9 years, Gulf Corvina is a fast-growing, short-lived species which reaches sexual maturity at 2 years of age (Gherard et al., 2013). However, the combination of highly efficient, size-selective gear and persistent overfishing have severely truncated the age structure of the population (Erisman et al., 2014; Ortiz et al., 2016). The mean age of capture of Gulf Corvina is 5 years (ca. 700 mm TL), and few individuals older than age 7 or younger than age 4 have been observed in the fishery (Gherard et al., 2013; Erisman et al., 2014; Ortiz et al., 2016).

Past studies of Gulf Corvina growth, which have relied solely on fishery-dependent data with incomplete sampling of all size and age classes, have produced different results due to differences in model selection approach. Based on the congruence of the model with the growth pattern of many species of the genus Cynoscion and other sciaenid fishes (Rutherford, Thue & Buker, 1982; Lowerre-Barbieri, Chittenden & Barbieri, 1995; Rodriguez & Hammann, 1997), Gherard et al. (2013) took a conservative, single model approach and fit the von Bertalanffy growth model to Gulf Corvina age–length data. Conversely, Aragon-Noriega (2014) chose a statistically-driven approach and fit several models to multiple datasets, concluding that Gulf Corvina grew in a biphasic pattern with slow growth in the beginning of life, rapid growth after age two, and slow growth after age four. Notably, Aragon-Noriega’s (2014) estimates for the L_∞ parameter varied greatly, from 735.0 to 1,126.6 mm, depending on which dataset was used. Given this variability, absence of biphasic growth patterns in similar sciaenids, and the distance from the maximum observed length of Gulf Corvina (1,013 mm; Gherard et al., 2013), Aragon-Noriega’s (2014) estimates may be biologically unrealistic. Mendivil-Mendoza et al. (2017) took a similar approach and found a similarly wide range of L_∞ values (666.7–1,306.0 mm). However, despite fitting models to similar data as Aragon-Noriega (2014), Mendivil-Mendoza et al. (2017) did not describe the biphasic growth pattern recorded by Aragon-Noriega (2014). The existence of discrepancies between the previous Gulf Corvina growth studies and the importance of the age–length relationship to the stock assessment of the fishery merit further investigation on the growth pattern of the species.

Here, we model the growth of Gulf Corvina and draw conclusions about data needs and fisheries assessments. Our specific objectives were to: (1) assess how representation of size and age classes affected growth parameter estimates and (2) compare the performance of multiple growth models for describing age-at-length data for Gulf Corvina. Through generating a more complete dataset than previous studies and testing for biases in our data with simple indicators, we addressed these objectives. Moreover, using the results of simulations with a per-recruit model, we discussed the implications of misrepresenting growth in highly-exploited, vulnerable marine fishes.

Materials and Methods

Data collection

Seven hundred and forty-nine Gulf Corvina were sampled from 2009 through 2013 at the three locations in the upper Gulf of California: El Golfo de Santa Clara (Sonora), San Felipe (Baja California), and El Zanjón (Baja California). Information on total length (TL) was recorded to the nearest mm for each fish collected, and the sagittal otoliths were removed, dried whole and stored until further use. Five hundred and thirty of these samples were collected from the commercial Gulf Corvina fishery and from bycatch from the shrimp fishery. These data were used by Gherard et al. (2013). In order to increase representation of size and age classes that were scarce in the dataset used by Gherard et al. (2013), we collected 219 additional samples in 2012–2013 from the bycatch of other fisheries (e.g., shrimp), fishery-independent sampling of small individuals (<30 cm TL), and the commercial Gulf Corvina fishery. All fish were deceased at the time of collection from fishers. The research protocol was approved under UCSD IACUC ID no. S13240, and data were collected under CONANP permit no. CNANP-00-007.

