Age and growth

What is a rate?

• Rate = “something” per time unit

What is the unit of F?

Z and M are also per time unit (years, months, days..)

F CB

F CB

C kg yr

B kgyr

11

Age and growth

• Why do we want to age fish?

changes in leng th or w eigh tdL

d tor

dW

dt

changes in num bersdN

dt

changes in b iom ass or yielddB

dtor

dY

d t

They are all values per time unit. We are working with rates. Therefore a measure of time (speed) is needed. Age - or relative age - of the fish is used to determine the time scale.

GrowthThere are two types of growth to be considered:

• Population growth in numbers or weight

• Individual growth in length or weight

time

Population growth in numbers Individual growth in lenght

time

len

gth

Individual growth in length

d N

d tr N

N

K

1

L L etK t t 1 0

dL

d ta b L t

Growth• Individual growth is - within wide limits - determined

genetically, but is influenced by several factors: • Environment

– Food availability (quality/quantity)

– Temperature (fish are poikilotherms)

– Oxygen (very important limiting factor in water)

• Behaviour and biology– Variable allocation of surplus energy (somatic or gonadal

tissue growth, locomotion or maintenance)

– Sexual differences

– Density and size distribution (hierarchical behaviour and/or competition)

Growth varies ..

Three approaches to ageing

• Direct observations of individual fish, either held in confinement or from marking/recapture experiments.

• Ageing of individual fish based on annual patterns in hard structures e.g. otoliths, scales, bones etc.

• Identification of mean length of cohorts based on length frequency distributions from one or several samples representing a wide range of the population.

A cohort of fish

1980 1981 1982 1983 1984 19850 2435 3456 2845 2010 1879 24561 679 1336 852 775 1103 9812 1282 354 733 423 405 6053 512 669 185 403 210 2114 140 267 349 97 221 1045 73 112 95 182 50 121

Cohorts, number of survivors

Age

The 1980 Year-class in 6 age groups [0..5]

Cod: Cohort change in length with age

10

20

30

40

50

60

70

80

90

100

110

1 2 3 4 5 6 7 8

time/age (years)

Len

gth

(cm

)

Birth and growth of a cohortLife history of 11 fish

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

Age (months)

Len

gth

(m

m)

Von Bertalanffy Growth Function (VBGF):

A growth trajectory in lenght

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

time

len

gth

asymptotic length

dL

dt

L L etK t t 1 0

the increase in length is a decreasing function of length

Growth and VBGF

• The increase in length is a function of length:

dL

d ta b L t

Von Bertalanffy Growth Function (VBGF):d L

d ta b L t

dL/dt as a function of mean lenght

y = -0.8978x + 269.33

0

50

100

150

200

250

50.00 100.00 150.00 200.00 250.00 300.00 350.00

Lt+(dt+Lt)/2

dL

/dt

d L

d tK L L t ( )

a K L

b K

-K

L∞

Von Bertalanffy Growth Function (VBGF):

d L

d tK L L t ( )

This equation can be integrated to the VBGF:

L L etK t t 1 0

One new parameter t0:

Also called the ‘initial condition factor’. It gives the start of the curve, i.e. the time where the theoretical length is zero

tK

L L

Ltt

L0

1

ln

K and L∞• L is called "L-infinity" or the "asymptotic length",

representing the maximum length of an infinitely old fish of the given stock. L can be estimated from graphical plots, or it can be approximated by the mean of a selection of the biggest specimens recorded from the population, or the relation L Lmax/0.95.

• K is called the "curvature parameter". It determines how fast the growth is relative to L, i.e. how fast the fish reaches its maximum size. An estimate of K is calculated from the slopes in the different graphical plots. Note that K is not a growth rate as it has the unit ‘per time’ only.

• Different K’s cannot be compared when L is different!

K and L∞

Growth of Tilapia under different oxygen conditions

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Age (months)

Len

gth

(m

m)

High

Medium

Low

All combined

K= 0.94, L∞ = 440

K= 0.98, L∞ = 389

K= 1.12, L∞ = 307

K and L∞ are inversely related !Which curve has the highest K?

Estimating K and L∞

K b

La

bLinear regression:

dL/dt as a function of mean lenght

y = -0.8978x + 269.33

0

50

100

150

200

250

50.00 100.00 150.00 200.00 250.00 300.00 350.00

Lt+(dt+Lt)/2

dL

/dt -K-K

L∞L∞

d L

d ta b L t

Gulland & Holt plot

Estimating to

K b

ta

b0 Linear regression:

ln 1

L

La btt

Von Bertalanffy Plot

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

t

-ln

(1-m

ea

nL

(t)/

Lo

o)

t o

K

Getting dL/dt and mean length

dL

dt

Estimating K and L∞Practical hints: Use young fish!!

