FTP

Surplus-Production Models

2 Overview

Purpose of slides: Introduction to the production model (revision) Overview of different methods of estimating the

parameters Go over some critique of the method

Source: Haddon 2001, Chapter 10 Hilborn and Walters 1992, Chapter 8

3 Introduction

Nomenclature: surplus production models = biomass dynamic models

Simplest analytical models available Pool recruitment, mortality and growth into a

single production function The stock is thus an undifferentiated mass

Minimum data requirements: Time series of index of relative abundance

Survey index Catch per unit of effort from the commercial fisheries

Time series of catch

Revision

4 General form of surplus production

Are related directly to Russels mass-balance formulation:

Bt+1 = Bt + Rt + Gt - Mt – Ct

Bt+1 = Bt + Pt – Ct

Bt+1 = Bt + f(Bt) – Ct

Bt+1: Biomass in the beginning of year t+1 (or end of t)

Bt: Biomass in the beginning of year t Pt: Surplus production

the difference between production (recruitment + growth) and natural mortality

f(Bt): Surplus production as a function of biomass in the start of the year t

Ct: Biomass (yield) caught during year t

Revision

5 Functional forms of surplus productions

Classic Schaefer (logistic) form:

The more general Pella & Tomlinson form:

Note: when p=1 the two functional forms are the same

1 tt t

Bf B rB

K

1p

tt t

r Bf B B

p K

Revision

6 Population trajectory according to Schaefer model

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100

0 5 10 15 20 25

Time t

Sto

ck b

iom

ass

Bt

K: Asymptotic carrying capacity

1 1t t t tB B rB B K Unharvested curve

Revision

7

Surplus production as a function of stock size

0

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0 10 20 30 40 50 60 70 80 90 100

Stock biomass Bt

Surp

lus

pro

duct

ion

K

K /2

Maximum production1 t

t

BrB

K

Revision

The reference points from a production model

• Biomass giving maximum sustainable yield:BMSY = K/2

• Fishing mortality rate leading to BMSY:FMSY = r/2

• Effort leading to BMSY:EMSY = FMSY/q = (r/2)/q = r/(2q)

• Maximum sustainable yieldMSY = FMSY * BMSY = r/2 * K/2 = rK/4

Revision

Add fishing to the system

• Recall that

• r, B and q are all parameters we need to estimates

• We also need an index of abundance

• most often assume:

ttttt CKBrBBB 11

ttttt BqEBFC

t tCPUE f B

tt t

t

CCPUE qB

E

A side step – what does the model do

Parametersr 0.4 Reference pointsK 180 Bmsy 90p 1 Fmsy 0.2B1 144 MSY 18

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200

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0.00

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0.35

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0.45

0.50

Biomass

Bmsy

K

CPUE

0.00

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0.35

0 20 40 60 80 100

Ft

Fmsy

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25

0 20 40 60 80 100

Catch

MSY

FTP

Estimating r, K and q

12 Estimation procedures

Three types of models have been used over time:

Equilibrium methods The oldest method Never use equilibrium methods!

Regression (linear transformation) methods Computationally quick Often make odd assumption about error structure

Time series fitting Currently considered to be best method available

13 The equilibrium model 1

The model:

The equilibrium assumption:

Each years catch and effort data represent an equilibrium situation where the catch is equal to the surplus production at the level of effort in that year. This also means:

If the fishing regime is changed (effort) the stock is assumed to move instantaneously to a different stable equilibrium biomass with associated surplus production. The time series nature of the system is thus ignored.

THIS IS PATENTLY WRONG, SO NEVER FIT THE DATA USING THE EQUILIBRIUM ASSUMPTION

1 1t t t t tB B rB B K C t

t tt

CCPUE qB

E

Ct = Production

KBrBC ttt /1

tt BB 1

14 The equilibrium model 2

Given this assumption: It can be shown that:

can be simplified to

Then one obtains the reference values from:

EMSY = a/(2b)

MSY = (a/2)2/b

1 1t t t t tB B rB B K Y

1t t tC rB B K

tt t

t

CCPUE qB

E

Ca bE

E 2C aE bE

Ct = Production

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Effort

Surp

lus

pro

duct

ion

15 The equilibrium model 3Ct = Production

16 Equilibrium model 4

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Effort

Yie

ldIf effort increases as the fisheries develops we may expect to get data like the blue trajectory

The trueequilibrium

Observationsand the fit

Ct = Production

17 The regression model 1

Next biomass = this biomass + surplus production - catch

1 (1 )tt t t t t

BB B rB qE B

K

BC P U E

q

U

qtt t

1

2

• Substituting in gives:12

1 11 1t t tt t t

t t

U rU U rr E q r U qE

U qK U qK

….and multiplying by

t t t t tY F B qE B

tU

qtt

tttt UEqK

U

q

Ur

q

U

q

U

11

18 The regression model 2

…this means:

Regress: U

Ut

t

1 1 on Ut and Et which is a multiple regression of the form:

0 1 1 2 2 1 2

00 1 2

1 2

, ,

t tY b b X b X where X U and X E

r bb r b b q K

qK b b

1 1tt t

t

U rr U qE

U qK

mortality

fishing

rategrowth

inreduction

dependent

density

rate

growth

instrinsic

biomassin

change

of rate

19 Time series fitting 1

The population model:

The observation model (the link model):

The statistical model

1 1 tt t t t

BB B rB Y

K

ˆ ˆ or it t i t tU qB U qB e

2 2

min min0 0

ˆ ˆ or ln lnt t

t t t tt t

SS U U SS U U

20 Time series fitting 2

The population model: Given some initial guesses of the parameters r and K and

an initial starting value B1, and given the observed yield (Yt) a series of expected Bt’s, can be produced.

