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Growth Function Parameterizations in FSA

Derek H. Ogle

2025-05-06

Source: vignettes/articles/Growth_Function_Parameterizations.qmd

Introduction

The most common growth models used in fisheries, such as von Bertalanffy, Gompertz, logistic, Richards, Schnute, and Schnute-Richards, are non-linear models. Most of these functions can be expressed in different forms, called parameterizations, with different parameters. Parameters in common between the different parameterizations and all predicted values will be the same across parameterizations. However, the different parameters may provide insights into different characteristics of growth (e.g., mean length at a specific age, or mean age at a specific length) or provide some benefits to fitting the non-linear function to real data.

The main growth-related analysis functions in FSA – makeGrowthFun(), showGrowthFun(), findGrowthStarts() – may be used for a variety of parameterizations of the common growth functions. This article shows those different parameterizations and defines the various parameters.


von Bertalanffy

Annual Length-at-Age Functions

The von Bertalanffy parameterizations for length and annual age data are in Table 1. In these equations, the response variable, LL, is length and the explanatory variable, tt is age, and E(Lt)E(L_t) is the “expected length at age tt” or the mean length at age tt.


Table 1: Parameterizations of the von Bertalanffy growth equation for length-at-age (annual) data available in FSA. Synonyms are “Beverton-Holt” for “Traditional”, “von Bertalanffy” for “Original”, “Ogle” for “Ogle-Isermann”, and “Laslett” or “Polacheck” for “Double”.
param pname Equation
1 Traditional E(Lt)=L∞(1−e−K(t−t0))E(L_t)=L_\infty\left(1-e^{-K(t-t_0)}\right)
2 Original E(Lt)=L∞−(L∞−L0)e−KtE(L_t)=L_\infty - (L_\infty-L_0)~e^{-Kt}
3 Gallucci-Quinn E(Lt)=ωK(1−e−K(t−t0))E(L_t)=\frac{\omega}{K}\left(1-e^{-K(t-t_0)}\right)
4 Mooij E(Lt)=L∞−(L∞−L0)e−ωL∞tE(L_t)=L_\infty - (L_\infty-L_0)~e^{-\frac{\omega}{L_\infty}t}
5 Weisberg E(Lt)=L∞(1−e−log(2)t−t0t50−t0)E(L_t)=L_\infty\left(1-e^{-log(2)\frac{t-t_0}{t_{50}-t_0}}\right)
6 Ogle-Isermann E(Lt)=Lr+(L∞−Lr)e−e−K(t−tr)E(L_t)=L_r + (L_\infty-L_r)~e^{-e^{-K(t-t_r)}}
7 Schnute E(Lt)=L1+(L3−L1)1−e−K(t−t1)1−e−K(t3−t1)E(L_t)=L_1+(L_3-L_1)\frac{1-e^{-K(t-t_1)}}{1-e^{-K(t_3-t_1)}}
8 Francis E(Lt)=L1+(L3−L1)1−r2t−t1t3−t11−r2E(L_t)=L_1+(L_3-L_1)\frac{1-r^{2\frac{t-t_1}{t_3-t_1}}}{1-r^2} where r=L3−L2L2−L1r=\frac{L_3-L_2}{L_2-L_1}
9 Double E(Lt)=L∞(1−e−K2(t−t0))(1+e−b(t−t0−a))(1+eab)−K2K1bE(L_t)=L_\infty\frac{\left(1-e^{-K_2(t-t_0)}\right)\left(1+e^{-b(t-t_0-a)}\right)}{\left(1+e^{ab}\right)^{-\frac{K_2K_1}{b}}}


Parameters in these models are:

  • L∞L_\infty = asymptotic mean length
  • KK = exponential rate of approach to L∞L_\infty
  • t0t_0 = nuisance parameter that is the hypothetical time/age when mean length is 0
  • L0L_0 = mean length at age-0 (i.e., hatching or birth)
  • ω\omega = growth rate near t0t_0
  • t50t_{50} = age when half of L∞L_\infty is reached
  • trt_r = mean age at LrL_r (sometimes this is a constant)
  • LrL_r = mean length at trt_r (sometimes this is a constant)
  • L1L_1 = mean length at t1t_1 (generally a younger age)
  • L2L_2 = mean length at t2t_2 (generally an intermediate age)
  • L3L_3 = mean length at t3t_3 (generally a older age)

Constant values (i.e., set by the user) are:

  • trt_r = mean age at LrL_r (sometimes this is a parameter)
  • LrL_r = mean length at trt_r (sometimes this is a parameter)
  • t1t_1 = a younger (generally) age
  • t2t_2 = an age halfway between t1t_1 and t2t_2
  • t3t_3 = an older (generally) age


Seasonal Length-at-Age Functions

The von Bertalanffy parameterizations for length and seasonal age data are in Table 2.


