1. Day 1 Session 3 Key concepts of stock assessment modelling
11/21/2018
Day 1 Session 3 Key concepts of stock assessment modelling

2. Overview of key concepts
11/21/2018
Overview of key concepts
What is a fish “stock”?
What is stock assessment?
What is a stock assessment model?
What are the key population and fishery processes we need to include/account for in a stock assessment model?
Equations in stock assessment
Fitting models to data
Types of model

3. Key Concept 1: The Stock Key Concept 1
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Key Concept 1: The Stock
Key Concept 1
A stock assessment model is used to assess a fish population that has little or no mixing or interbreeding with other populations.
What is a fish stock?
“A unit stock is an arbitrary collection [of a single species] of fish that is large enough to be essentially self reproducing (abundance changes are not dominated by immigration and emigration) with members of the collection showing similar patterns of growth*, migration and dispersal.
The unit should not be so large as to contain many genetically distinct races of subpopulations within it.”
Hilborn and Walters (1991)

4. Key Concept 1: The Stock Stock
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Key Concept 1: The Stock
Why do we manage and assess fisheries at the level of a stock?
1. Self contained
2. Management convenience
3. Scientifically meaningful
4. Little or no external influences
Stock

5. Key Concept 1: The Stock How do we identify a fish stock?
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Key Concept 1: The Stock
How do we identify a fish stock?
It’s a very difficult task ………often little clear information. We can use:
Genetics
Tagging
CPUE analyses
Morphometrics
Often stock assessments are conducted on “stocks” where there is some uncertainty regarding the boundaries of the stock (e.g. WCPO v EPO bet/yft; SWPO v SPO v PO stm)

6. WCPO tuna “stocks” Yellowfin tuna
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WCPO tuna “stocks”
Yellowfin tuna
Limited mixing and genetic variation found
Bigeye tuna
Mixing less limited and no genetic variation found
Schaefer, K.M. and D.W. Fuller (2009). Horizontal movements of bigeye tuna (Thunnus obesus) in the Eastern Pacific Ocean, as determined from conventional and archival tagging experiments initiated during IATTC Bull. 24(2):

7. WCPO tuna “stocks” Albacore tuna
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WCPO tuna “stocks”
Albacore tuna
Low catch and CPUE in equatorial waters suggests limited adult mixing
No tag exchange between north and south PO tagged fish
Discrete spawning areas (based on larval surveys) between North and south Pacific

8. WCPO tuna “stocks” Skipjack tuna
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WCPO tuna “stocks”
Skipjack tuna
There is uncertainty regarding skipjack stock structure in the Pacific, but given a lack of evidence for trans basin movements and generally localised tag returns in general, mixing is thought to be limited within generations (short lived species) and in the medium term, meaning the WCPO “stock” is assessed as such for management purposes.

9. WCPO tuna “stocks” Striped marlin
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WCPO tuna “stocks”
Striped marlin
Equatorial CPUEs low, genetic variation between SWPO and NEPO, discrete spawning areas?, no transbasin tag returns, no transbasin movement indicated by 50 PSAT tags, all indicate limited mixing and a potential southwest Pacific stock (for management purposes)
Bromhead et al (2004)
Courteousy Michael Domeier, Pfelger Institute, 2006
Bromhead et al (2004)

10. Key Concept 2: Stock Assessment
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Key Concept 2: Stock Assessment
Stock assessment is a multi-step process that starts with management questions, and includes processes involved in data collection, model selection, stock assessment modelling, and subsequent advice to decision makers.

11. Key Concept 3: The stock assessment model
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Key Concept 3: The stock assessment model
A stock assessment model provides a simplification of a very complex system (fish and fishery), to help us estimate population changes over time in response to fishing

12. Key Concept 3: The stock assessment model
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Key Concept 3: The stock assessment model
What is a model?
A mathematical representation (or description) of a system that is used to help us understand the system and how the system works
e.g.
Bt+1=Bt+R+G-M-C
This is a simple mathematical model of our fish population….it describes a system (fish population) and the processes that effect that system (recruitment, growth, mortality)…it provides us a simplification of a complex system to help us understand that system, predict how it will react to different conditions, so we can make informed decisions regarding its management.
Growth
Recruitment
Biomass
Death
(Natural mortality)
Catch
(Fishing mortality)
Internal Movement

