Growth in Length and Weight

Rainer FroeseSS 2008

Why is Growth Important?

• Determines increase in body mass

• Determines maturation and generation time

• Determines population increase

• Determines sustainable yield

Fish Grow Forever

von Bertalanffy Growth

Von Bertalanffy’s (1934) Growth Function (VBGF)

dW/dt = H * W2/3 – B * Wwhere W=weight, H ~ metabolism, B ~ catabolism

Solving the differential equation results in

Wt = Winf (1 – e-K * t)3

Lt = Linf(1 - e-K * t)

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

Using VBGF

Wt = Winf (1 – exp(-K * (t – t0)))b

Where Wt = weight (g) at age t (years)

Winf = asymptotic weight if t = infinite

b is the exponent of the length-weight relationship

W = aLb

K and t0 same as with VBGF for length

Age at Length

• In species with indeterminate growth there is a unique relationship between age and length, and vice versa:

tL = -LN(1 – L / Linf) / K + t0

Understanding K

K describes the steepness 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

Understanding Linf

Linf is similar to maximum size (e.g. mean of

three largest specimens) reached in an unfished population

Linf = Lmax / 0.95 = Lmax + 0.05Lmax

Linf as a Function of Lmax

Froese, R. and C. Binohlan 2000. 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. 56:758-773.

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Maximum length (Lmax; log cm)

Asy

mpt

otic

leng

th (

L;

log

cm)

log10L = 0.044 + 0.9841 * log10(Lmax) (n = 551, r2 = 0.959)

Understanding t0

• Length at birth t=0 is larger than zero

• t0 is used to correct for that and improve the fit of the curve by moving it to the left

• t0 is thus the hypothetical age at L=0 if VBGF would apply to larvae

• t0 does not change K or Linf

• Growth curves without t0 give length at ‘relative’ age

Growth and Maturity

• VBGF in weight has an inflection point at 0.296 Winf ~ 2/3 Linf = Lopt

Fish Grow Forever

Growth and Maturity

• VBGF in weight has an inflection point at 0.296 Winf ~ 2/3 Linf = Lopt

• At Lopt production of tissue and gonads is fastest

• Semelparous fish mature at Lopt

• Iteroparous fish mature so that Lopt is reached within the average duration of the reproductive phase, typically between 1/3 and ½ Linf

Length at Maturity vs Linf

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0 0.5 1 1.5 2 2.5 3 3.5Asymptotic length (Loo; log cm)

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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 2000. 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. 56:758-773.

Log10Lm = 0.8979 * log10Linf -0.0782 r2= 0.888 n=467

Grow Fast, Die Young

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Age (years)

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Linf = 64K = 0.53tmax= 5.6

Linf = 73K = 0.4tmax= 7.5

Linf = 88K = 0.27tmax= 11

Linf = 120K = 0.13tmax= 23

Interrelationship between K and Linf

K and Linf are inversely correlated

-> FishBase, Micropterus salmoides, Growth, Auximetric graph

K is NOT a growth-per-time indicator:Anchovy K >1.0 reach 20 cm in second yearCod K ~ 0.13 reaches 30 cm in second year Size reached in 1 or 2 years is better growth indicator

Growth within Species is similar(e.g. Cod)

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Age (years)

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Linf = 64K = 0.53tmax= 5.6

Linf = 73K = 0.4tmax= 7.5

Linf = 88K = 0.27tmax= 11

Linf = 120K = 0.13tmax= 23

Growth across Species

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AnchovyLinf = 20K = 1.2tmax= 2.5Ø' = 2.7

CodLinf = 120K = 0.13tmax= 23Ø' = 3.3

White sharkLinf = 653K = 0.06tmax= 51Ø' = 4.4

Whale sharkLinf = 14mK = 0.05tmax= 60Ø' = 5.0

Bluefin tunaLinf = 330K = 0.1tmax= 30Ø' = 4.0

Maximum Size is a function of Lifespan

Maximum age as function of K

• The age corresponding to 95% Linf is a good predictor of maximum age tmax

tmax = - LN(1-0.95) / K + t0 ~ 3/K

Maximum age as function of K

VBGF Fits Many Species

Whale shark vs Fin whale

• The largest fish, the Whale shark (Rhincodon typus) needs about 9 years to reach maturity at about 5.5 m and 750 kg, and about 60 years to reach a maximum length of 14 m and 12 tons.

• The fin whale (Balaenoptera physalus) needs about 7 years to reach maturity at a size of about 20 m and about 36 tons, with a maximum size of about 25 m and 70 tons.

Whale shark vs Fin whale

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Age (years)

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Fin whale

Whale shark

White shark vs Killer whale

• The Great white shark females take about 12 years to reach maturity at 4.5-5 m and about 0.8 tons; they need 36 years to reach a maximum size of 7.2 m and 3.4 tons;

• Killer whales (Orcinus orca) reach maturity in 6-10 years at 5-6 m length and about 1.8 tons, with the typical size of about 7 m and 3.8 tons reached a few years later.

Great white shark vs Killer whale

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Age (years)

Len

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Great white shark

What You Need to Know

• Lt = Linf(1 - e-K * (t - to))

• Linf ranges from 1 cm to 20 m

• K ranges from 0.05 to 2.0

• Linf and K are inversely related

• 3/K is a good predictor of maximum age

Thank You