Otolith preparation and ageing protocols were followed according to the methods developed by Gherard et al. (2013) for Gulf Corvina. Whole sagittal otoliths were first mounted on wood blocks with a cyanoacrylate adhesive and a 0.5 mm dorsal-ventral cross-section was cut through the otolith focus using a double-bladed Buehler Isomet 1000 precision saw (Allen et al., 1995). Sub-sections were then mounted on a glass slide using thermoplastic glue and submerged in a glass petri dish with water and a black background. Transmitted light under a Zeiss Stemi 2000-C microscope with a Zeiss Axiocam 105 color camera at 6.25× total magnification was used to count the alternating opaque and translucent growth zones that define an annulus (Fig. 1). For the purposes of this study, an annulus was defined as one full opaque and translucent zone of growth (Cailliet et al., 1996), which was validated for Gulf Corvina by previous studies (Román-Rodríguez, 2000; Rowell et al., 2005; Gherard et al., 2013). Each otolith was aged by two independent readers from digital images of cross-sections, as direct observation through the scope did not distort band pattern and did not affect age estimates. Samples were excluded from analysis when discrepancies between readers occurred.

Figure 1: Transverse section of a sagittal otolith from a five-year old Gulf Corvina.
Annuli are numbered and marked by white rectangles. Transmitted light under a Zeiss Stemi 2000-C microscope with a Zeiss Axiocam 105 color camera at 6.25× total magnification was used to count the alternating opaque and translucent growth zones that define an annulus.

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DOI: 10.7717/peerj.5582/fig-1

Model fitting and assessment of fit

Growth modelling

A suite of growth models was fit to age data determined from otoliths as described, and length data obtained in the field. Model parameters were estimated using non-linear least squares regression with the Levenberg–Marquardt algorithm, and confidence limits were placed around parameter estimates in R studio (using the R packages Ogle, 2017; Elzhov et al., 2015 and Baty et al., 2015).

The specialized von Bertalanffy growth model (Von Bertalanffy, 1938) is given by: (1) $L (t) = L_{\infty} [1 - e^{- K (t - t_{0})}]$ where L(t) is size (in mm TL) at age t, L_∞ is the maximum average length (in mm TL), K is the growth rate coefficient (in year⁻¹), and t ₀ is the theoretical age at which length is zero (in years).

The Gompertz growth model (Gompertz, 1825) is given by: (2) $L (t) = L_{\infty} e^{(- (\frac{1}{K}) e^{- K (t - t_{0})})}$ where the consistent parameters are the same as described for Eq. (1), K₁ is the rate of exponential decrease of the relative growth rate with age (with units of yr⁻¹), and t₁ is 1∕k₄ ln λ, where λ is the theoretical initial relative growth rate at zero age (with units of yr⁻¹).

The logistic model (Ricker, 1975) is given by: (3) $L (t) = L_{\infty} {[1 + e^{- K (t - t_{0})}]}^{- 1}$ where the consistent parameters are the same as described for Eqs. (1) and (2), K₂ is a relative growth rate parameter (with units of yr⁻¹), and t₂ is the inflection point of the curve.

The Schnute model where a and b are not equal to zero (Schnute, 1981) is given by: (4) $L (t) = {[L_{1}^{b} + (L_{2}^{b} - L_{1}^{b}) \frac{1 - e^{- a (t - T_{1})}}{1 - e^{- a (T_{2} - T_{1})}}]}^{1 ∕ b}$ where T₁ is the first specified age, T₂ is the second specified age, L ₁ is size at age T₁, L₂ is size at age T₂, a is the constant relative rate of relative growth (in year⁻¹), and b is the incremental relative rate of relative growth (dimensionless),

Finally, the Schnute–Richards model (Schnute & Richards, 1990) is given by: (5) $L (t) = L_{\infty} {(1 + α e^{- a t^{c}})}^{1 ∕ b}$ where α, b, and c are dimensionless parameters, and a has the unit of year⁻^b.

Statistical measures of fit

Model fit was assessed with the bias-corrected Akaike Information Criterion (AICc) (Shono, 2000; Burnham & Anderson, 2004), and Bayesian Information Criterion (Schwarz, 1978) in R Studio (using the R package (Mazerolle, 2017)).

The formula for AICc is given by: (6) $A I C_{c} = A I C + \frac{2 k (k + 1)}{n - k - 1}$ where: (7) $A I C = - 2 log (L) + 2 k$ and n is the number of observations, k is the number of model parameters, and L is the likelihood.

The formula for BIC is given by: (8) $B I C = 2 ln (L) + k log (n)$ where parameter definitions are the same as described for Eq. (7).

