Size-at-age (length or mass) is an important metric about animals and so there is a large body of literature about which growth models fit best to given size-at-age data. The von Bertalanffy [1] and Pütter [2] differential Equation (1) provides a comprehensive framework for the most common models for mass-growth:

d m ( t ) d t = p ⋅ m ( t ) a − q ⋅ m ( t ) b (1)

It describes body mass m(t) > 0 as a function of age t, using the following five model parameters: The exponent-pair a < b (“metabolic scaling exponents”) and the constants p and q are assumed to be non-negative; m₀ > 0 is an initial value, i.e. m(0) = m₀. In the case of equal exponents, Equation (1) is replaced by the generalized Gompertz [3] differential Equation (2), the limiting case of Equation (1), when b approaches a:

d m ( t ) d t = p ⋅ m ( t ) a − q ⋅ ln ( m ( t ) ) ⋅ m ( t ) a (2)

This equation uses four model parameters: a, p, q (non-negative) and m₀ > 0. In general, the solutions of (1) and (2) are expressed in terms of non-elementary functions, namely hypergeometric functions [4] and exponential integrals [5] , respectively. The present paper used numerical solutions [6] [7] . For several “named models” these equations could be solved by elementary means, namely for the exponent-pairs a = 2/3 and b = 1 of von Bertalanffy [1] , a = 3/4, b = 1 of West [8] , a = 1, b = 2 for logistic growth of Verhulst [9] , and a = b = 1 of Gompertz [3] . There are also elementary solutions for Richards’ [10] model (a = 1 and b > 1 is a free parameter) and for the generalized von Bertalanffy model (b = 1 and 0 ≤ a < 1 is a free parameter). The von Bertalanffy growth function for length (VBGF) fits to the present framework, too: It is a special case of Equation (1) using the exponent pair a = 0, b = 1 (bounded exponential growth). Other special cases are power-laws between mass and time (q = 0, p > 0), in particular linear growth (a = b = q = 0, p > 0, m₀ > 0).

Figure 1 plots the exponent-pairs for these named models. Several of these models were motivated by competing biological theories about the relation between growth and metabolism; whereby different authors proposed different exponent-pairs [11] . However, these models “have only a loose connection to the biology behind the actual growth processes” [12] . Therefore, no single growth model may be exactly correct for all species [13] . For fish, in particular, the optimal parameters of the growth models may depend on environmental factors [14] [15] . Where there is no information about other biological factors for growth, the models may nevertheless be used to extract relevant information from the data, as is current practice fisheries management [16] .

The models differ in the assumed exponents and in the number of free parameters (these are optimized to obtain the best fit to given data): For the “named models” with given exponent-pairs the parameters p, q and m₀ are free; for the more general model classes in addition one or two exponents are free. The typical

growth curves (solutions of the model equations) are increasing, bounded and sigmoidal (S-shaped): Initially the rate of growth increases, until the inception point is reached. Subsequently, it decreases to zero in the limit, when the asymptotic mass (m_max) is reached. The latter condition (m' = 0) provides the following equation for the asymptotic mass:

m max = ( p q ) 1 b − a for a < b , and m max = exp ( p q ) for a = b (3)

The parameters a, b, p, q come from Equation (1) and Equation (2), respectively. For exceptional exponents and parameters, the growth curves may be non-sigmoidal (a = 0) or unbounded (p > 0, q = 0). For instance, the most common growth model in fisheries literature, VBGF for length (Google search for “VBGF, fish”: ca. 15,000 results), is not sigmoidal. However, it is equivalent to a sigmoidal model for mass-growth in the following sense [1] : If length-growth follows VBGF, and if mass is assumed to be proportional to a power L^k of length L (k > 1), then mass-growth is described by the exponent-pair a = 1 − 1/k > 0, b = 1 (sigmoidal growth curve).

