growth in length and weight rainer froese ss 2008
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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.
0
0.5
1
1.5
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2.5
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0 0.5 1 1.5 2 2.5 3
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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0.5
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1.5
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0 0.5 1 1.5 2 2.5 3 3.5Asymptotic length (Loo; log cm)
Le
ng
th a
t fir
st m
atu
rity
(L m;
log
cm
)
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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0 2 4 6 8 10 12
Age (years)
Le
ng
th (
cm
)
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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0 2 4 6 8 10 12
Age (years)
Le
ng
th (
cm
)
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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Age (years)
Le
ng
th (
cm
)
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)
Len
gth
(cm
)
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
gth
(cm
)Killer whale
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
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