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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?