sustainalbe fishery management / fish population dynamics
DESCRIPTION
Recruitment : increase of the fish in to the stock at the age a fish can be caught Fish experience Mass Mortality at the early life stage. The magnitude will be less than 1/1000. The S-R relationship is not clear and sometimes looks like no relationship between them. S-R models are used for describing ideal relationship Beverton and Holt Model Ricker Model MSY will be calculated from S-R curve. VBGC is often used for describing the fish growth Weight is converted by the allometric equation. Instantaneous rate of mortality is used Total mortality Z is observed from age composition. Usually Natural Mortality Mis estimated from Empirical Equations Fishing mortality F is estimated as Z minus MTRANSCRIPT
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Fish Population Dynamics
Takashi Matsuishi At SERD, AIT, Thailand 24Feb-14Mar, 2014
1
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Russell’s Equation
2
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Basic Idea of Population Dynamics of
Exploited Stock
3
Closed stock : without Immigration / Emigration
Stock size will be increase only by
Recruitment
Growth
Stock size will decrease only by
Natural Mortality
Fishing Mortality
If increasing factor and decreasing factor balance, then the
stock size will be stable
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Russell’s Equation (Russell 1931)
Russel, E. S. 1931. Some
theoretical considerations on the
‘overfishing’ problem.
Journal du Conseil International
pour l’Exploration de la Mer, 6: 3-
20.
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Morta
lity
Yie
ld
Bio
mass a
t t
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Stock Growth
6
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Morta
lity
Yie
ld
Bio
mass a
t t
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Stock Growth
8
Stock Growth is the increasing factors of the stock.
It is divided Recruitment and Individual Growth.
Recruitment is a factor of stock growth, which adding
number of individuals in the stock.
Individual Growth (or simply Growth) is a factor of
growth, which adding weight of each individual in the
stock.
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Recruitment
9
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Morta
lity
Yie
ld
Bio
mass a
t t
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Definition of Recruitment
11
Recruitment is defined as the increase of the number of
individual into the stock at the age a fish CAN be caught.
Cf. Age at First Capture
Recruitment will be affected both with life history and
fishery.
Migration from Nursery ground to Fishing ground
Body length reaches to the minimum size for fishing
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Schematic Display of Population Dynamics
12
Mass M
ortality
Recru
itment
First C
aptu
re
Matu
ration
Longe
vity
#Fish
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Mass Mortality
13
In the life history of fish from the hatching to the recruit, most fish species experience mass mortality.
The magnitude of the survival rate will be less than 0.001 level.
The factor of the mass mortality will be
Feeding
Unsuccessful transportation
Competition on feeding
Mismatch of the prey species
Predation
Physical Environment
Sea water temperature
Etc
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14
Rough calculation of the early mass
mortality of Walleye Pollock P-stock
Spawing stock 4×108 ind . (stock assessment)
Fecundity 1×106eggs (observation)
Hatching Rate 10% (observation)
#hatched juvenile 4×1013 ind.
# Recruitment at age 1 1×109 (Stock assessement)
Survival Rate from Hatched juvenile to Age 1 fish
0.000025
The survival rate will fluctuate widely.
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Two approach for dealing recruitment
for fishery management
15
Estimate the relationship of recruitment and various
factors
Spawning biomass
Physical environmental factors
Biological environmental factors (#prey, #predator)
Assume recruitment can not estimate or independent to
the spawning stock, and only consider the ratio to
recruitment
Per recruit analysis
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Examples of stock-recruitment relationship
16
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17
Sardine Pacific Stock
http://abchan.job.affrc.go.jp/digests19/details/1901.pdf
Spawning Stock (1000t)
Recr
uitm
ent
(mill
ion)
R=19.85S
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18
Chub mackerel, Pacific Stock
http://abchan.job.affrc.go.jp/digests19/details/1905.pdf
Recr
uitm
ent
(100m
illio
n)
SSB(1000t)
Recr
uitm
ent
(100m
illio
n)
SSB(1000t)
Curves are best fitted Ricker Curve
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19
Walleye Pollock P-stock
http://abchan.job.affrc.go.jp/digests19/details/1913.pdf
SSB (1000t)
Recr
uitm
ent
(mill
ion a
t ag
e 0
)
Dominant YC
Acceptable
Level
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20
Japanese Flying Squid – J stock
http://abchan.job.affrc.go.jp/digests19/details/1919.pdf
Number of Spawners (100million)
Num
ber
of N
ext
Genera
tion 1
00m
illio
n)
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Stock and Recruitment Relationship
21
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22
Stock Recruitment Relationship
Quantitative relationship between the number of parents
(t) generation and children (t+1) generation
It would be simple if the number is measured at same age
in different generation.