K b

La

b

Linear regression:dL

dta b

L Lt t d t

2

Gulland & Holt Plot:

dL/dt as a function of mean lenght

0

50

100

150

200

250

50.00 100.00 150.00 200.00 250.00 300.00 350.00

(Lt+dt+Lt)/2dL

/dt

Loo = fixed

-K

dL/dt as a function of mean lenght

0

50

100

150

200

250

50.00 100.00 150.00 200.00 250.00 300.00 350.00

(Lt+dt+Lt)/2

dL/d

t

Loo

-K ?

Old fish Young fish

d L

d tK L L t ( )

Relative age and t0

• In most length-based stock assessment models absolute age is not used, only in relative age. When computing the time it takes to grow from L1 to L2 we use the inverse VBGF:

• Subtracting two such equations in order to find the time interval (dt) between the length interval L1 and L2 (dL) will give

tK

L L

LtL

t

1

0ln

tK

L L

L LL L2 1

1 1

2

ln t0 no longer used

Length instead of ageLength (cm) Jan-92 Feb-92 Mar-92 Apr-92 May-92 Jun-92 Jul-92 Aug-92 Sep-92 Oct-92 Nov-92 Dec-92

8 19

10 111 7 10 8 4 6 2 1012 1 1 2 19 25 18 16 24 7 17 1213 1 3 15 25 15 11 30 32 38 2214 1 9 19 11 14 30 23 59 2715 2 1 1 9 11 21 7 38 31 25 2416 11 18 3 17 18 37 28 46 2217 21 6 4 9 8 11 9 25 40 48 2318 14 3 7 2 3 4 8 2 17 23 35 2719 14 7 16 8 5 4 2 3 13 22 38 2220 8 13 22 6 9 4 4 1 9 11 28 1721 10 10 32 30 26 2 4 3 8 10 18 1822 9 12 33 22 29 6 6 4 7 3 14 1323 5 7 42 29 39 11 4 8 3 5 6 524 14 7 28 25 39 18 5 11 12 5 4 725 15 10 13 18 23 22 9 19 10 11 8 426 24 7 22 8 26 16 10 10 10 8 4 227 26 12 25 11 25 9 2 14 11 14 9 628 17 13 26 13 12 3 6 8 12 10 13 529 10 12 25 10 8 6 2 7 2 10 11 830 8 13 27 14 19 3 3 1 3 6 9 1531 6 7 12 13 12 8 6 7 1 6 12 1032 7 10 16 11 19 8 1 7 3 6 6 433 4 4 12 8 9 2 4 3 3 6 1034 5 6 2 9 7 2 4 6 6 435 2 7 3 3 1 6 3 8 236 2 2 3 3 4 4 4 6 2 4 4 437 2 2 6 2 7 3 3 6 6 2 3 338 2 4 6 5 1 1 3 5 339 2 1 1 2 4 2 3 3 240 1 2 1 3 1 2 3 241 1 1 1 2 1 1 142 1 2 2 1 1 1

Growth ?

Growth ?

Length frequencies over time

?

Length frequencies over timeOne observation = a composite distribution of 1..n cohorts

Length frequency analysis- composite cohorts

1980 1981 1982 1983 1984 19850 2435 3456 2845 2010 1879 24561 679 1336 852 775 1103 9812 1282 354 733 423 405 6053 512 669 185 403 210 2114 140 267 349 97 221 1045 73 112 95 182 50 121

Cohorts, number of survivors

Age

The 1980 Year-class in 6 age groups [0..5]

Cod: Length and age composition in survey, march 2002

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90 100

Length (cm)

Abundance (

num

bers

)

1 2 3 4 5 6 7 <-- age (years)

LFQ analysis – 1 sample

Composite lenght frequency distribution - how many cohorts?

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

N1

N2

N3 N4N5

N6

The normal distribution

• Described by 3 parameters:– n (number)– s (SD)– (mean)

2

i

1

2

L

1n d

2

i

Li

X X

s

n es

0

100

200

300

400

500

20 25 30 35 40 45 50 55 60 65 70 75 80

Length (mm)

Num

bers

X

Bhattacharya method

Converting a normal distribution to straight line

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

5 10 15 20 25 30 35 40 45 50

-10.00

-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

y = ln(f(x))

Y = ln(f(x+dl))-ln(fx)) slope = SD

Mean

Bhattacharya method

Based on:

• Assumed normal distributions of the components in a composite length frequency distribution.

• Transformation of the normal distributions into straight lines.

