1. Surplus Production Models
Michael A. Rutter
Penn State Erie

2. 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

3. 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

4. Logistic Growth Model R
1.45 K
100000 Bo
30000

5. Adding Harvest
Constant rate of harvest

6. Low Initial Biomass R
1.45 K
100000 Bo
30000
Harvest
5000

7. High Initial Biomass R
1.45 K
100000 Bo
98000
Harvest
5000

8. 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)

9. Solve for Bt
Bt=87267 kg for this example

10. Population Crash Surplus production can’t handle the harvest R 1.45 K
100000 No
12000
Harvest
5000
Surplus production can’t handle the harvest

11. A More Realistic Harvest
Commercial harvest is rarely constant
Harvest is a function of the fishing effort
Example: Gill Nets

12. 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

13. Modeling Harvest
Simple model: Harvest is proportional to biomass/abundance

14. Modeling Harvest Simple model E is the effort (1000s feet/day)
q is catchability
The proportion of fish caught for one unit of effort

15. Constant Effort
R=1.45, K=100,000, B0=30,000, E=1000, q=0.0001

16. Our Model

17. Estimating Parameters
In the real world, the only quantities measured/observed are effort and harvest

18. 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

19. 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

20. Statistical Stuff Effort is assumed known, without error
Harvest is also known, but has measurement error
Assume that the measurement error is lognormally distributed

21. 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

22. Horrible Equations
Use the board

23. Estimating Parameters
We adjust the values of q, Bo, R, and K to maximize the likelihood
Why do we ignore s2?
We assume that s2 is a measure of the reliability of the data
With only one data source (harvest), all the data is equally good (or bad)

24. 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

25. Maximizing Surplus Production
Max occurs when biomass is:
For our parabola (it can be shown…)

26. 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

27. Our original example

28. 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

29. 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?