1. Stock Assessment Workshop 19th June -25th June 2008 SPC Headquarters Noumea New Caledonia

2. Day 3 Session 1 Parameter Estimation – Natural Mortality and Fishing Mortality

3. Overview What is mortality? What is natural mortality?
How and why does it vary with age and size?
Why do we estimate it in stock assessments
How is M estimated (outside assessment models)
How is M used and estimated inside assessment models
What is fishing mortality and why is it important to assessment models?
How and why does it vary with size and age?
How is F estimated outside and inside models, and in MULTIFAN-CL
How can F estimates be used to provide information to fishery managers
Summary

4. Bt+1=Bt+R+G-M-C Our model = Z (Total Mortality) Death
(Natural mortality - M)
Recruitment
(+)
Whole population
(-)
Catch
(Fishing mortality - F)
= Z (Total Mortality)
(-)
Growth
(+)
Movement

5. Introduction – Age/size structured models
RECRUITMENT BIOMASS +
1 year olds
(+) Growth
(-) Natural mortality
(-) Fishing mortality
Whole population
2 year olds
(+) Growth
(-) Natural mortality
(-) Fishing mortality
3 year olds
(+) Growth
(-) Natural mortality
(-) Fishing mortality

6. What is mortality (Z)?
Simply, the process of mortality (death) of fish
Total mortality = fishing mortality + natural mortality
Z = F + M
Think of it as the removal of fish from a population from all causes
Reduces the number of fish in subsequent age classes
F and M are generally treated separately in stock assessment models, as the implications for management of high F or high M can be very different
F can be managed
M generally cant be controlled.

7. How are mortality estimates incorporated into age based models?
Bt+1=Bt+R+G-M-C
Age/size-specific maturity
Age/size-specific growth
Age/size-specific movement
Age/size-specific natural mortality
Age/size-specific habitat
Age/size-specific fishing mortality
Age/size-specific selectivity
Ba+1,t+1 = Ba,t + Ra,t + Ga,t - Ma,t - Ca,t
Bt =  Bt,axwa

8. Natural Mortality

9. What is natural mortality (M)?
The process of mortality (death) of fish due to natural causes;
predation
disease
senescence
Starvation
other
Would occur with or without fishing

10. How and why does M vary with age and size?
M tends to decrease with age
[Fish ‘out-grow’ predators]
May increase again in older fish
[‘Stress’ associated with reproduction, old age?]
BET
SKJ
YFT

11. How and why does M vary with age and size?
Natural mortality varies throughout the life-cycle of a species
Lower condition factor and reduced capacity to survive periods without food
Size/age – fish may “out-grow” predators (e.g. range of predators of larval v juvenile v adult marlin, plus cannabalism factor)
Senescence processes
Reproductive stresses
Movement away from areas of high mortality
Behavioural changes (e.g. formation of schools)
Changes in ecosystem status (e.g. prey availability, habitat availability)
Changes in abundance (e.g. density-dependence influences, like cannibalism, prey limitations, older fish outcompeting younger fish)

12. Why do we estimate M in stock assessment models?
Natural mortality rates are critical in understanding stock dynamics of fishery species
Bt+1=Bt+R+G-M-C
Allows an understanding of the relative impacts of fishing (e.g. compare natural v fishing mortality rates)
Zt=Mt+Ft
Permits an understanding of the “robustness” of a stock to fishing
Allows an understanding of the impacts of fishing of a stock (overall and by age/size class)

13. Why do we estimate M in stock assessment models?
M has direct and indirect impacts on populations and fisheries which are important to be able to understand and account for within models
Direct
M will affect the number of fish surviving to a given size/age that become available to a fishery
Thus M may influence the abundance of fish available to fisheries

14. Why do we estimate M in stock assessment models?
Indirect impacts
Need to ensure that an adequate number/proportion of each size class survive through to the next age class (and ultimately to contribute to reproduction of a stock)
If M is very high then F may need to be relatively low (as you cannot control M)
M may limit/restrict total fishing mortality rate (F) of an age/size-class or stock if M is extremely high
This could potentially mean that fishing on a certain component of the stock may be restricted