Table 2: Parameterizations of the von Bertalanffy growth equation for length-at-age (seasonal) data available in FSA. Synonyms are “Somers1” for “Somers”.
param pname Equation
10 Somers E(Lt)=L∞(1−e−K(t−t0)−S(t)+S(t0))E(L_t)=L_\infty\left(1-e^{-K(t-t_0)-S(t)+S(t_0)}\right) where S(t)=CK2πsin(2π(t−ts))S(t)=\frac{CK}{2\pi \text{sin}(2\pi(t-t_s))}
11 Somers2 E(Lt)=L∞(1−e−K(t−t0)−R(t)+R(t0))E(L_t)=L_\infty\left(1-e^{-K(t-t_0)-R(t)+R(t_0)}\right) where R(t)=CK2πsin(2π(t−WP+0.5))R(t)=\frac{CK}{2\pi \text{sin}(2\pi(t-WP+0.5))}
12 Pauly E(Lt)=L∞(1−e−K′(t′−t0)−V(t′)+V(t0))E(L_t)=L_\infty\left(1-e^{-K'(t'-t_0)-V(t')+V(t_0)}\right) where V(t)=K′(1−NGT)2πsin(2π(1−NGT)(t−ts))V(t)=\frac{K'(1-NGT)}{2\pi}\text{sin}\left(\frac{2\pi}{(1-NGT)(t-t_s)}\right)


New parameters in these growth functions are:1

  • CC = proportional growth depression at “winter peak”
  • tst_s = time from t=0t=0 until first growth oscillation begins
  • WPWP = “winter peak” (point of slowest growth)
  • K′K' = exponential rate of approach to L∞L_\infty during the growth period
  • NGTNGT = length of “no-growth period”


Tag-Recapture Functions

The von Bertalanffy parameterizations for use with tag-recapture data are in Table 3. Note that the response variable is generally the change in length (i.e., growth increment) from time of marking (i.e., tagging) to time of recapture, Lr−LmL_r-L_m. Some models are parameterized to have LmL_m on the right-hand-side though. The explanatory variable is the change in time between the time of marking and recapture, δt\delta t.


Table 3: Parameterizations of the von Bertalanffy growth equation for tag-recaputre data available in FSA. Synonyms are “Fabens1” for “Fabens” and “Wang1” for “Wang”.
param pname Equation
13 Fabens E(Lr−Lm)=(L∞−Lm)(1−e−Kδt)E(L_r-L_m)=(L_\infty-L_m)\left(1-e^{-K\delta t}\right)
14 Fabens2 E(Lr)=Lm+(L∞−Lm)(1−e−Kδt)E(L_r)=L_m + (L_\infty-L_m)\left(1-e^{-K\delta t}\right)
15 Wang E(Lr−Lm)=(L∞+β(L‾m−Lm)−Lm)(1−e−Kδt)E(L_r-L_m)=(L_\infty+\beta(\bar{L}_m-L_m)-L_m)\left(1-e^{-K\delta t}\right)
16 Wang2 E(Lr−Lm)=(α+βLm(1−e−Kδt)E(L_r-L_m)=(\alpha+\beta L_m\left(1-e^{-K\delta t}\right)
17 Wang3 E(Lr)=Lm+(α+βLm(1−e−Kδt)E(L_r)=L_m+(\alpha+\beta L_m\left(1-e^{-K\delta t}\right)
18 Francis2 E(Lr−Lm)=[L2g1−L1g2g1−g2−Lm][1−(1+g1−g2L1+L2)dt]E(L_r-L_m)=\left[\frac{L_2g_1-L_1g_2}{g_1-g_2}-L_m\right]\left[1-\left(1+\frac{g_1-g_2}{L_1+L_2}\right)^{dt}\right]


New parameters in these growth functions are:

  • β\beta = a measure of individual fish variability
  • α\alpha = a nuisance parameter related to L∞L_\infty and an individual’s LmL_m
  • g1g_1 = mean annual growth rate at the (relatively small) reference length L1L_1
  • g2g_2 = mean annual growth rate at the (relatively large) reference length L2L_2


Seasonal Tag-Recapture Functions

One von Bertalanffy parameterization for seasonal tag recapture data is in Table 4.


Table 4: Parameterizations of the von Bertalanffy growth equation for seasonal tag-recapture data available in FSA.
param pname Equation
19 Francis3 E(Lr−Lm)=[L2g1−L1g2g1−g2−Lm][1−(1+g1−g2L1+L2)t2−t1+S(t2)−S(t1)]E(L_r-L_m)=\left[\frac{L_2g_1-L_1g_2}{g_1-g_2}-L_m\right]\left[1-\left(1+\frac{g_1-g_2}{L_1+L_2}\right)^{t_2-t_1+S(t_2)-S(t_1)}\right] where S(t)=usin(2π(t−w)2π)S(t)=u\text{sin}\left(\frac{2\pi(t-w)}{2\pi}\right)


New parameters in this growth function are:

  • uu = “the extent of seasonality” (uu=0 is no seasonality)
  • ww = time of year for maximum growth rate


Gompertz

Annual Length-at-Age Functions

Gompertz parameterizations for length and annual age data are in Table 5.