Summarizing, there seems to be a consensus in literature that the mass-growth of fish is bounded and sigmoidal. However, catch data for fish seem not to support this statement. The authors [17] investigated a set of 60 data for different species (37 for fish) and fitted Equation (1) with the exponent b = 1 (generalized von Bertalanffy model). They found that for 17 of 37 fish-data, but only for one non-fish species, any exponent a could be used to model mass-growth without affecting the fit to the data significantly (when the other free parameters p, q, m₀ were optimized). In particular, a fit by a non-sigmoidal growth curve (a = 0) was acceptable. Conversely, for 18 of 20 data, where a = 0 was acceptable, any other exponent was acceptable, too. As the authors noted, such a high variability has implications for data-fitting, as standard search strategies may not always find the right direction towards the optimum parameters, if there is no significant difference in the fits. Indeed, fisheries management literature reported problems with convergence [18] . The authors attributed the high variability of the exponents to differences in data quality, as most non-fish data were average values from controlled studies, where measurements were conducted repeatedly for the same group of animals, while most fish-data were about wild-caught fish, where each fish contributed only once in its lifetime to the measurements. Such data were also affected by gear-selectivity, where there may be few data about young (small) fish, as these could escape from the trawler nets, or as anglers preferred to harvest large fish, and few data about older fish, as most of them did not survive long enough to come close to their maximum size, resulting in many different possible shapes for the catch-curve that describes the age structure of the collected data [19] . As a consequence, several data supported unbounded growth [20] ; [21] reported this for about 1/3 of their fish data. This was considered as an indication for too few data about older animals. By similar considerations, where data supported bounded non-sigmoidal growth this was considered as an indication for too few data about young fish.

The conclusion about the high variability for fish-data was drawn from data-fitting to average weights. The present paper therefore reconsidered this issue and asked, if data-fitting to raw data would change the picture. More specifically, the paper aimed at identifying data that could be fitted well by a bounded sigmoidal mass-growth curve and where this fit was clearly superior to the fit by typically non-sigmoidal model functions. These data were deemed insofar as suitable for the application of sigmoidal growth models, as the variability of the exponents was more restricted.

To achieve this goal, the paper compared the fit of three typical models: Linear growth, as it represents unbounded models, bounded exponential growth, as it represents bounded non-sigmoidal models, and the von Bertalanffy exponent-pair, as it represents sigmoidal growth. This comparison was applied to various data for wild-caught fish, whereby the model curves were fitted to the raw data from [22] . Further, data-fitting took into account the typically heteroscedastic distributions of size (meaning a larger variance for a higher mass). The paper tested, if there was an acceptable fit by the sigmoidal model, but no acceptable fit by the other two models. The paper then explored, based on this test, how many of the considered catch data were suitable for the application of sigmoidal growth functions. For these data it determined the best fitting sigmoidal model amongst models described by Equation (1) and Equation (2) and searched for a pattern of the optimal exponent-pairs.

The term “acceptable fit” needs a technical definition, explained in the Methods. For instance, a visual inspection alone did not suffice to recognize that the fit of the bounded exponential model was acceptable for Figure 2, but not acceptable for Figure 3, when compared to the von Bertalanffy exponent-pair model. For, the plot did not reveal, for how many data-points a good/poor fit was achieved.

The authors conducted a literature search for growth data of wild-caught fish, referring to the following secondary sources: 39 data-sets came from a repository by Derek Ogle [22] listed under “growth” (Table 1 for the file names). The conversion of length-at-age data into mass-at-age was based on Fish Base [23] . All data were processed in Mathematica 11.3 of Wolfram Research^Ò; Appendix explains the used code. As it is generally accepted in fisheries research that model comparison is a data-hungry exercise, only data for population-level studies with initially at least 500 data-points (different fish) were considered. Further, data were removed if after the elimination of incomplete data-points (e.g. size at an unknown age) and filtering (e.g. by sex) there remained fewer than about 100 data-points or less than six points of time (The former condition was not strict, but the latter was, as the paper compared models with up to 6 free parameters).

Literature in fisheries management prefers to fit model curves directly to the raw data, assuming a heteroscedastic distribution of size. Most common is the assumption of a lognormal distribution [24] , as by Equation (4) for a lognormal distribution the standard deviation stdev of size is proportional to the expected

^a#n is the numbering of the data that were retrieved from the selected dataset.

value ev of size with the shape parameter as (approximate) constant of proportionality:

e v = exp ( m + s 2 2 ) and s t d e v = e v ⋅ exp ( s 2 2 ) − 1 ≈ e v ⋅ ( s + O ( s 2 ) ) (4)

Here, m is the location parameter and s > 0 is the shape parameter of a lognormal distribution and the rightmost term in Equation (4) assumes s » 0 (whereby O(∙) refers to the Taylor series remainder). The present paper used this approach in order to conform to established practice in fisheries management literature (A different approach assumes a normal distribution for each age class, but an age dependent variance; e.g. [25] ).

