Growth in Length and Weight Rainer Froese SS

Growth in Length and Weight Rainer Froese SS 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) d. W/dt = H * W 2/3 – B * W where 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 – t 0))) Where Lt = length (cm) at age t (years) Linf = asymptotic length if t = infinite K = parameter indicating how fast Linf is approached (1/year) t 0 = hypothetical age at L = 0

Using VBGF Wt = Winf (1 – exp(-K * (t – t 0)))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 = a. Lb K and t 0 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: t. L = -LN(1 – L / Linf) / K + t 0

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 is similar to maximum size (e. g. mean of three largest specimens) reached in an unfished population Linf = Lmax / 0. 95 = Lmax + 0. 05 Lmax

Linf as a Function of Lmax log 10 L = 0. 044 + 0. 9841 * log 10(Lmax) (n = 551, r 2 = 0. 959) 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.

Understanding t 0 • Length at birth t=0 is larger than zero • t 0 is used to correct for that and improve the fit of the curve by moving it to the left • t 0 is thus the hypothetical age at L=0 if VBGF would apply to larvae • t 0 does not change K or Linf • Growth curves without t 0 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 Log 10 Lm = 0. 8979 * log 10 Linf -0. 0782 r 2= 0. 888 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 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.

Grow Fast, Die Young

Interrelationship between K and Linf are inversely correlated -> Fish. Base, Micropterus salmoides, Growth, Auximetric graph K is NOT a growth-per-time indicator: Anchovy K >1. 0 reach 20 cm in second year Cod 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)

Growth across Species

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 = - LN(1 -0. 95) / K + t 0 ~ 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

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

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