What it does
Most fish grow fast when young and slow down as they approach a maximum size. The von Bertalanffy growth function describes this:
L(t) = L∞ × (1 − e^(−K (t − t₀)))
- L∞ is the asymptotic length, the mean length fish would reach if they lived for ever.
- K is the growth coefficient (per year): how quickly length approaches L∞. A fish covers about half the distance to L∞ in ln 2 ÷ K years.
- t₀ is the theoretical age at which length would be zero; usually slightly negative and of little biological meaning.
The tool fits the curve to your fish by nonlinear least squares (Levenberg-Marquardt) and gives each parameter with its standard error and 95 % interval (Student’s t). It also gives:
- the growth performance index φ’ = log₁₀ K + 2 log₁₀ L∞ (L∞ in cm), used to compare growth between stocks and species, because K and L∞ trade off against each other;
- the Ford-Walford plot: mean length at age t + 1 against mean length at age t. For von Bertalanffy growth the points lie on a straight line L(t+1) = a + b L(t), giving K = −ln b and L∞ = a ÷ (1 − b), where the line crosses the 1:1 line. It is a quick check and supplies the starting values for the fit.
How to use it
- Paste or open one row per fish: age in years and length. Fractional ages are fine.
- Pick the length unit (cm, mm or inches). It labels the results and converts L∞ to cm for φ’.
- Read L∞, K and t₀, check the curve against the points, and look at the table of mean length by age for systematic misfit.
Worked example (made-up fish)
Fifty fish aged 1 to 10 years, from 14.7 to 55.3 cm. The fit gives L∞ = 60.9 cm (± 3.11; 95 % interval 54.6 to 67.1), K = 0.212 per year (± 0.0303; 0.151 to 0.273) and t₀ = −0.697 years (± 0.22). The residual standard deviation is 2.45 cm, and φ’ = 2.9. The Ford-Walford line through the mean lengths at ages 1 to 10, L(t+1) = 12.1 + 0.800 × L(t), gives a close check: L∞ 60.4 cm and K 0.223.
Tips
- Old, large fish carry most of the information about L∞. With few of them, L∞ is uncertain and strongly correlated with K, which is why φ’ is often compared instead.
- Gear that misses small young fish, or only catches the fastest-growing ones, biases the curve. Sample the full size range if you can.
- Compare sexes or stocks by fitting each separately and comparing the intervals.
FAQs
Why is t₀ negative? Larval and juvenile growth does not follow the adult curve, so extrapolating back to length 0 rarely lands at age 0.
Can I fit weight at age? Not here; weight follows a related curve with an exponent of about 3.
Is my data uploaded? No. Everything runs in your browser.
Where the numbers come from
- Von Bertalanffy growth function L(t) = L∞(1 − e^(−K(t − t0))) (L. von Bertalanffy 1938; R. J. H. Beverton and S. J. Holt 1957); Ford-Walford plot (E. Ford 1933; L. A. Walford 1946); growth performance index φ' = log10 K + 2 log10 L∞ (D. Pauly and J. L. Munro 1984, Fishbyte 2: 21). Method only.
- Fitting by Levenberg-Marquardt with standard errors from s²(JᵀJ)⁻¹. Checked in the tests against SciPy's scipy.optimize.curve_fit (parameters, standard errors, t intervals) and scipy.stats.linregress (Ford-Walford line). The example fish are made up.
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Last updated 2026-11-09.