The smallest AICc and BIC values indicate the best model. The difference between the two criteria is that AICc is designed to select the model that describes reality the best while treating no models as true, which is consistent with an information theory approach, whereas BIC is designed to select the true model. Practically, BIC penalizes for the number of parameters more heavily than AICc. AICc was used instead of AIC as it is bias-corrected at small n values or high k:n ratios; AICc converges to AIC at large n values (Burnham & Anderson, 2004). AICc and BIC values were calculated to show the absolute difference between model fits. Next, AICc weights were calculated for model averaging of parameter estimates; the AIC weighting formula is given by: (9) $w_{i} = \frac{e (- 0.5 Δ_{i})}{\sum_{k = 1}^{5} e (- 0.5 Δ_{k})}$ where parameter definitions are the same as described for Eqs. (7) and (8).

Simple indicators of biased data

Simulation of an ideal sampling outcome

To test for the influence of sampled population structure on growth model output, different amounts of simulated data were added to raw data so that each age observed (1–8) had 200 total observations. Data were simulated from a normal distribution with the same mean and standard deviation as the raw data at each age class. This simulation was not intended to generate the true population structure of Gulf Corvina in the Gulf of California, but rather to generate an equal number of samples in each age and size class. This simulation did not explicitly account for selectivity or limits in sampling effort, but filled in gaps left by these factors and others that prevented more equal representation of each size and age class in the raw data. Models were fit to the new dataset and goodness of fit was assessed in the same manner as was described above.

Froese and Binohlan’s empirical relationship

Froese & Binohlan’s (2000) empirical relationship between the longest fish in the data set (L_max) and L_∞ was used to specifically test for the influence of the lack of large and old fish in the raw dataset, which is likely due to heavy exploitation. If large and old fish are insufficiently represented in the dataset, it stands to reason that the L_∞ predicted by this relationship will be greater than the modelled L_∞. This relationship is given by: (10) $log L_{\infty} = 0.044 + 0.9841 * log (L_{max})$

Literature review

A brief literature review of sciaenid growth modelling was conducted to assess how the results of this study compared with other studies on fishes closely related to the Gulf Corvina (e.g., other species in the genus Cynoscion). In conjunction with Froese and Binohlan’s empirical relationship and the simple simulation of an ideal sampling scenario, this brief literature review was intended to check if the samples used in this study produced a biologically plausible growth pattern when growth was modelled.

Simulations with a per-recruit model

To be able to discuss the implications of misrepresenting growth in Gulf Corvina, we ran simulations with a per-recruit model detailed in Appendix S1. In brief, this per-recruit model estimates the female spawning-stock-biomass-per-recruit (SSBR; a proxy of reproductive capacity) and yield-per-recruit (YPR; exploitable biomass) of Gulf Corvina in relation to the annual exploitation rates of the old adults (≥5 year-old individuals) of the species (E_OA). In this per-recruit model, Gulf Corvina are assumed to grow according to one of five alternative growth models: (1) the von Bertalanffy model developed in Gherard et al. (2013), referred to as the “Gherard model”; (2) the von Bertalanffy model fit to raw data in the present study; (3) the von Bertalanffy model fit to raw data bolstered by simulation values in this study; (4) the Schnute–Richards model fit to raw data in the present study; and (5) the Schnute–Richards model fit to raw data bolstered by simulation values in this study. The current E_OA was estimated to be 0.825 year⁻¹ (Appendix S1). We first ran simulations with the per-recruit model to determine the maximum value of the YPR of Gulf Corvina (Y PR_max) and the natural SSBR of Gulf Corvina (NSSBR), i.e., its SSBR in the absence of fishing (Appendix S1). Then, we estimated the current fraction of NSSBR (current FNSSBR, i.e., the ratio of current SSBR to NSSBR) and the current YPR over Y PR_max of Gulf Corvina, when each of the five abovementioned growth models is used to represent the growth in length of Gulf Corvina.

Results

Length and age structure

A bimodal distribution was observed in the length and age structure of the fish used in this study (Figs. 2 and 3). The first mode of the distribution represents Gulf Corvina caught as bycatch, whereas the second represents Gulf Corvina caught in the targeted fishery. Lengths ranged from 141–1,013 mm TL, and ages ranged from 1–8 years.