The hypothesis of lognormally distributed data was not always exactly correct: For the data of Figure 3 the Cramér-von Mises distribution fit test [26] identified significant deviations from a normal distribution. This was due to the large number of data points. However, for practical purposes (convenience for data-fitting) the hypothesis of a lognormal distribution could be retained, as a probability plot (Figure 4) confirmed that the lognormal distribution was a reasonable approximation to the true distribution (closeness to the diagonal). Further, the paper avoided to draw conclusions that would essentially depend on this distribution assumption.

There are several approaches for heteroscedastic data fitting, amongst them weighted least-squares for the mass-at-age data, using the reciprocals of the standard deviations (of mass at each age class) as weights. The present paper uses instead transformed data. For, technically data-fitting to lognormally distributed mass-at-age data (i.e. maximum likelihood estimation of the optimal distribution and growth function parameters) is equivalent to using a logarithmic transformation of mass and the method of least squares to fit the transformed growth function u(t) = ln(m(t)) to these transformed data (Thereby also the parameters for linear growth, m = m₀ + p×t, were determined from a nonlinear regression for u).

Further, (for the transformed data) the least squares method is equivalent to the minimization of the lack-of-fit sum of squares LFSS, where LFSS is the weighted sum of the deviations of the model curve from the averages at each age (the weight is the count of data at this age). The present paper used this reformulation of the least squares method to speed up computations (It has been used earlier in fisheries management, e.g. [27] ). The use of LFSS has the following justification: The method of least squares assumes that the errors (deviations from the model curve) are random, meaning independent (normally distributed) random variables with expected value 0. Therefore, the sum of the squared errors, SSE, may be decomposed into SSE = LFSS + PESS, where PESS, the sum of squares of pure errors, is the sum of the squared deviations of the data at each age from the averages at each age. Curve fitting can only minimize LFSS, while PESS remains unchanged and can be ignored for the purpose of data-fitting. Summarizing, data fitting was reduced to minimizing LFSS, a weighted sum of the squared differences between the average of the logarithmic weights at this age and the logarithm of the growth curve at this age.

The data fit to the three test models (linear, bounded exponential, and von Bertalanffy exponent-pair) used the Mathematica function NMinimize and simulated annealing [28] as optimization method. Further, the parameters of the bounded models were restricted so as to ensure an “empirically meaningful” asymptotic mass, defined from a comparison of the asymptotic mass m_max of Equation (3) with the maximal (arithmetic) mean value of the mass at different ages. If m_max was in the interval between half and twice that maximal average mass, as could be observed e.g. for the model curves in Figure 3, then the corresponding model curve was accepted as empirically meaningful. This constraint helped to avoid that optimization was trapped at clearly false growth functions (e.g. constant functions). Note that the paper did not simplify optimization by using literature values for the initial condition m₀ (e.g. natal mass) or for the asymptotic mass m_max.

For the subsequent optimization of the general models (1) and (2), the paper used a custom-made variant of simulated annealing to minimize LFSS, but without constraints. This approach is explained below for the search of the parameters for the general von Bertalanffy-Pütter model (1). The following computations were repeated in a loop of 500,000 steps. Using starting values for the parameters a, b, m₀, p, q (starting from the optimal parameters of the von Bertalanffy exponent-pair or, Equation (2), from random numbers), it multiplied them by random numbers between 1 ± dist, but close to 1 (e.g. dist = 0.01); this deviated from the usual simulated annealing, where small random numbers are added. If the altered parameters improved LFSS, they were accepted. Otherwise, the increase of ln(LFSS/N) was compared with an exponentially distributed random number (e.g. mean value exm = 1; N = number of data points); this function was motivated by Akaike’s [29] index explained below. If the increase was below this random number, the altered parameters were accepted. Otherwise, the previous parameters were retained. The parameters dist and exm were set by the authors so as to obtain a reasonable convergence (several experments). After each 10,000 steps, dist and exm were reset: They were reduced by 10% in order to avoid moving too fast too far away from a good candidate for the optimum and the computations were restarted with the best hitherto found parameters. For several data the loop was repeated. In case that the optimal exponents were close to the diagonal (distance below 0.1), the optimization was repeated for the generalized Gompertz Equation (2). For the present data this did not occur. For three data the custom-made optimization approach was needed also for the von Bertalanffy exponent-pair, as the general purpose procedures produced a stack-overflow.