Table 5: Parameterizations of the Gompertz growth equation for length-at-age (annual) data available in FSA. Synonyms are “Gompertz” for “Original”, “Quinn-Deriso1” for “Ricker2”, and “Quinn-Deriso2” for “Ricker3”.
param pname Equation
1 Original E(Lt)=L∞e−ea1−gitE(L_t)=L_\infty e^{-e^{a_1-g_it}}
2 Ricker1 E(Lt)=L∞e−e−gi(t−ti)E(L_t)=L_\infty e^{-e^{-g_i(t-t_i)}}
3 Ricker2 E(Lt)=L0ea2(1−e−git)E(L_t)=L_0 e^{a_2(1-e^{-g_it})}
4 Ricker3 E(Lt)=L∞e−a2e−gitE(L_t)=L_\infty e^{-a_2e^{-g_it}}
5 Quinn-Deriso3 E(Lt)=L∞e−1gie−gi(t−t0)E(L_t)=L_\infty e^{-\frac{1}{g_i}e^{-g_i(t-t_0)}}


Within FSA …

  • L0L_0 = mean length at age 0
  • L∞L_\infty = mean asymptotic length
  • tit_i = age at the inflection point
  • gig_i = instantaneous growth rate at the inflection point
  • a1a_1 = nuisance parameter with no real-world interpretation
  • a2a_2 = nuisance parameter with no real-world interpretation
  • t0t_0 = nuisance parameter with no real-world interpretation. The use of t0t_0 here implies the same meaning at in the von Bertalanffy functions. However, the Gompertz function has a horizontal asymptote at L=0L=0 such that there is no “x-intercept.” Thus, t0t_0 here does not have the same interpretation as for the von Bertalanffy functions.

The parameterizations and parameters for the Gompertz function are varied and confusing in the literature. To address this confusion the uniform set of parameters described above are used in FSA. However, this provides some challenges when comparing the equations used in FSA to those used in common literature sources. Thus, some comments to aid comparisons to the literature are below.

  • In the Ricker (1979)[p. 705] functions (parameterizations 2-4), aa here is kk there and gig_i here is gg there. Also note that their ww is LL here.
  • In the Ricker (1979) functions (parameterizations 2-4), as presented in Campana and Jones (1992), aa here is kk there and gig_i here is GG there. Also note that their XX is LL here.
  • The function in Ricker (1975)[p. 232] is the same as the third parameterization here where a2a_2 here is GG there and gig_i here is gg there. Also their ww is LL here.
  • In the Quinn and Deriso (1999) functions (parameterizations 3-5), aa here is λK\frac{\lambda}{K} there and gig_i here is KK there. Also note that their YY is LL here.
  • The function in Quist et al. (2012)[p. 714] is the same as parameterization 2 where gig_i here is GG there and tit_i here is t0t_0 there.
  • The function in Katsanevakis and Maravelias (2008) is the same as parameterization 2 where gig_i here is k2k_2 there and tit_i here is t2t_2 there.


Tag-Recapture Functions

The Gompertz parameterizations for tag-recapture data are in Table 6.


Table 6: Parameterizations of the Gompertz growth equation for tag-recapture data available in FSA. Synonyms are “Troynikov1” for “Troynikov”.
param pname Equation
6 Troynikov E(Lr−Lm)=L∞[LmL∞]e−giΔt−LmE(L_r-L_m)=L_{\infty}\left[\frac{L_m}{L_{\infty}}\right]^{e^{-g_i\Delta t}}-L_m
7 Troynikov2 E(Lr)=L∞[LmL∞]e−giΔtE(L_r)=L_{\infty}\left[\frac{L_m}{L_{\infty}}\right]^{e^{-g_i\Delta t}}


Logistic

Annual Length-at-Age Functions

The logistic parameterizations for length and annual age data are in Table 7.


Table 7: Parameterizations of the Logistic growth equation for length-at-age (annual) data available in FSA.
param pname Equation
1 Campana-Jones1 E(Lt)=L∞1+e−g−∞(t−ti)E(L_t)=\frac{L_\infty}{1+e^{-g_{-\infty}(t-t_i)}}
2 Campana-Jones2 E(Lt)=L∞1+ae−g−∞tE(L_t)=\frac{L_\infty}{1+ae^{-g_{-\infty}t}}
3 Karkach E(Lt)=L0L∞L0+(L∞−L0)e−g−∞tE(L_t)=\frac{L_0L_\infty}{L_0+(L_\infty - L_0)e^{-g_{-\infty}t}}


New parameters in these growth functions are:

  • g−∞g_{-\infty} = instantaneous growth rate at t=−∞t=-\infty
  • aa = nuisance parameter with no real-world interpretation


Tag-Recapture Functions

The logistic parameterizations for tag-recapture data are in Table 8.


