Transcript Slide 1

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 * W2/3 – 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 – 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
log10L = 0.044 + 0.9841 * log10(Lmax)
(n = 551, r2 = 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 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
3
Length at first maturity (L
m; log cm)
Log10Lm = 0.8979 * log10Linf -0.0782
r2= 0.888 n=467
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
Asymptotic length (Loo; log cm)
3
3.5
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
Linf = 120
K = 0.13
tmax= 23
100
90
Linf = 88
K = 0.27
tmax= 11
80
Length (cm)
70
Linf = 64
K = 0.53
tmax= 5.6
60
Linf = 73
K = 0.4
tmax= 7.5
50
40
30
20
10
0
0
2
4
6
Age (years)
8
10
12
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 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)
Linf = 120
K = 0.13
tmax= 23
100
90
Linf = 88
K = 0.27
tmax= 11
80
Length (cm)
70
Linf = 64
K = 0.53
tmax= 5.6
60
Linf = 73
K = 0.4
tmax= 7.5
50
40
30
20
10
0
0
2
4
6
Age (years)
8
10
12
Growth across Species
Whale shark
Linf = 14m
K = 0.05
tmax= 60
Ø' = 5.0
200
180
White shark
Linf = 653
K = 0.06
tmax= 51
Ø' = 4.4
Bluefin tuna
Linf = 330
K = 0.1
tmax= 30
Ø' = 4.0
160
Length (cm)
140
120
100
80
Cod
Linf = 120
K = 0.13
tmax= 23
Ø' = 3.3
60
40
Anchovy
Linf = 20
K = 1.2
tmax= 2.5
Ø' = 2.7
20
0
0
2
4
6
Age (years)
8
10
12
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
2500
Fin whale
Length (cm)
2000
1500
1000
Whale shark
500
0
0
10
20
30
Age (years)
40
50
60
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
800
Killer whale
700
600
Length (cm)
Great white shark
500
400
300
200
100
0
0
5
10
15
20
Age (years)
25
30
35
40
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