• Calculation of N, , and SD by regression analysis.

x

Bhattacharya method • From a composite length-

frequency distribution (a)

• Identify, separate and remove (peel off) one cohort at a time starting from the left (b, c)

• Each cohort is identified by transforming the ‘normal’ distribution into a straight line and find mean and SD by regression

Bhattacharya method N1+

0

5

10

15

20

25

30

35

40

45

Lenght intervals

Fre

qu

enci

es

Bhattacharya method – step 1Transformation of a normal distribution to a straight line step 1

y = -0.1578x2 + 1.8441x - 1.6794

R2 = 0.9962

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 21-22

Length

Ln

co

nve

rted

fre

qu

enci

es

ln(N1+)

Poly. (ln(N1+))

Taking natural logarithm (ln) of the function will make a parabola

A parabola can be transformed into a straight line by calculating the difference of two adjacent function values y = f(x+dl) – f(x) and plotting this against a new independent value z = (x +(x+dl))/2

f(x+dl)

f(x)

z z z z

Bhattacharya method – step 2Transformation of a normal distribution to a straight line step 2

y = -0.3338x + 5.7507

R2 = 0.9929

-1.5

-1

-0.5

0

0.5

1

1.5

2

10 12 14 16 18 20 22

Length

ln(y

+1)

- l

n(y

)

z

Linear (z)

X

SD

Bhattacharya method – step 3

• From the linear regression coefficients we can now calculate the expected function values

• Use this to back-calculate the expected normal distribution of the cohort in the area of the composite distribution where there is overlap with (contaminates) the next cohort

Y a b X

Bhattacharya in Excel

A B C D E F G H ILength(x) N1+ ln(N1+) ln(x+1)-ln(x) z Calculated ln(N1) N1 N2+

12-13 1 0.00 12 y = a+b*z 1 0 13-14 4 1.39 1.39 13 1.35 4 0 14-15 11 2.40 1.01 14 1.04 11 0 15-16 24 3.18 0.78 15 0.73 24 0 16-17 38 3.64 0.46 16 0.43 3.64 38 0 17-18 42 3.74 0.10 17 0.12 3.76 42.90 -0.90 18-19 33 3.50 -0.24 18 -0.19 3.57 35.65 -2.65 19-20 20 3.00 -0.50 19 -0.49 3.08 21.81 -1.81 20-21 7 1.95 -1.05 20 -0.80 2.28 9.82 -2.82 21-22 3 1.10 -0.85 21 -1.10 1.18 3.25 -0.25 22-23 3 1.10 0.00 22 -1.41 -0.23 0.79 2.21 23-24 5 1.61 0.51 23 -1.72 -1.95 0.14 4.86 24-25 8 2.08 0.47 24 -2.02 -3.97 0.02 7.98 25-26 11 2.40 0.32 25 11 26-27 14 2.64 0.24 26 14 27-28 17 2.83 0.19 27 17 28-29 16 2.77 -0.06 28 16 29-30 15 2.71 -0.06 29 15 30-31 14 2.64 -0.07 30 14 31-32 11 2.40 -0.24 31 11

'clean'

Observation Parabola Y-values X-values

regression

Bhattacharya plot

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 10 20 30 40 50 60

(x+(x+dl))/2

ln(x

+d

l)-l

n(x

)Bhattacharya plot

Bhattacharya plot

y = -0.3064x + 5.3301

r2 = 0.976

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 10 20 30 40 50 60

(x+(x+dl))/2

ln(x

+d

l)-l

n(x

)

Bhattacharya in Excel

A B C D E F G H ILength(x) N1+ ln(N1+) ln(x+1)-ln(x) z Calculated ln(N1) N1 N2+

12-13 1 0.00 12 y = a+b*z 1 0 13-14 4 1.39 1.39 13 1.35 4 0 14-15 11 2.40 1.01 14 1.04 11 0 15-16 24 3.18 0.78 15 0.73 24 0 16-17 38 3.64 0.46 16 0.43 3.64 38 0 17-18 42 3.74 0.10 17 0.12 3.76 42.90 -0.90 18-19 33 3.50 -0.24 18 -0.19 3.57 35.65 -2.65 19-20 20 3.00 -0.50 19 -0.49 3.08 21.81 -1.81 20-21 7 1.95 -1.05 20 -0.80 2.28 9.82 -2.82 21-22 3 1.10 -0.85 21 -1.10 1.18 3.25 -0.25 22-23 3 1.10 0.00 22 -1.41 -0.23 0.79 2.21 23-24 5 1.61 0.51 23 -1.72 -1.95 0.14 4.86 24-25 8 2.08 0.47 24 -2.02 -3.97 0.02 7.98 25-26 11 2.40 0.32 25 11 26-27 14 2.64 0.24 26 14 27-28 17 2.83 0.19 27 17 28-29 16 2.77 -0.06 28 16 29-30 15 2.71 -0.06 29 15 30-31 14 2.64 -0.07 30 14 31-32 11 2.40 -0.24 31 11

'clean'

Observation Parabola Y-values X-values Predicted Parabola N1 isolated

regression

Substract N1

Go backwards

A clean value is one that does not overlap with the next cohort

Limitations to length-frequency analysis

• It is can difficult to separate the components of a composite frequency distribution. – In the older parts where the overlaps become increasingly bigger. – If continuous spawning (cohorts not discrete)

• To assess the reliability of resolving the components a separation index has been introduced (it is an automatic feature in the Bhattacharya method implemented in FiSAT)

I

L L

S D S Da a

a a

1

1 2

If the separation index (I) is less than 2 it is more or less impossible to properly separate the two components

Modal Progression Analysis (MPA)

Time

Leng

ht

?