Table 8: Parameterizations of the Logistic growth equation for tag-recapture data available in FSA.
param pname Equation
4 Haddon E(Lr−Lm)=ΔLmax1+elog(19)Lm−L50L95−L50E(L_r-L_m)=\frac{\Delta L_{max}}{1+e^{log(19)\frac{L_m-L_{50}}{L_{95}-L_{50}}}}


New parameters in these growth functions are:

  • ΔLmax\Delta L_{max} = maximum growth increment over the duration of observation
  • L50L_{50} = length-at-marking that produce a growth increment of 50% of ΔLmax\Delta L_{max}
  • L95L_{95} = length-at-marking that produce a growth increment of 95% of ΔLmax\Delta L_{max}


Richards (Annual Length-at-Age)

The Richards parameterizations for length and annual age data are in Table 9.


Table 9: Parameterizations of the Richards growth equation for length-at-age (annual) data available in FSA.
param pname Equation
1 Tjorve4 E(Lt)=L∞[1−1be−k(t−ti)]bE(L_t)=L_\infty\left[1-\frac{1}{b}e^{-k(t-t_i)}\right]^b
2 Tjorve3 E(Lt)=L∞(1+e−k(t−t0))bE(L_t)=L_\infty\left(1+e^{-k(t-t_0)}\right)^b
3 Tjorve7 E(Lt)=L∞[1+((L0L∞)1b−1)e−kt]bE(L_t)=L_\infty\left[1+\left(\left(\frac{L_0}{L_\infty}\right)^{\frac{1}{b}}-1\right)e^{-kt}\right]^b


New parameters in these growth functions are:

  • kk = slope at the inflection point; i.e., maximum relative growth rate
  • bb = a nuisance parameter that controls the vertical position of the inflection point


Only 4-parameter parameterizations from Tjorve and Tjorve (2010) that seemed useful for modeling fish growth are provided here. In Tjorve and Tjorve (2010) their AA, kk, W0W_0, TiT_i,and dd are L∞L_\infty, kk, L0L_0, tit_i, and bb, respectively, in FSA. The number at the end of respective pname corresponds to the equation number in Tjorve and Tjorve (2010). However, note that I modified bb in parameterizations 2 and 3 so that each equation appeared as L∞L_\infty times a quantity raised to a simple (i.e., non-negative and not a fraction) power. Further note that previous versions of FSA had two other parameterizations of the Richards function that differed only from parameterization 1 by simple additions or multiplications of bb. As bb has no biological meaning, these parameterizations were removed from FSA.


Schnute (Annual Length-at-Age)

The four cases for the Schnute model for simple length and annual age data are in Table 10.


Table 10: Cases of the Schnute growth equation for length-at-age (annual) data available in FSA.
param case Equation
1 a≠0a\neq 0, b≠0b\neq 0 E(Lt)=[L1b+(L3b−L1b)1−e−a(t−t1)1−e−a(t3−t1)]1bE(L_t)=\left[L^b_1+(L^b_3-L^b_1)\frac{1-e^{-a(t-t_1)}}{1-e^{-a(t_3-t_1)}}\right]^{\frac{1}{b}}
2 a≠0a\neq 0, b=0b=0 E(Lt)=L1elog(L3L1)1−e−a(t−t1)1−e−a(t3−t1)E(L_t)=L_1e^{log\left(\frac{L_3}{L_1}\right)\frac{1-e^{-a(t-t_1)}}{1-e^{-a(t_3-t_1)}}}
3 a=0a=0, b≠0b\neq 0 E(Lt)=[L1b+(L3b−L1b)t−t1t3−t1]1bE(L_t)=\left[L^b_1+(L^b_3-L^b_1)\frac{t-t_1}{t_3-t_1}\right]^{\frac{1}{b}}
4 a=0a=0, b=0b=0 E(Lt)=L1elog(L3L1)t−t1t3−t1E(L_t)=L_1e^{log\left(\frac{L_3}{L_1}\right)\frac{t-t_1}{t_3-t_1}}


Schnute-Richards (Annual Length-at-Age)

The Schnute-Richards model for simple length and annual age data is E(Lt)=L∞(1−ae−ktc)1/bE(L_t)=L_\infty\left(1-ae^{-kt^c}\right)^{1/b}. Note that this function is slightly modified (a ++ was changed to a −- so that aa is positive) from the original in Schnute and Richards (1990).