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) dW/dt = H * W 2/3 – B * W where W=weight, H ~ metabolism, B ~ catabolism Solving the differential equation results in W t = W inf (1 – e -K * t ) 3 L t = L inf (1 - e -K * t )

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

Using VBGF W t = W inf (1 – exp(-K * (t – t 0 ))) b Where W t = weight (g) at age t (years) W inf = asymptotic weight if t = infinite b is the exponent of the length-weight relationship W = aL b 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 / L inf ) / K + t 0

Understanding K K describes the steepness of the growth curve, i.e., how fast L inf is reached: K <= 0.05 in large, long-lived fishes K > 1 in small, short-lived fishes

Understanding L inf L inf is similar to maximum size ( e.g. mean of three largest specimens ) reached in an unfished population L inf = L max / 0.95 = L max L max

L inf as a Function of L max 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: log10L  = * log10(L max ) (n = 551, r 2 = 0.959)

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 L inf Growth curves without t 0 give length at ‘relative’ age

Growth and Maturity VBGF in weight has an inflection point at W inf ~ 2/3 L inf = L opt

Fish Grow Forever

Growth and Maturity VBGF in weight has an inflection point at W inf ~ 2/3 L inf = L opt At L opt production of tissue and gonads is fastest Semelparous fish mature at L opt Iteroparous fish mature so that L opt is reached within the average duration of the reproductive phase, typically between 1/3 and ½ L inf

Length at Maturity vs Linf 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: Log10L m = * log10L inf r 2 = n=467

Grow Fast, Die Young

Interrelationship between K and L inf K and L inf 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 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% L inf is a good predictor of maximum age t max t max = - 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 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 L t = L inf (1 - e -K * (t - to) ) L inf ranges from 1 cm to 20 m K ranges from 0.05 to 2.0 L inf and K are inversely related 3/K is a good predictor of maximum age

Thank You