For example; pink salmon
Come back to the original river exactly
Come back at 2 years old
Easy to count in the river.
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23
Stock-Recruitment Curve
Theoretical Curve to describe the Parents Generation and Children Generation
On the replacement line (45-degree line), the number of t generation and t+1 generation is same
Cross point of the S-R curve and replacement line is the equilibrium point, here population does not increase and decrease in long term average.
t+1 g
en
era
tio
n
t generation
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Example of S-R Curve
24
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25
Beverton-Holt Recruitment Model
Sb
aSR
0
5000
10000
15000
20000
0 5000 10000 15000 20000
t+1 g
en
era
tio
n
t generation
a, b: constant
S: Spawning stock (t)
R: Recruitment (t+1)
Contest Competition a=5, b=0.000267
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26
Example
S-R Relationship of Sea Bream (Okada 1974)
Spawning Stock
Recr
uitm
ent
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27
Ricker Model
bSaSeR
Scramble Competition
0
5000
10000
15000
20000
0 5000 10000 15000 20000
t+1 g
en
era
tio
n
t generation
a, b: constant
S: Spawning stock (t)
R: Recruitment (t+1)
a= 4.482, b=0.0001
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28
Sockeye Salmon in Kurlak River Alaska
(Tanaka 1960)
Spawning Stock
Recr
uitm
ent(
100,0
00)
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29
Sustainable Yield inferred from S-R Curve 1
Without Exploitation S1 is
the equilibrium point.
Recruit will be R1=S1
S1 S2
R1
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C2
30
Sustainable Yield inferred from S-R Curve 2
If S2 ,recruit will be R2
which is S2+C2.
If C2 is caught, the rest of
the stock is S2 and in the
next generation R2 will
come back.
You can catch C2 for ever.
It is Sustainable Yield.
S1 S2
R2
R1
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C3
31
Sustainable Yield inferred from S-R Curve 3
At S3, vertical distance
between S-R curve and
replacement curve is max.
C3 is also sustainable yield,
and you can catch C3 for
ever.
It is Maximum Sustainable
Yield (MSY).
S3
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Growth
32
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Morta
lity
Yie
ld
Bio
mass a
t t
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Growth of individuals
34
Growth is another component of stock production.
Growth is usually described by using theoretical growth
curve.
Usually growth curve describe the relationship between
age and length.
Weight growth curve can be used, but sometimes the
weight is converted from length by using the allometric
equation.
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Von Bertalanffy growth curve
35
Von Bertalanffy growth curve (VBGC) is most popular.
Lt : length at age t
L∞: asymptotic average maximum body size
K : growth rate coefficient
t0: hypothetical age which the species has zero length
01ttK
t eLL
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Von Bertalanffy Growth Curve
36
0
10
20
30
40
50
0 1 2 3 4 5 6 7 8 9 10
Len
gth
Age L∞=50, K=0.2, t0=-0.5
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Length – Weight relationship
37
Usually the relationship between weight and length follow
the allometric equation
wt: weight at age t
Lt: length at age t
a: scaling constant
b: allometric growth parameter (close to 3)
b
tt aLw
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Example
38
0 1 2 3 4 5 6 7 8 9
10
0 5 10 15 20 25 30 35 40
Weig
ht(
kg)
Length(cm) a=0.00015, b=3
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Von Bertalanffy growth equation
for body weight
39
Combined with VBGC and allometric equation
VBGC for body weight ;
wt: weight at age t
w∞: asymptotic average maximum body weight
K : growth rate coefficient
t0: hypothetical age which the species has zero length
b: allometric growth parameter (often set to 3)
bttK
t eww 01
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Example of VBGC for body weight
40
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10
Weig
ht(
kg)
Age w∞=18.75, K=0.2, t0=-0.5, b=3
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VBGC for Length vs. Weight
41
0
10
20
30
40
50
0 1 2 3 4 5 6 7 8 9 10
Len
gth
Age
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10 W
eig
ht(
kg)
Age
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Mortality
42
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Mo
rtality
Yie
ld
Bio
mass a
t t
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Total Mortality
44
Total Mortality is the factor reducing the stock.