Figure 2: Total length frequency of Gulf Corvina from raw data represented in 10 mm bins.
A bimodal distribution was observed, with the first consisting of Gulf Corvina caught as bycatch, and the second largely consisting of fish from the directed fishery. Few fish larger than 750 mm are present in this dataset.

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DOI: 10.7717/peerj.5582/fig-2

Figure 3: Age frequency of Gulf Corvina from raw data.
A bimodal distribution was observed, with the first consisting of Gulf Corvina caught as bycatch, and the second largely consisting of fish from the directed fishery. Few fish older than age 6 are present in this dataset.

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DOI: 10.7717/peerj.5582/fig-3

Model fitting and assessment of fit for models fit to raw data

Growth patterns and parameter estimates for models fit to raw data

The Gompertz, logistic, and von Bertalanffy models yielded asymptotic growth patterns, while the Schnute–Richards model described biphasic growth and the Schnute model described near-linear growth after 1.5 years of life (Fig. 4). Modelled length at age was most similar among models at intermediate ages, where samples were most abundant (Fig. 4). Conversely, modelled length at age was most variable at young and old ages, where samples were most scarce (Fig. 4). Estimates of L_∞ ranged from 730.91 mm (Schnute–Richards model) to 916.05 mm (von Bertalanffy model). All parameter estimates are summarized in Table 1, while confidence intervals around parameter estimates are reported in Appendix S2.

Table 1:

Parameter estimates for growth models fit to raw age-length data for Gulf Corvina.

Estimates of L_∞ were variable, but not as variable as those reported in previous multi-model studies of Gulf Corvina growth (Aragon-Noriega, 2014; Mendivil-Mendoza et al., 2017). Confidence intervals around parameter estimates may be found in the Supplemental Information.

Model name	Model equation when fit to raw data
von Bertalanffy	$L (t) = 916.05 [1 - e^{- 0.28 (t - (- 0.17))}]$
Gompertz	$L (t) = 820.64 e^{(- (\frac{1}{0.51}) e^{- 0.51 (t - 1.29)})}$
Logistic	$L (t) = 778.88 {[1 + e^{- 0.76 (t - 1.92)}]}^{- 1}$
Schnute	$L (t) = {[14 1^{- 0.33} + (101 3^{- 0.33} - 14 1^{- 0.33}) \frac{1 - e^{- 3.36 (t - 1)}}{1 - e^{- 3.36 (8 - - 1)}}]}^{1 ∕ - 0.33}$
Schnute–Richards	$L (t) = 730.91 {(1 + (- 0.003) e^{- (0.12) t^{2.18}})}^{1 ∕ 0.003}$

DOI: 10.7717/peerj.5582/table-1

Measures of statistical fit for models fit to raw data

AICc and BIC values indicated that the Schnute–Richards model described the raw data best, followed by the logistic, Gompertz, von Bertalanffy, and Schnute models (Table 2). The AIC weighting formula gave full support to the Schnute–Richards model, so no model averaging of parameters was necessary.

Table 2:

Statistical measures of fit for growth models fit to raw age-length data for Gulf Corvina.

The Schnute–Richards model fit the data best according to AICc and BIC values, but is only marginally better than the logistic, Gompertz, and von Bertalanffy models. Note: K indicates the number of parameters.

Model	K	AICc	ΔAICc	AICc weight	BIC	ΔBIC
Schnute–Richards	6	8,759.82	0.00	1	8,787.42	0.00
Logistic	4	8,773.62	13.80	0	8,792.04	4.62
Gompertz	4	8,789.69	29.87	0	8,808.11	20.69
von Bertalanffy	4	8,813.66	53.84	0	8,832.08	44.66
Schnute	3^a	9,148.78	388.96	0	9,162.61	375.19

DOI: 10.7717/peerj.5582/table-2

Notes:

aThree parameters were estimated by nonlinear least squares, but four additional parameters were manually inputted (maximum and minimum ages and lengths) for the Schnute model.

Figure 5: Growth models fit to raw Gulf Corvina age-length data bolstered by simulated values.
All models except for the Schnute described asymptotic growth, and only showed slight differences in modelled size at age. Differences in modelled size at age were most pronounced at the beginning and end of life.

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DOI: 10.7717/peerj.5582/fig-5