Sustainalbe Fishery Management / Fish Population Dynamics

Fish Population Dynamics

Takashi Matsuishi At SERD, AIT, Thailand 24Feb-14Mar, 2014

1

Russell’s Equation

2

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

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.

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

Stock Growth

6

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

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.

Recruitment

9

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

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

Schematic Display of Population Dynamics

12

Mass M

ortality

Recru

itment

First C

aptu

re

Matu

ration

Longe

vity

#Fish

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

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.

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

Examples of stock-recruitment relationship

16

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

18

Chub mackerel, Pacific Stock


Recr

uitm

ent

(100m

illio

n)

SSB(1000t)

Recr

uitm

ent

(100m

illio

n)

SSB(1000t)

Curves are best fitted Ricker Curve

19

Walleye Pollock P-stock


SSB (1000t)

Recr

uitm

ent

(mill

ion a

t ag

e 0

)

Dominant YC

Acceptable

Level

20

Japanese Flying Squid – J stock


Number of Spawners (100million)

Num

ber

of N

ext

Genera

tion 1

00m

illio

n)

Stock and Recruitment Relationship

21

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.

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

Example of S-R Curve

24

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

26

Example

S-R Relationship of Sea Bream (Okada 1974)

Spawning Stock

Recr

uitm

ent

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

28

Sockeye Salmon in Kurlak River Alaska

(Tanaka 1960)

Spawning Stock

Recr

uitm

ent(

100,0

00)

29

Sustainable Yield inferred from S-R Curve 1

Without Exploitation S1 is

the equilibrium point.

Recruit will be R1=S1

S1 S2

R1

C2

30


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

C3

31


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

Growth

32

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

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.

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

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

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

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

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

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

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

Mortality

42

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

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.

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” .

Equations of Mortality

46

MFZ

Total M

ortality

Fish

ing M

ortality

Natu

ral Mortality

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

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

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

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

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

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.

Realistic Example

53

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10

ln(C

t)

Age

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

)

Natural Mortality

55

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

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…

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

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)

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

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)

Methods and Results of Hewitt et al. (2007)

63

Frequency distribution of the range of the

results

64

Value currently used for

stock assessment

(Hewitt et al. 2007)

Yield / Fishing Mortality

65

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

Fishing Mortality

67

Fishing mortality is estimated from Z and M

MFZ Total

Mortality

Fish

ing

Mortality

Natu

ral

Mortality

MZF

Related Equations

68

Total Mortality

Survival Rate

Mortality Rate

Catch Equation

MFZ ZeS

SD 1

tt DNZ

FC

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

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

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

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

Key Points of This Section

73

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

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.

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

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