Surplus Production Models

Michael A. Rutter

Penn State Erie

Motivation

• Fishery managers are usually only interested in fish populations that are being exploited

• Data on the fish population is usually limited to catch information

• Managers wish to estimate that size of the population and set fishing levels

Modeling Biomass

• Often model the biomass (kg) of the population as opposed to the number of fish

• Harvest is measured in kg

• Difficult to specify fishing regulations based on number of fish (size issues)

• Model works the same

Logistic Growth Model

0

20000

40000

60000

80000

100000

120000

0 5 10 15 20 25 30

Years

Biomass

K

BBRBB tttt 111

R 1.45

K 100000

Bo 30000

Adding Harvest

• Constant rate of harvest

tt

ttt HK

BBRBB

111

0

20000

40000

60000

80000

100000

0 5 10 15 20 25 30

Years

Biomass

R 1.45

K 100000

Bo 30000

Harvest 5000

Low Initial Biomass

High Initial Biomass

0

20000

40000

60000

80000

100000

0 5 10 15 20 25 30

Years

Biomass R 1.45

K 100000

Bo 98000

Harvest 5000

Surplus Production

• Surplus production is defined as the biomass remaining after the previous years biomass has been replaced

• Biomass stabilizes when surplus production equals harvest (assuming constant harvest)

K

BBR tt 11

HK

BBR tt

11

0)1(1 2

HBRBK

Rtt

• Solve for Bt

• Bt=87267 kg for this example

)1(2

442()1( 2

R

HHRKKRKRKRKBt

Population Crash

• Surplus production can’t handle the harvest

-3000

0

3000

6000

9000

12000

15000

0 2 4 6 8 10

Years

Biomass

R 1.45

K 100000

No 12000

Harvest 5000

A More Realistic Harvest

• Commercial harvest is rarely constant

• Harvest is a function of the fishing effort

• Example: Gill Nets

Measuring Effort

• Depending on the harvesting method, effort can be measured in different units

• 1000s of feet per day – Gill net

• Number of hours angling – Hook and line

• Length of trawl – Trawling

• Number of net sets

Modeling Harvest

• Simple model: Harvest is proportional to biomass/abundance

Modeling Harvest

• Simple model

• E is the effort (1000s feet/day)

• q is catchability

– The proportion of fish caught for one unit of effort

qEBH

Constant Effort

R=1.45, K=100,000, B0=30,000, E=1000, q=0.0001

0

20000

40000

60000

80000

100000

0 5 10 15 20 25 30

Year

Biomass

0200040006000800010000120001400016000

Harvest

BiomassHarvest

Our Model

ttt

ttt BqEK

BBRBB

111

Estimating Parameters

• In the real world, the only quantities measured/observed are effort and harvest

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30

Year

Effort

0

5000

10000

15000

20000

25000

Catch

EffortCatch

What is measured with error?

• Unless the fisherpersons are lying, the effort is assumed to be recorded accurately

• The amount of biomass harvested is measured with some error– Lognormal error– Try weighing hundreds of wet slippery

fish on a boat in the middle of the ocean/lake

What needs to be estimated?

• From a fisheries perspective, we are interested only in R and K

• Also need to know q and an initial biomass

ttt

ttt BqEK

BBRBB

111

Statistical Stuff

• Effort is assumed known, without error

• Harvest is also known, but has measurement error

• Assume that the measurement error is lognormally distributed

Maximum Likelihood

• In order to find the best estimates of the model parameters, we need to find the likelihood of the observed harvest given the model parameters and the known effort

• Use numerical methods to find the maximum likelihood estimates of the model parameters

Horrible Equations

• Use the board

Estimating Parameters

• We adjust the values of q, Bo, R, and K to maximize the likelihood

• Why do we ignore 2?

– We assume that 2 is a measure of the reliability of the data

– With only one data source (harvest), all the data is equally good (or bad)

Parameter Estimates

• Only R and K are of interest– q and Bo are only needed for the model

• How can we use this to manage the fishery?– Recall the surplus production

tt BRBK

R)1(

1 2

Maximizing Surplus Production

• Max occurs when biomass is:

• For our parabola (it can be shown…)

2

KB

a

b

2

Maximum Sustainable Yield

• The largest amount of biomass that can be removed and maintain the biomass at a constant value

• Occurs when biomass is at K/2

• Use R to determine harvest at that point

• Usually set a harvest level at 90% or 85% MSY to prevent crashing the population

Our original example

)1(4

111

11

22

RKK

R

K

BBRH

KK

tt

Exercise 3

• Fit a surplus production model to actual Tuna data from the South Atlantic (based on Polacheck et al. 1993)– “exercise3.xls”

• Determine– Maximum sustainable yield– Harvest at MSY

But wait…

• As with all statistical things, there is error

• How do we describe the error so we can prevent the extinction of the fishery?