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Unformatted text preview: EXST7025 : Biological Population Statistics
Special Growth Topics Page 1 Given a growth model, there are occasions when we want to estimate age from size of the change
in age between two sizes.
Starting with a von Bertalanffy growth model, Lt L 1 e K t t0 t or Lt L L e K t t0 t we derive a from previously used in a linearized form of the model, Lt L L e
L Lt L e K t t0 K t t0 L Lt
K t t e 0
L L Lt
ln L K t t0 This form we previously used as a linearized version of the von Bertalanffy model. Now we
solve for t to get the estimated age. L Lt
ln L K t t0 1 L Lt
ln K L
t t t0 1 L Lt
ln K L t0 if we get a t1 for l1 and a t2 for an l2, subtract and get t t2 t1 1 L l1 ln K L l2 EXST7025 : Biological Population Statistics
Special Growth Topics Page 2 Growth when AGE is unknown
TYPICAL SITUATION - Tagging studies where we know a size at some initial time (t), and
some time later (t)
t may or may not be constant
we will define
t as the initial time (measurement is lt)
t+t as the later time (measurement is lt+t)
t = t1 – t2
l = l1 – l2
note: negative values may occur because of measurement error, but their deletion causes bias.
Parrack (1978) derives equations for l for a number of growth models. These are best described in
Phares (1980). All of the equations are for lt on lt+t, and retain the original growth parameters.
Each observation with a t = t1 – t2 also will have an initial length l1 taken at t1 and a length
l2 taken at t2
The previously discussed models can then be fitted as; l2 Logistic L
1 von Bertalanffy l1
l1 k ( t )
e l2 L L l1 e k t L 1 e k ( t ) Gompertz l2 l1e Richards l2 L1 m L1 m l11 m e (1 m ) k t Phares adds one other consideration, DEGREE DAYS. 1 m EXST7025 : Biological Population Statistics
Special Growth Topics Page 3 Seasonally adjusted growth curves
To any of the models previously discussed, a seasonal adjustment can be added to the growth rate
Seasonal adjustments can be added as any function. The most common is the sine curve, but
polynomials and other functions can be added as well.
The sine of numbers from 0 to 2, shown below, complete one cycle.
0.6 Sine curve 0.4
1 2 3 4 5 6 7 -0.2
-1 In order to simulate one annual cycle, the period of the cycle must be adjusted to range from 0
For example, a 365 day cycle can be adjusted to range from 0 to 2 by calculating
where “date” is the julian date (a day number between 1 and 365).
The sine curve amplitude ranges from –1 to 1, but can also be adjusted to any desired range.
The whole curve can also be shifted vertically or horizontially.
A completely adjusted sine curve would be
This calculation can be added to a growth model such that the K value becomes the amplitude of
the sine curve. k shift amplitude*sin( 2 ( date )
t t 365 0 Lt L 1 e t Note that the age (t) of the fish is not changed. The seasonal adjustment on date in this case occurs
within an annual cycle. Polynomials and other curves can be added in a similar fashion.
The example below represents a sine curve with an amplitude of 7, a horizontal shift of 60 days and
a vertical shift of 2. EXST7025 : Biological Population Statistics
Special Growth Topics Page 4 10
50 100 150 200 250 300 350 400 450 -2
-4 Sine curve adjustment
-6 The complete vonBertalanffy curve would look like the following. 100
20 von Bertalanffy with sine curve 10
0 0 1 2 3 4 5 Age 6 EXST7025 : Biological Population Statistics
Special Growth Topics Page 5 Heavily exploited fisheries often do not have the larger sizes. The L is difficult to determine
under these conditions
70 Length (cm) 60
20 Length of Cod at Age 10
0 2 Age (years) 4 6 Some situations are just too complicated for a 3-parameter growth model.
140 Height (cm) 120
40 Height of Belgian males 20
0 2 4 6 8 10 12 14 16 18 20 Age (years) 25 30 40 50 60 ...
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- Spring '08