13. Key Concept 3: The stock assessment model
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Key Concept 3: The stock assessment model
Example of a simple stock assessment model
Nt+1,a+1 = Nt,ae-(Ma + Ft,a)
Ft,a = qtEtsa
Ct,a = Nt,aFt,awa
Rt = (ASt)/(b+St)
Nt+1,1 = Rt
Bt = ΣNt,awa
St = ΣNt,awaoa
VBt = ΣNt,awasa
Nt+1,a+1 = Number of fish of age+1 at time+1
Ma = natural mortality rate at age a
Fa = fishing mortality rate at age a
q = catchability
E = fishing effort (units)
s = age specific vulnerability to the gear (selectivity of the gear)
Ct,a = Catch at time t and age a
wa = Mean weight at age a << (Growth)
Rt = Recruitment at time t
A = maximum recruitment
b = Stock size when recruitment is half the maximum recruitment
wa = weight at age a
oa = proportion mature at age a
Bt = population biomass at time t
St = spawning stock biomass at time t
VB = vulnerable biomass at time t

14. Key Concept 4: Processes within the model
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Key Concept 4: Processes within the model
Stock size fluctuations can be estimated by accounting for four key processes, additive processes (growth, recruitment) and subtractive processes (fishing mortality, natural mortality) over time.
Bt+1=Bt+R+G-M-C
Death
(Natural mortality)
Recruitment
Biomass
Catch
(Fishing mortality)
Growth

15. Review of concepts so far…..
11/21/2018
Review of concepts so far…..
Concept 1 – A stock is a fish population that has little or no mixing or interbreeding with other populations.
Concept 2 – Stock assessment is a multistep process that starts with management questions regarding the impact of fishing on the stock, and includes processes involved in data collection, model selection, stock assessment modelling, and subsequent advice to decision makers.
Concept 3 - A stock assessment model provides a simplification of a very complex system (fish and fishery), to help us estimate population changes over time in response to fishing, and predict population changes in future in response to management actions
Concept 4 - Stock size fluctuations can be estimated by accounting for four key processes, additive processes (growth, recruitment) and subtractive processes (fishing mortality, natural mortality) over time.

16. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
To understand how stock assessments work at a technical level, it is important to understand the key types of equation used and how they are used….
…..the following section provides a very brief and basic overview of the use of equations in stock assessment modeling

17. Key Concept 5: Stock Assessment involves equations!
11/21/2018
Key Concept 5: Stock Assessment involves equations!
A quick word about equations and model building!
An equation can be thought of as simply being a sentence, but the words have been replaced by symbols….for example:
Bt+1=Bt+R+G-M-C
If we know what the symbols mean, we can read the sentence!
“Population biomass next year is equal to the biomass this year, plus biomass of new recruits in one years time, plus biomass of additional growth of this years fish, minus biomass of fish that died of natural causes, minus the biomass of fish killed by fishing”.
Large stock assessment models can involve complex equations expressing mathematical and statistical functions which attempt to describe the interacting fishery and fish population processes. Interpreting those interlinking equations requires training in maths, statistics and computer programming…

18. Key Concept 5: Stock Assessment involves equations!
11/21/2018
Key Concept 5: Stock Assessment involves equations!
A quick word about equations and model building!
However, we don’t need to have all that training to get a basic understanding of how stock assessment models work
We will focus on a very basic model of an exploited fish population during this workshop, which incorporates nearly all the major elements that comprise far more complex assessments, such as those conducted for tuna with MULTIFAN-CL in the Western and Central Pacific.
If you can gain an understanding of how our workshop model works, you will be a long way towards understanding the key principles and mechanics that underpin the Pacific tuna assessment models.

19. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
The equations making up our simple stock assessment model
Nt+1,a+1 = Nt,ae-(Ma + Ft,a)
Ft,a = qtEtsa
Ct,a = Nt,aFt,awa
Rt = (ASt)/(b+St)
Nt+1,1 = Rt
Bt = ΣNt,awa
St = ΣNt,awaoa
VBt = ΣNt,awasa
Nt+1,a+1 = Number of fish of age+1 at time+1
Ma = natural mortality rate at age a
Fa = fishing mortality rate at age a
q = catchability
E = fishing effort (units)
s = age specific vulnerability to the gear (selectivity of the gear)
Ct,a = Catch at time t and age a
wa = Mean weight at age a << (Growth)
Rt = Recruitment at time t
A = maximum recruitment
b = Stock size when recruitment is half the maximum recruitment
wa = weight at age a
oa = proportion mature at age a
Bt = population biomass at time t
St = spawning stock biomass at time t
VB = vulnerable biomass at time t

20. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
What types of equation are used in stock assessment models?
Differential equations – measure rates of change
Difference equations – predict values at fixed point in time

21. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
Calculating rates (of change)
..we do it every day..
Car speed (km/hour)
Interest rates
Weight loss (kg/week)
Typing (words/minute)
What is a rate?
It is the extent to which a change in one quantity affects a change in another related quantity. This is called a rate of change. 70 60 50
Distance (km) 40 30 20 10
Time (hours)

22. Key Concept 5: Stock Assessment involves equations!
11/21/2018
Key Concept 5: Stock Assessment involves equations!
Example
Two cars, A and B
Car A travels 80km after 4 hours
Car B travels 40km after 8 hours
How do we calculate the rate of travel?
Rate = change in y change in x
Car A: Rate = 80/ = 20km/hour
Car B: Rate = 40/8
= 5km/hour
Car A
So….
Car A is travelling at a faster rate than Car B 70 60 50
Distance (km)
Car B 40 30 20 10
Time (hours)

23. Key Concept 5: Stock Assessment involves equations!
11/21/2018
Key Concept 5: Stock Assessment involves equations!
Relevance to stock assessment?
e.g. fish growth rates
Growth rate impacts population size
Natural mortality
Size at maturity
Biomass increase in existing pop
Therefore estimating growth rates is important in predicting population change over time
This is just one example of how rates may be used in stock assessment
Species A
So….
Species A grows at a faster rate than species B 70 60 50
Length (cm)
Species B 40 30 20 10
Time (years)

24. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
Relevance to stock assessment?
Often the rates are not constant as in this example
e.g. fish growth rates (in length) rates slow as the fish get older
e.g. fish survival rates may increase as fish get older and then decrease when very old.
Species A 70 60 50
Species B
Length (cm) 40 30 20 10
Time (years)

25. Key Concept 5: Stock Assessment involves equations!
11/21/2018
Key Concept 5: Stock Assessment involves equations!
What types of equation are used in stock assessment models?
Differential equations – measure rates of change
Difference equations – predict values at fixed point in time

26. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
Difference Equations
Predicting values at fixed points in time.
Example: Two populations, A and B. Population A grows at fish per year. Population B grows at 5000 fish per year.
How do we calculate the population size 4 years into the future?
Rate =
change in y/change in x
But we want to know y!
Species A
20000=y/4
4*20000=y
80000=y
Species B
5000=y/4
4*5000=y
20000=y
Species A
So….
Population A grows at a faster rate than Population B 70 60 50
Species B
Population size (x1000) 40 30 20 10
Time (years)

27. Key Concept 5: Stock Assessment involves equations!
11/21/2018
Key Concept 5: Stock Assessment involves equations!
Equations which estimate a value at a fixed point (e.g. in time) are called difference equations
We already know one very well!
Bt+1=Bt+R+G-M-C
Many stock assessments these days are based on difference equations….they are easier to understand and more intuitively logical
However, each of the components of such an equation may require estimation by another equation, and often these equations can involve the calculation of rates (ie. Use of differential equations)

28. Key Concept 5: Stock Assessment involves equations!
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Key Concept 5: Stock Assessment involves equations!
For example, a basic logistic growth model
We can write it to calculate change in biomass over time:
dB/dt = rB(1-Bt/k)-C
Or we can write it to predict biomass at some time in the future
Bt+1 = Bt + rB(1-Bt/k)-Ct

29. Review of concepts so far…..
11/21/2018
Review of concepts so far…..
Concept 1 – A stock is a fish population that has little or no mixing or interbreeding with other populations.
Concept 2 – Stock assessment is a multistep process that starts with management questions regarding the impact of fishing on the stock, and includes processes involved in data collection, model selection, stock assessment modelling, and subsequent advice to decision makers.
Concept 3 - A stock assessment model provides a simplification of a very complex system (fish and fishery), to help us estimate population changes over time in response to fishing
Concept 4 - Stock size fluctuations can be estimated by accounting for four key processes, additive processes (growth, recruitment) and subtractive processes (fishing mortality, natural mortality) over time.
Concept 5 - The estimation of biomass and the above processes within the model relies on various types of equations, in particular difference (predictions of values at fixed points) and differential (predictions of rates of change) equations.

30. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Key processes in stock assessment modeling include:
Developing a realistic mathematical description (model) of population processes and their interaction with the fishery and,
“Fitting” that model to real (observed) data which indexes changes in population size, structure and movement, to ensure the model can provide…
Realistic estimates of uncertain or unknown parameters within the model, so enabling….
Use of the model in predicting current and future fishing impacts upon the fish population and fishery
“Fitting” is typically achieved via either minimization of the sums of squares of errors or maximum likelihood approaches.

31. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Introduction to model fitting
A model is a mathematical simplification of a real process, which when accurate, will allow us to understand and make predictions regarding the real process.
Unfortunately:
“Counting fish is just like counting trees…except that they are invisible and they move (ref*)”
…..fish populations are not like animals or plants on land which we can visually count and estimate….
…..so how do we know that our exploited fish population model is a good representation of the real population and its interaction with the fishery?

32. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
A simple stock assessment model
Nt+1,a+1 = Nt,ae-(Ma + Ft,a)
Ft,a = qtEtsa
Ct,a = Nt,aFt,awa
Rt = (ASt)/(b+St)
Nt+1,1 = Rt
Bt = ΣNt,awa
St = ΣNt,awaoa
VBt = ΣNt,awasa
Nt+1,a+1 = Number of fish of age+1 at time+1
Ma = natural mortality rate at age a
Fa = fishing mortality rate at age a
q = catchability
E = fishing effort (units)
s = age specific vulnerability to the gear (selectivity of the gear)
Ct,a = Catch at time t and age a
wa = Mean weight at age a << (Growth)
Rt = Recruitment at time t
A = maximum recruitment
b = Stock size when recruitment is half the maximum recruitment
wa = weight at age a
oa = proportion mature at age a
Bt = population biomass at time t
St = spawning stock biomass at time t
VB = vulnerable biomass at time t

33. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Well, while we cant see the fish population or how it changes over time, we do have a good general understanding of the key processes that influence how populations (generally) change over time, from many studies of many organisms (terrestrial and sea) over many years. We can describe the interaction of these processes mathematically to build our basic model framework. For example:
Once we have a basic model which describes the key interacting processes, we are left with the task of collecting the data to inform each parameter in the model, so we can use the model to understand the current/future fishery.
The problem is this: Some parameters we can collect data for (catch, fishing effort, sizes, growth rates, maturity etc) but some parameters we may not be able to collect data for or estimate outside the model.

34. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
This creates significant uncertainty in our models ability to predict changes in population size and structure due to fishing and other processes.
How do we get around this?
Fortunately, it is possible to use the model itself (both its mathematical and statistical components) to estimate the value of the unknown parameters, through the process of “fitting” the model to observed data from the fishery
How does model fitting work? An example….
Collect data which will index changes in the fish populations size over time. Typically this is catch rate or catch per unit effort (CPUE) data. We assume that CPUE is directly proportional to population size, (if CPUE goes up, the population has gotten bigger; if it goes down, it has gotten smaller) and is therefore an accurate index of population change over time.

35. Key Concept 6: Fitting models to data
11/21/2018
Key Concept 6: Fitting models to data
How does model fitting work? An example….
Use a statistical model* (and computer) to search for and find the combination of “uncertain or unknown” parameter values which allow the model to most closely predict the observed CPUE data trend. (Essentially the computer tests across 1000s or 10000s of different possible parameter values until it finds the combination that gives the best “fit” between observed and predicted CPUE)
Why do we use this approach?
Because we believe the CPUE trend is proportional to and accurately reflects population (biomass) trends. So if our model can predict our CPUE trend, and CPUE relates directly to biomass, then it can predict our biomass trend.
Models can also be fit to size data (to ensure the model is realistically predicting population size/age structure) and to tagging data (to ensure the model is realistically predicting fish movements).

36. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Summary of model fitting
We use our knowledge of population processes to build a model of the population that has equations to describe all the processes, how they link together, and how they influence population size over time.
Each equation will be made up of different components (or parameters)
Some of the parameter values we will know already (e.g from biological research, from fisheries catch effort data collection, etc). Some of the parameters will have unknown values.
We use a statistical model (and computer) to go through all the different combinations of possible values for those unknown parameters, until it finds a combination that allows the model to accurately predict the observed CPUE. In other words, produce a CPUE time series that fits or matches (i.e. differs very little from..) the real CPUE time series.
** This description describes some of the core principles only

37. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
There are two main methods by which stock assessment scientists fit models to observation data:
Minimisation of Sums of Squares of Errors
Basically, this approach asks “What combination of values result in there being the smallest difference (degree of error) between the model estimated CPUE series and the real CPUE series?”
Maximum likelihood approach
This approach asks “What combination of values for all of these parameters would most likely result in the observed CPUE values occuring?”

38. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
1. Minimisation of Sums of Squares of Errors
This approach involves a search for the parameter values which minimise the sums of squared differences between the observed data and the data as predicted by the model and parameters.
It is almost impossible in any slightly complex system to create a model that exactly fits the real data….there is always some error. The objective of the SSE approach is to find parameter values that minimise the total error.

39. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Models used to deduce relationship and find best fit
Observed values Y X

40. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Models used to deduce relationship and find best fit
Predicted values
SSE = Sum (Observed-Predicted)2
Observed values
Difference between the observed and predicted value is the “residual error”

41. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Models used to deduce relationship and find best fit 25
Square of the error
= 102 = 100 20
Sums of Squares of the error
= = 200 15
Square of the error
= 102 = 100 10 5

42. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
Models used to deduce relationship and find best fit 25
Square of the error
= 52 = 25 20
Sums of Squares of the error
= = 250 15
Square of the error
= 152 = 225 10 5

43. Key Concept 6: Fitting models to data
11/21/2018
Key Concept 6: Fitting models to data
Models used to deduce relationship and find best fit
Predicted values
SSE = Sum (Observed-Predicted)2
Observed values
Difference between the observed and predicted value is the “residual error”

44. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
2. Maximum Likelihood Method
For this approach, parameters are selected which maximise the probability or likelihood that the observed values (the data) would have occurred given the particular model and the set of parameters selected (the hypothesis being tested)
The set of parameter values which generate the largest likelihood are the maximum likelihood estimates:
So..
Likelihood = P{data|hypothesis}
Which means “the probability of the data (the observed values) given the hypothesis (the model plus the parameter values selected)”.
E.g. Think of the flip of a coin
Whats the probability of getting heads? Of getting tails?
Stock assessment models can use fairly complex mathematics to determine the probability of, for example, the observed CPUE series occuring, given a particular model.

45. Key Concept 6: Fitting models to data
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Key Concept 6: Fitting models to data
SUMMARY
The process of creating a model that is reflective of the real fish population involves three phases:
1. Creating a mathematical model of the system (population and interaction with fishery) using knowledge of basic population and fisheries dynamics.
2. Fixing parameter values for which the values are known (Predetermined through other research perhaps). Where parameters have predetermined values these are called constant or fixed values. In some instances an exact value might not be fixed but a range within which the model is allowed to search for the best value might be specified. This is called setting constraints.
3. Simultaneously fitting the model to the observed data (via a statistical model), with unknown parameters being estimated at the same time to be values that ensure the best fit between model and data. This processes requires that there is some kind of criterion by which to judge the quality of the fit (e.g. SSE or Likelihood).

46. Key Concept 7: Model types and selection
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Key Concept 7: Model types and selection
Concept 7: There are many different types of fish stock assessment model that can be used and selecting an appropriate model is dependant on the management question being asked and the data that is available.
What are the various types of stock assessment model? How do they differ?
Age structure
Fishermen dynamics
Biomass Dynamics Models
Ecosystems and multispecies models
Spatial models
Modified from Hillborn and Walters, 1992
WE ARE GOING TO FOCUS ON AGE STRUCTURED MODELS IN THIS WORKSHOP