For the definition of an acceptable fit, the paper defined a measure of fit that was motivated by the Akaike information criterion [29] , namely the following pseudo-Akaike-weight pprob defined from a pseudo-Akaike index PAIC. In terms of this index, the most acceptable model has the least PAIC, defined below. The best fitting model (least LFSS) needs not be most acceptable (penalty for additional parameters).

P A I C ( model ) = A C ⋅ ln ( L F S S ( model ) N ) + 2 ⋅ K + 2 ⋅ K ⋅ ( K + 1 ) A C − K − 1 (5)

The relation of formula (4) to the usual AIC is explained below. PAIC was defined from the lack-of-fit sum of squares for the model, LFSS (model), from the number N of data-points, from the number AC of age classes and from the number K of optimized parameters. Thereby, K = 4 for the test models (except for linear growth: K = 3), as m₀, p, q and implicitly the shape parameter s of the lognormal distribution were optimized. Further, K = 5 and K = 6 for the generalized Gompertz model (2) and the von Bertalanffy-Pütter model (2), respectively. The index pprob compares a given model with the most acceptable one.

p p r o b ( model ) = e − Δ / 2 1 + e − Δ / 2 , (6)

Here, D = PAIC (model) ? PAIC (most acceptable model) > 0. The paper defined a fit as acceptable, if pprob ≥ 2.5%. As the maximal value is pprob = 50%, inacceptable fits were linked to the lowest 5% of possible values of pprob. For example, in Figure 3 and Figure 2 the von Bertalanffy exponent-pair model had the best fit amongst the three compared models. For Figure 3, but not for Figure 2, the fit was insofar significantly better, as none (respectively both) of the non-sigmoidal models had an acceptable fit. Further, for both figures the best fitting model (amongst those considered) did not reduce LFSS enough to minimize also pprob. Thus, its predictive power was highest amongst the considered models, but the loss in simplicity may not have been justified by the gain in predictive power.

The following outline explains, why the authors decided to use pprob as a measure of the goodness of fit. PAIC is a modification of the Akaike index AIC_c for small samples [30] [31] ; for variants c.f. [32] . Recall AIC_c = N∙ln(SSE/N) + 2∙K + 2∙K∙(K + 1)/(N - K - 1); in this formula the authors replaced N by AC and SSE by LFSS×AC/N. Thus, PAIC assessed the fit of the models in terms of their fit to the averages (of the logarithmically transformed data) at age, weighted by the percentage of data represented by this age. Therefore, pprob was the Akaike weight relative to this fit to averages.

Clearly, PAIC penalized complex models more, than AIC_c did, because for the averages there were only few degrees of freedom. However, in the context of fitting growth models the degrees of freedom attainable for large sample sizes may be illusory, if data come from a few age classes. For instance, data for thousands of fish, but all at age 0, do not confer any information about growth. PAIC has the advantage that with a high number of data points a slightly better fitting model with more parameters is not deemed as better, unless the number of age classes is high enough to support more parameters. Further, the Akaike measures depend heavily on the assumption of normally distributed data, which is not exactly true for all fish data, as was noted previously. By contrast, for (weighted) averages of large samples (of logarithmically transformed mass) there is a theoretical justification for assuming a normal distribution, whence for PAIC and pprob the usual interpretations for the Akaike index and the Akaike weight may be assumed. In particular, pprob may be interpreted as a probability that a model is true, assuming that one of the considered models is true. In view of this interpretation, pprob has the same meaning, independently of the considered data, whence it could be used to define the notion of an acceptable fit.

The traditional explanation of differential Equation (1) proposed that the rate of growth would be proportional to the difference between anabolism and catabolism, both of which would be proportional to a power of mass. Specific values of the exponents were then derived from biophysical arguments; e.g.: b = 1, as catabolism would be proportional to mass (number of sustained cells) and a = 2/3, as anabolism would be proportional to the gills’ surface (oxygen consumption) and therefore to the 2/3^th power of mass [1] .