15. How is M estimated? One of the more difficult parameters to estimate
Confounded with the effects of recruitment and fishing mortality
Measuring the ‘disappearance’ of fish that can’t be attributed to other sources - fishing mortality, movements
Often, total mortality (Z) and fishing mortality (F) are estimated first;
Z = F + M, then
M = Z – F

16. How is M estimated (outside the assessment models)?
1. Maximum age (Hoenig)
There is a relationship between the maximum age of a species and total mortality
The higher the estimated maximum age, the lower the mortality must be
ln(Z) = ln tmax
2. Length- based (Beverton and Holt)
Extends the relationship between growth rate (K) and size, incorporating the mean size and smallest size of captured fish
Z = K* [(Linf- Lmean)/(Lmean - Lsmallest in catch)]

17. How is M estimated (outside the assessment models)?
3. Application of the relationship between M and K
Ratio of M:K has been tested and shown to be between 1.5 and 1.6, (with a standard error of 0.58)
[A result of biological ‘trade-offs’ between growth and mortality, due to the influences on reproduction and survival].
Therefore, if you have an estimate of K (from growth), then you also have a starting point for M
e.g. K = 0.4, M = 0.6
[Can incorporate variations between M and K by using the SE to define a prior distribution]

18. How is M estimated (outside the assessment models)?
4. Length frequency analyses (catch-curves for Z)
develop length/age-frequency plots
look for declines in frequency of older age/size classes
Estimate regression parameters for the decline in frequency in older age classes
-1 * slope= mortality rate
in the absence of fishing this would equate to natural mortality [rare for the WCPO tuna species]
in the presence of fishing this would equate to total mortality (Z)

19. How is M estimated (outside the assessment models)?
5. Otolith based studies
Calculations to similar to length-frequency estimates of M except that age classes are used.
Labour intense in order to generate enough ageing data
Usually rely on a (much) smaller sample size than length-frequency analyses.
Used widely on non-fishery species

20. How is M estimated (outside the assessment models)?
6. Tagging studies
Known number of returns of tagged fish (from fishers)
[Estimated return rate from fishers can be included]
Reduction in the number of returns through time
Estimate slope of regression: Z if fished; M if unfished
More tagged fish = higher number of returns = better estimates of mortality
(and other parameters – movement, biomass, growth)

21. How is M estimated (outside the assessment models)?
Many estimates of mortality on fished stocks result in estimates of total mortality (Z)
i.e. Fishing mortality (F) + Natural mortality (M)
Z = M+ F
Need to split into estimates of M and F
Splitting can be done if contrasting effort levels are available
Plot Z estimates (from length-based) against effort
Estimate slope
Intercept on y-axis gives and estimate of M

22. How is M used in stock assessments?
Simply, to remove fish from a stock due to natural sources of mortality
Allows for the “removal” of fish in the model not related to fishing
Nt+1= Nt+ (Rt+ Gt) – (Zt+ Et); Zt=Mt+Ft
‘Removes’ an age/size specific proportion of fish from each age/size class at each time step in age/size structured models (i.e. incorporated into a rate)
Provides for more realistic population dynamics
M can be fixed or size/age dependant

23. How is M incorporated into assessments?
Estimates of M (e.g. from previous studies) can be used and fixed for all age classes
An estimate of M is fixed in the model for all age-classes
At each time step, M-proportion of fish from each age-class are “removed” from further calculations
This allows the model to incorporate total mortality for each age-class at each time-step
[Can also test the sensitivity of the analyses to M by changing the value of M]

24. How is M incorporated into assessments?
2. Age/size specific estimates of M-at-age can be used for each age-class
Mortality estimates for some ages/stages of fish for a species may be available
These are also applied at each time step to the appropriate age-classes of fish, to remove fish from the model
Biomass estimates and other outputs incorporate age-specific M

25. Age-specific estimates of M from MFCL
BET
(SC-s SAWP-2)
YFT
(SC-1 SAWP-1)
SKJ
(SC-1 SAWP-4)

26. How is M incorporated into assessments?
3. M can be estimated by MFCL
Still require a starting value (seed-value) and range (prior distribution) [usually from previous studies]
Seed values are usually available from published literature and/or from previous assessments; constrain possible values by limiting range (priors)
MFCL can calculate average overall M
Age-specific deviations from average M can also be calculated by MFCL, allowing identification of age-classes that may be more or less susceptible to fishing

