Assessment of fish populations
III: fish metrics and growth.
Earlier, we talked about some of the applications of
age data to fish populations:
K. Limburg, lecture notes, Fisheries Science
• constructing age-length keys
• computing the age structure of a population
• back-calculating growth rates (several
methods)
• back-calculating hatch-dates
King 1995
A phenomenon that is often seen in fish population
analysis: Lee’s phenomenon
Today, we’ll look more carefully at some of the
common metrics we use, and learn about two important
growth models.
Metrics for characterizing fish within a population.
1. Size characteristics
Total length
Fork length
Standard length
Weight (wet or dry)
Year Class
2007
2006
2005
2004
2003
2002
Why might this occur?
Source: Devries and Frie 1996
1
2. Condition indices.
Length-weight relationships: usually one of the best
statistical relationships you will find!
Condition factors are
“fatness indices”
W = a • Lb  ln(W) = ln(a) + b • ln(L)
Length-weight relationship
Log-transformed length and weight
14
3.00
2.00
10
ln(wet weight)
wet weight (g)
12
8
6
4
2
0
0
20
40
60
80
total length (mm )
100
120
y = 3.1469x - 12.452
R2 = 0.984
1.00
0.00
-1.00
-2.00
3.00
3.50
4.00
4.50
http://www.hooked-in.com/catches/show/5508
5.00
ln(total length)
 Fulton’s condition factor.
K
(typical form, TL in mm,
W in g)
W
100,000
L3
 Relative weight.
Wr = W/W*,
W* = weight predicted from length/weight relationship
Can also express condition
as the deviation from the
expected W:
Condition = (W-W*) / W*
Redbreast sunfish, Catlin Lake, Fall 2008
800
700
Weight, g
Starving lake trout
in Canadian lakes –
an indirect result of
acid rain
600
500
400
300
200
300
Photos: Experimental Lakes Area, Univ. of Manitoba
350
400
450
500
Total length, mm
2
Example of use of Fulton K: blueback herrings on spawning run
 Gonadosomatic index (GSI)
GSI = WGonads/Wbody
Used to assess reproductive state
K. Limburg
Limburg and Blackburn unpublished data
Example of use
of a
gonadosomatic
index:
blueback
herring
II. Models of growth.
We can express growth as a(nother) model. One of the most
widely used models for quantifying growth in fish is the
so-called
von Bertalanffy growth curve:
Lt  L (1  e (  K ( t t0 )) )
River kilometer (distance from Atlantic Ocean)
Limburg and Blackburn unpublished data
3
Lt  L (1  e
(  K ( t  t 0 ))
What does this growth curve look like?
)
von Bertalanffy growth curve
L
Model parameters (constants to be determined)
where
60.0
L = theoretical maximum length (asymptotic)
K = growth coefficient, proportional to rate at which L is
reached
length (cm)
50.0
Lt = length at age t
40.0
30.0
Lt  L (1  e (  K ( t t0 )) )
20.0
10.0
0.0
t0 = theoretical age at L = 0 (often negative, or zero)
0
2
4
6
8
10
12
age (years)
The weight equivalent is very similar, but with a twist:
Wt  W (1  e (  K ( t t0 )) ) 3
How to estimate the parameters L and K? We
do this by constructing a
Ford-Walford plot
von Bertalanffy growth curve in weight
(see mathematical derivation, pp 195-196, in chapter by
M. King)
5.00
weight (kg)
4.00
The Ford-Walford plot is a graph of the following equation:
3.00
2.00
Lt+1 = L (1 – e-K) + Lt e-K
1.00
0.00
0
2
4
6
8
10
12
(it assumes that t0 = 0)
age (years)
4
Lt+1 = L (1 –
e-K)
+ Lt
Thus, if we know the ages of fish, and their mean
lengths at age, we can construct this graph and estimate
the parameters:
e-K
Ford-Walford plot
(intercept)
another constant
(slope)
Age 1
Age 2
etc
Yet again, we have the equation for a straight line.
We obtain the graph by plotting the length at year
(t+1) against the fish’s length the previous year (t)
L_t+1
27.5
34.9
39.9
43.2
45.5
47.0
48.0
48.6
49.1
60.0
Age 2
Age 3
50.0
slope b = exp(-K)
K = -ln(b)
40.0
L_inf = a/(1-b)
etc
L_t+1
a constant
L_t
16.5
27.5
34.9
39.9
43.2
45.5
47.0
48.0
48.6
49.1
30.0
L
a
20.0
10.0
1:1 Line
0.0
0.0
10.0
20.0
30.0
40.0
50.0
60.0
L_t
Indep.
Estimating the
von Bertalanffy
growth
parameters:
Dep.
One more growth model: Gompertz
Although the von Bertalanffy model works extremely
well when analyzing fish populations at the annual level,
it doesn’t always work so well if you want to look at
growth within the year.
Lt+1 = a + b Lt
K = - ln(b)
L = a / (1-b)
Lt  L (1  e
(  K ( t  t 0 ))
It turns out that another model, developed by
Benjamin Gompertz, works well for within-year
growth.
)
Source: King 1995
5
Otolith increment widths track growth rates!
5-day growth increments in a single juvenile fish
5-day otolith increment widths
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
5/24
7/13
9/1
10/21
12/10
Date
Add up these increment widths to obtain a cumulative graph 
Lt  L e
Cumulative growth in the same fish
cumulative otolith growth
50
 (e t )
45
40
35
L = asymptotic length of the fish
30
25
20
Lt  L e
15
10
5
0
5/24
7/13
9/1
Date
10/21
 (e t )
 and  are constants
This model is more difficult to parameterize – you have to do
it with a statistical program. (Hint:  and  are negative #s)
12/10
The point is, though, that it does a good job of fitting the
observed data.
6