Total Mortality is divided to Natural Mortality and Fishery
Mortality (= Yield, Harvest)
Usually, only total mortality can be observed from age
composition.
Natural Mortality can be estimated from various method,
and also estimated from various empirical equations.
Fishing Mortality is estimated from total mortality and
natural mortality.
The estimated fishing mortality contains errors in
estimating total mortality, and natural mortality.
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Index of Mortality
45
Usually mortality is measured by the instantaneous rate.
“Instantaneous rate of mortality” is simply called as
“mortality”.
If you use the percentage of the died individuals to the
population at the beginning of the year, it is called
“mortality rate”, and is different to “instantaneous rate of
mortality” .
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Equations of Mortality
46
MFZ
Total M
ortality
Fish
ing M
ortality
Natu
ral Mortality
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Z
Zt
ZZt
Zt
tZ
t
t
eeN
eeN
eN
eN
N
NS
0
0
0
1
01
Mortality and Survival Rate
47
Zt
t eNN 0
Survival Rate
Mortality Rate SD 1
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Mortality and Population Dynamics
48
0
200
400
600
800
1,000
0 2 4 6 8 10
Po
pu
lati
on
Age
t Nt S 0 1,000 0.7
1 700 0.7
2 490 0.7
3 343 0.7
4 240 0.7
5 168 0.7
6 118 0.7
7 82 0.7
8 58 0.7
9 40 0.7
10 28
tt NNS 1Z=0.357
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Cf Constant Death
49
0
200
400
600
800
1,000
0 2 4 6 8 10
Po
pu
lati
on
Age
KtNNt 0
t Nt S 0 1,000 0.90
1 900 0.89
2 800 0.88
3 700 0.86
4 600 0.83
5 500 0.80
6 400 0.75
7 300 0.67
8 200 0.50
9 100 0.00
10 0
K=100
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Linear Scale Log Scale
50
0
200
400
600
800
1,000
0 2 4 6 8 10
Age
1
10
100
1,000
0 2 4 6 8 10
Age Z=0.357
Po
pu
lati
on
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0
1
2
3
4
5
6
0 2 4 6 8 10
Age Z=0.357
ln(Nt)
51
Po
pu
lati
on
y = -0.357x + 6.908
t Nt ln(Nt)
0 1,000 6.91 1 700 6.55 2 490 6.19 3 343 5.84 4 240 5.48 5 168 5.12 6 118 4.77 7 82 4.41 8 58 4.05 9 40 3.70 10 28 3.34
X Y
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Estimation of Total Mortality
52
1. Get the N1, N2, N3, ..., NT or its index, from a same year
class.
2. If impossible, and if you can assume the recruit and
fishery is stable, use C1, C2, C3, ..., CT from a same year.
3. Calculate ln(Ci) (i=1,...,T)
4. Confirm that it declines monotonously. If not, omit it. It
would be affected by gear selectivity.
