1. Biodiversity of Fishes Growth
Rainer Froese
GEOMAR, Kiel

2. Most Species Grow Throughout their Lifes
(Exception: birds and mammals)

3. Karl Ludwig von Bertalanffy
Professor Uni Wien
Later London, Canada, USA
Concept of ‘Fließgleichgewicht’ (steady state of open systems)

4. von Bertalanffy Growth
Von Bertalanffy’s (1934) Growth Function (VBGF)
dW/dt = H * Wt 2/3 – B * Wt
where W = body weight, H W2/3= total available energy (metabolism),
B W = energy needed for processes other than growth, t = age,
dW/dt = growth rate at age t = energy available for growth at age t
Solving the differential equation results in
Wt = Winf (1 – e-K * t)3
Lt = Linf (1 - e-K * t)
where Winf and Linf are asymptotic weight and length, and K describes how fast these are approached (the exponential decrease of the distance to Linf)

5. ⅔ Versus ¾ Scaling The ¾ scaling often observed
in metabolic comparisons across
species (black curve) gives a similar
but slightly less convincing fit than
the 2/3 geometric scaling (red curve)

6. Using VBGF Lt = Linf (1 – exp(-K * (t – t0))) Where
Lt = length (cm) at age t (years)
Linf = asymptotic length if t = infinite
K = parameter indicating how fast Linf is approached (1/year)
t0 = hypothetical age at L = 0

7. Understanding K
K describes the curvature of the growth curve, i.e., how fast Linf is reached:
K <= 0.05 in large, long-lived fishes
K > 1 in small, short-lived fishes
2.996/K + to is the time needed to reach 0.95 Linf which is a proxy for maximum age

8. Understanding Linf
Linf is similar to maximum size (e.g. mean of three largest specimens) reached in an unexploited population

9. Linf as a Function of Lmax
log10L = * log10(Lmax) (n = 551, r2 = 0.959)
Froese, R. and C. Binohlan Empirical relationships to estimate asymptotic length,
length at first maturity and length at maximum yield per recruit in fishes, with a simple
method to evaluate length frequency data. J. Fish Biol. 56:

10. Understanding t0
Age t = 0 is at hatching or birth, when newborns already have a length

11. Understanding t0
Age t = 0 is at hatching or birth, when newborns already have a length
t0 is used to account for that and improve the fit of the curve by moving it to the left t0

12. Understanding t0
t0 is thus the hypothetical age at L=0 if VBGF applies (not for larvae)
t0 is usually small and negative
t0 moves the curve left (t0 is typically negative) without changing K or Linf
Growth curves without t0 give length at ‘relative’ age; for true age add t0

13. Growth and Maturity
VBGF in weight has an inflection point at 0.3 Winf = 2/3 Linf (if growth is isometric with b ~ 3)
Fish mature before or at that size
First maturity
max dW/dt

14. Length at Maturity vs Linf
Log10Lm = * log10Linf r2= n=467
Relationship between length at first maturity and asymptotic length for all records representing 265 species of
fish. Regression lines are for females (----) and males ().
Froese, R. and C. Binohlan Empirical relationships to estimate asymptotic length, length at first maturity and length at maximum yield per recruit in fishes, with a simple method to evaluate length frequency data. J. Fish Biol. 56: (173 Web of Science citations 11/2015)

15. Grow Fast, Die Young

16. Interrelationship between K and Linf
Largemouth black bass, freshwater North America

17. How to Compare Growth K is NOT a growth-per-time indicator: Example:
Anchovy K >1.0 reach 20 cm in second year
Cod K ~ 0.13 reaches 30 cm in second year

18. How to Compare Growth
Compare the time needed to reach a certain length
Compare the time needed to reach a certain weight
find corresponding length from length-weight relationship
L = 10^((logW – log a) / b)
2. find corresponding age from tL

19. Time to Reach 200 g Whale shark 6.2 months White shark 9.5 months
Bluefin tuna
8.3 months
Cod
24 months
Anchovy
Never
50 g
3 years

20. Exercises Find a species with at least 5 growth studies
Discuss the variability of Linf and K and the value of t0
Select a study that describes growth well and justify your selection
How long will it take to reach 200 g?
W = a Lb