The observation model (“the link model”): The expected value of Bt’s is used to produce a

predicted series of Ût (CPUEt) by multiplying Bt with a catchability coefficient (q)

The statistical model: The predicted series of Ût are compared with those

observed (Ut) and the best set of parameter r,K,B0 and q obtained by minimizing the sums of squares

21

Example: N-Australian tiger prawn fishery 1

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7000

1970 1975 1980 1985 1990 1995

Catch

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40000

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Effort

History of fishery:1970-76: Effort and catch low, cpue high1976-83: Effort increased 5-7fold and catch 5fold, cpue declined by 3/41985-: Effort declining, catch intermediate, cpue gradually increasing.

Objective: Fit a stock production model to the data to determine state of stock and fisheries in relation to reference values.

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1970 1975 1980 1985 1990 1995

CPUE

Haddon 2001

22

Example: N-Australian tiger prawn fishery 2

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0.8

1.0

1.2

1970 1975 1980 1985 1990 1995

Et/Emsy

0.00.20.40.60.81.01.21.41.61.82.0

1970 1975 1980 1985 1990 1995

Bt/Bmsy

The observe catch rates (red) and model fit (blue).Parameters: r= 0.32, K=49275, Bo=37050Reference points:MSY: 3900Emsy: 32700Bcurrent/Bmsy: 1.4

The analysis indicates that the current status of the stock is above Bmsy and that effort is below Emsy.Question is how informative are the data and how sensitive is the fit.

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1970 1975 1980 1985 1990 1995

CPUE

Haddon 2001

23 Adding auxiliary information

In addition to yield and biomass indices data may have information on:

Stock biomass may have been un-fished at the beginning of the time series. I.e. B1 = K. Thus possible to assume that

U1 = qB1=qK However data most often not available at the beginning of the

fisheries Absolute biomass estimates from direct counts, acoustic

or trawl surveys for one or more years. Those years may be used to obtain estimates of q.

Prior estimates of r, K or q q: tagging studies r: basic biology or other similar populations

In the latter case make sure that the estimates are sound K: area or habitat available basis

Greenland halibut

Bootstrap uncertainty – history and predictions

FTP

Some word of caution

27 One way trip

“One way trip” Increase in effort and decline in CPUE with time

A lot of catch and effort series fall under this category.

Time

Effort

Time

Catch

Time

CPUE

28 One way trip 2

Fit# r K q MSY Emsy SSR 1.24 3 105 10 10-5 100,000 0.6 106 ----T1 0.18 2 106 10 10-7 100,000 1.9 106 3.82T2 0.15 4 106 5 10-7 150,000 4.5 106 3.83T3 0.13 8 106 2 10-7 250,000 9.1 106 3.83

Identical fit to the CPUE series, however: K doubles from fit 1 to 2 to 3

Thus no information about K from this data set Thus estimates of MSY and Emsy is very unreliable

29 The importance of perturbation histories 1

“Principle: You can not understand how a stock will respond to exploitation unless the stock has been exploited”. (Walters and Hilborn 1992).

Ideally, to get a good fit we need three types of situations:

Stock size Effort get parameterlow low rhigh (K) low K (given we know q)high/low high q (given we know r)

Due to time series nature of stock and fishery development it is virtually impossible to get three such divergent & informative situations

steel JK slide from the first lecture on the development

31 One more word of caution

Equilibrium model: Most abused stock-assessment technique – hardly used

Published applications based on the assumption of equilibrium should be ignored

Problem is that equilibrium based models almost always produce workable management advice. Non-equilibrium model fitting may however reveal that there is no information in the data.

Latter not useful for management, but it is scientific

Efficiency is likely to increase with time, thus may have:

Commercial CPUE indices may not be proportional to abundance. I.e.

Why do we be default assume that the relationship is proportional?

And then K may not have been constant over the time series of the data

t tCPUE qB

tqqtfq add 1tq e.g )(

32

CPUE = f(B): What is the true relationship??

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Biomass

com

merc

ial CPU

E

HyperstabilityProportionalHyperdepletion

Why do we by default assume that the relationship is proportional?

33 Accuracy = Precision + Bias

Not accurate and not precise

Accurate but not precise

(Vaguely right)

Accurate and precise

Precise but not accurate

(Precisely wrong)

It is better to be vaguely right than precisely wrong!