The present alternative explanation of Equation (1) hypothesizes that body mass would be a function of the total food intake F(t) since birth (t = 0), whereby the (individual) asymptotic mass m_max would be approached at a rate dependent on instantaneous food intake:

d m / d t m max d − m ( t ) d = k 1 ⋅ d F d t (7)

In the formula, k₁ > 0 a constant of proportionality and d is a constant that explains the speed of growth: With the same total food intake F, with a larger d the asymptotic mass m_max is approached faster. This model, using d = 1, was proposed by Parks ( [35] at p. 26), who supported it by evidence from farmed animals. The food consumption, which cannot be observed for wild-caught fish, can be eliminated by the additional hypothesis that the instantaneous food intake dF/dt would be proportional to the energy needs E (i.e. dF/dt = k₂∙E), which in turn would be proportional to a power of body mass (i.e. E = k₃∙m^a). Substituting this into Equation (7), then after a multiplication and renaming of constants (p = k₁∙k₂∙k₃∙(m_max)^d, q = k₁∙k₂∙k₃) this simplifies to Equation (8), which is just another parametrization of Equation (1):

d m ( t ) d t = p ⋅ m ( t ) a − q ⋅ m ( t ) a + d (8)

This parametrization suggests to consider model classes that are defined by assuming a value of d. As the original motivation comes from Parks, this paper calls this the Parks-d model class. For instance, Parks’ assumption would define the Parks-1 model class with bounded exponential growth (a = 0: constant energy needs) and logistic growth (a = 1: energy needs proportional to mass) as special cases. As the von Bertalanffy exponent-pair is a special case of the Parks-1/3 model class, the present paper investigates this class.

Figure 1 suggests that the Parks-1/3 model class may come close to providing the best-fitting description of the mass-growth of several species of fish. Figure 7 illustrates the good fit compared to the best-fitting and the von Bertalanffy models for the six considered data. The optimization used the custom-made approach, starting with a, m₀, p, q of the optimal model (setting b = a + 1/3). Table 4 summarizes the outcome of the data-fit for this model class. In comparison to the von Bertalanffy exponent-pair model and the best fitting von Bertalanffy-Pütter model, the Parks-1/3 model had the most acceptable fit for three data and an acceptable fit for the three other data, where the von Bertalanffy exponent-pair model had the most acceptable fit. However, the fit of the von Bertalanffy exponent-pair model was not acceptable for data #10 (It was acceptable,

when compared with the non-sigmoidal models, but the consideration of the best-fitting model changed the assessment). Further, the fit of the Parks-1/3 model was nearly optimal, as seen in Figure 7: For all data the Parks-1/3 curve coincided with the best-fitting von Bertalanffy-Pütter curve. Consequently, the Parks-1/3 model was more acceptable, as it used fewer parameters.

There were surprisingly few data that supported the hypothesis of a sigmoidal shape of mass growth, and those did support the new Parks-1/3 model (exponent-pairs b = a + 1/3).

The 39 datasets allowed to define 70+ data using additional stratifications. When testing the suitability of these data for modeling sigmoidal growth, more than 3/4 of these data were removed due to poor data quality (insufficient size). Of the remaining 15 data almost 2/3 were removed due to the insufficient fit of a sigmoidal growth function. A speculative explanation for the insufficient fit may be data quality, again. There were some data with notable differences between the arithmetic and geometric mean (Figure 6). This difference was caused by the presence of atypically small fish. In case that this difference has a biological meaning (e.g. slow and fast growth as different survival strategies), then it may be meaningful to split the data and model the growth of slow and of fast growing fish separately.

For all six data that were suitable for modeling sigmoidal growth, the Parks-1/3 model provided an acceptable fit; for 50% of the data it improved upon the von Bertalanffy exponent-pair model (one significant improvement) despite the penalty for the additional free parameter. Therefore, when compared to the model classes of Richards, Gompertz or the generalized von Bertalanffy model, the Parks-1/3 model may provide the most realistic description of the mass growth of fish. In particular, the advantage of this model is its consideration food consumption and the flexibility gained by an additional parameter that for all data resulted in a near optimal fit.