?

dt dt dt

Computerised versions of length frequency analysis

• ELEFAN (Electronic LEngth Frequency ANalysis) developed by Pauly & David (1981) and with later refinements and extensions (ELEFAN I..IV). (BASIC)

• LFSA (Length Frequency Stock Assessment) developed by P. Sparre (1987a) (BASIC).

• The MAXIMUM-LIKELIHOOD-METHOD: NORMSEP developed by Tomlinson (1971) and later extensions and modifications by MacDonald & Pitcher (1979), Schnute & Fournier (1980) and Sparre (1987b). (FORTRAN)

• FiSAT (FAO/ICLARM Stock Assessment Tools) (Gayanilo and Pauly 1997) is a package combining ELEFAN and LFSA together with additional features and a more user friendly interface. FiSAT is now available in upgraded Windows version http://www.fao.org/fi/statist/fisoft/fisat/index.htm

ELEFAN and FiSAT

• Automatic search routine (works like Solver) on restructured length-frequency data

• Requires reasonable input (seed) values to avoid local minima

• Has a reputation for overestimating L∞

• Good tool if used with critical precaution

The restructuring principles of ELEFANrunning restructured ASP=

Length(x) x average values 2.00 12-13 1 5.3 -0.81 TRUE 13-14 4 10.0 -0.60 mean 14-15 11 15.6 -0.29 15-16 24 23.8 0.01 16-17 38 29.6 0.28 17-18 42 31.4 0.34 17.3 18-19 33 28.0 0.18 19-20 20 21.0 -0.05 20-21 7 13.2 -0.47 21-22 3 7.6 -0.61 22-23 3 5.2 -0.42 23-24 5 6.0 -0.17 24-25 8 8.2 -0.02 25-26 11 11.0 0.00 26-27 14 13.2 0.06 27-28 17 14.6 0.16 27.9 28-29 16 15.2 0.05 29-30 15 14.6 0.03 30-31 14 13.4 0.04 31-32 11 12.2 -0.10 32-33 11 11.0 0.0033-34 10 10.2 -0.0234-35 9 10.2 -0.1235-36 10 10.0 0.0036-37 11 10.0 0.10 35.337-38 10 10.4 -0.0438-39 10 10.6 -0.0639-40 11 10.2 0.0840-41 11 9.6 0.15 40.241-42 9 9.0 0.0042-43 7 7.8 -0.1043-44 7 6.8 0.03 43.344-45 5 6.0 -0.1745-46 6 5.2 0.1546-47 5 4.2 0.19 45.547-48 3 3.6 -0.1748-49 2 2.8 -0.2949-50 2 2.0 0.0050-51 2 1.8 0.14

0

5

10

15

20

25

30

35

40

45

12

-13

14

-15

16

-17

18

-19

20

-21

22

-23

24

-25

26

-27

28

-29

30

-31

32

-33

34

-35

36

-37

38

-39

40

-41

42

-43

44

-45

46

-47

48

-49

50

-51

observed frequencies

running average

restructured frequencies

-0.90

-0.70

-0.50

-0.30

-0.10

0.10

0.30

0.50 1

2-1

3

14

-15

16

-17

18

-19

20

-21

22

-23

24

-25

26

-27

28

-29

30

-31

32

-33

34

-35

36

-37

38

-39

40

-41

42

-43

44

-45

46

-47

48

-49

50

-51

1

2 3 45

6?

=(x/running average) - 1

A S P

FiSAT - ELEFANFitted growth curve on restructured length-frequencies

Normal VBGF fitted Seasonal VBGF fitted

m a xE SP

A SP

E xp la ined sum o f peaks

A va ilab le sum o f peaks

Variable time intervals

1993 1994

General comments:Can you see growth? If not don’t try!!

General comments I

• What you cannot see you cannot fit.• If there is no reasonable clear visual

indications of growth in the data, do not try to fit a model.

• Software packages will always give a result for any dataset

• Never show results without superimposing the growth curve on the frequencies.

• Sometimes migrations can be misinterpreted as growth

General comments II

• For length based estimation of growth you need:

• Relative large sample over relatively short intervals of sampling.

• Representative proportions of young fish in the sample (commercial data often useless)

• ‘Non-selective’ sampling gear

• Distinct spawning season(s) so that cohorts are size-segregated.