5. Plot and make linear regression.
6. The coefficient for tangent is –Z.
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Realistic Example
53
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10
ln(C
t)
Age
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Schematic Display of Population Dynamics
54
Mass M
ortality
Recru
itment
First C
aptu
re
Matu
ration
Longe
vity
#Fish
M M+F ln(N
)
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Natural Mortality
55
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Mo
rtality
Yie
ld
Bio
mass a
t t
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Natural Mortality 1
57
Natural Mortality is a part of mortality caused by natural
reason
Various Factors
Disease
Predation
Prey Shortage
Physical Environment
Competition
Unexpected Emigration
Unreported Fishery
Etc…
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Estimation of the Natural Mortality
58
Mark- Recapture Method
In Captivity
Total Mortality of Unexploited Stock
Estimated from the change of Fishing Effort
Empirical Method
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59
Natural Mortality Estimation
Fishing mortality will be
proportional to fishing
effort f with coefficient q
Z and f has linear relation
Plot Z and f
M is estimated as the y-
intercept of the regression
line
qfF
MqfMFZ (Silliman 1943) Age
ln(C
t) 1st period (1925-33)
2nd period (1937-42)
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Empirical Method 1
60
It is very difficult to conduct direct measurement of
Natural mortality for each commercial species.
No enough data for analysis
The range of the fishing effort variation is small
Difficult to conduct mark-recapture experiment because of the
lack of budget and man-power
Many empirical method are proposed
Collecting the results of the direct measurements
Find some relationship with available parameters
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Empirical Method
61
Mainly estimated from
Growth curve parameter,
Water Temperature,
Life history Parameter
Longevity,
age at Mature
etc
Results may have large variety.
Use
Common methods in consensus
Compare the results.
Parameters used in Hewitt et al. (2007)
tm = age at maturity (years)
X = a constant taken from the given sources
K = von Bertalanffy growth coefficient (per year)
tmax = longevity(years)
CW∞ = asymptotic maximum carapace width (cm) from VBGC
T = grand annual mean of water temperature (degree Celcius)
W ∞ = asymptotic maximum weight (g) from VBGCw
W = wet weight (g)
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62
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Methods and Results of Hewitt et al. (2007)
63
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Frequency distribution of the range of the
results
64
Value currently used for
stock assessment
(Hewitt et al. 2007)
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Yield / Fishing Mortality
65
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Russell's Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Morta
lity
Yie
ld
Bio
mass a
t t
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Fishing Mortality
67
Fishing mortality is estimated from Z and M
MFZ Total
Mortality
Fish
ing
Mortality
Natu
ral
Mortality
MZF
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Related Equations
68
Total Mortality
Survival Rate
Mortality Rate
Catch Equation
MFZ ZeS
SD 1
tt DNZ
FC
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Catch Equation
69
The relationship between Population, Mortality, and Catch
tt DNZ
FC
Catch
in N
um
ber
Populatio
n in
Num
ber
Mortality R
ate
Portio
n o
f die
d fish
by fish
ery
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Catch Equation
70
The relationship between Population, Mortality, and Catch
t
MF
tt
NeMF
F
DNZ
FC
1
C is a function of F, M, and N
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Feature of Fishing Mortality
71
Fishing Mortality will be calculated from Z and M
Fishing Mortality will be proportional to the fishing effort.
Fishing Mortality is not proportional to Catch.
MN
NMZF
t
t
1
ln
qfF
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Yield
72
Yield / Fishing Mortality is the only controllable
component in the Russell's Equation
Given recruit, growth and natural mortality, if you would
like to increase the stock more, the only way is to reduce
yield.
To optimize the sustainable yield,
Monitor the stock biomass Stock assessment
Optimize the fishing effort MSY and other fishery models
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Key Points of This Section
73
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Key Points 1: Russell’s Equation
YDGRBB tt 1
Bio
mass a
t t+1
Re
cru
itme
nt
Gro
wth
Na
tura
l Morta
lity
Yie
ld
Bio
mass a
t t
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Key Points 2 : Recruitment
75
Recruitment : increase of the fish in to the stock at the
age a fish can be caught
Fish experience Mass Mortality at the early life stage.
The magnitude will be less than 1/1000.
The S-R relationship is not clear and sometimes looks
like no relationship between them.
S-R models are used for describing ideal relationship
Beverton and Holt Model
Ricker Model
MSY will be calculated from S-R curve.
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Key Points 3 : Growth Mortality
76
VBGC is often used for describing the fish growth
Weight is converted by the allometric equation.
Instantaneous rate of mortality is used
Total mortality Z is observed from age composition.
Usually Natural Mortality Mis estimated from
Empirical Equations
Fishing mortality F is estimated as Z minus M
01ttK
t eLL