27. Fishing Mortality

28. What is Fishing mortality (F)?
The process of mortality (death) of fish due to fishing;
Catch
Discard mortalities
Think of it as the removal of fish from a population due to fishing activities only

29. Why is F important to stock assessment models?
Fishing mortality (or catch) is the entire reason you are here today! It is the primary reason that we undertake stock assessments!
We wish to understand the past, present and future probable impacts of fishing upon the fish stocks that we exploit.
With age structured models we go one step further, identifying which components (age classes) within the stock are the most impacted.
In situations where the resource is being overexploited, we can simulate different management options to help the stock to recover, by simulating different fishing mortality rates by different gears on different age classes within the stock.

30. How and why does fishing mortality vary with age and size?
Fishing mortality often varies by size or age class for one main reason….fishing gears tend to be size selective, that is, more likely to catch fish of a certain size and less likely to catch fish of other sizes
For example, small bigeye tuna tend to be caught by purse seine sets on floating objects, but large (adult) bigeye tuna are much less frequently caught. In contrast, adult bigeye are caught on longline, but very small juvenile bigeye are not often caught …**More on Selectivity this afternoon!
YFT (SC-1 SAWP-1)
BET (SC-1 SAWP-2)
F at Age
Proportion at Age
F at Age
Proportion at Age

31. How is F estimated outside and inside the models?
Estimating F is different to estimating M as you have the data  CATCH
Observer data, port sampling data (size data)
Can estimate the proportion of fish in each size/ age class in the catch (selectivity, catchability) and then the impact on each size class

32. …….or as a differential equation…..
How is F estimated?
So where does F fit in stock assessment models?
Schaefer model:
dB/dt = rB(1-B/k)-C
…….or as a differential equation…..
Bt+1 = Bt + rB(1-Bt/k)-Ct
Fishing mortality rate is the proportion of the population killed by fishing each time step (e.g. year, quarter)

33. How is F estimated? dB/dt = rB(1-B/k)-C Schaefer model:
There is an assumption that catch rate C is proportional to biomass and to fishing effort;
C=qEB
Catch rate* = Catchability x Effort x Biomass
(*per unit time)

34. How is F estimated?
Firstly, lets consider what are the main factors that will effect catch?
What happens to catch if we increase the number of hooks (effort)?(E)
What happens to catch if biomass (B) decreases?
What happens to catch if the fish swim deeper? Catchability (q) decreases.

35. How is F estimated?
What happens to catch if we increase the depth of our hooks to target the deep swimming fish?

36. How is F estimated?
We can rearrange this equation to show that CPUE is proportional to biomass (abundance)
C/E = qB
And Catchability is the proportion of the stock caught by one unit fishing effort (e.g. one set, 100 hooks etc)….
q = C/EB
And fishing mortality rate is the proportion of the population removed by fishing over time, (e.g. a year’ a quarter)
F = C/B

37. How is F estimated?
Then using the previous equations, fishing mortality rate will be the product of catchability and fishing effort
F = qE = C/B
Therefore, we can also state a relationship between catch and fishing mortality rate:
C = FB
In age-structured models we calculate F at age, and this requires an additional parameter, Selectivity:
Fa = qEsa

38. How is F estimated?
Fishing mortality rate can be estimated within a model or outside of it
[Can be confounded with the effects of (and variations in) recruitment and natural mortality]
Outside of model
Tagging studies
2. Effort based series (remember back to natural mortality?]:
Plot Z estimates (from length-based data) against effort
Estimate slope
Intercept on y-axis gives an estimate of M. Difference is F

39. F estimation in age structured models
In age/length structured models, F is critical for the estimation of C (catch).
Nt+1,a+1 = Nt,a(1-m) – Ct,a
Ct,a = Nt,aftva
Bt = ∑aNt,awa
Nt+1,1 = (aEt)/(b+Et)
F is critical to estimation of catch, which is required in predicting biomass in the future.
Number of fish in age class a in one years time
(Number of fish of age a now
Survival rate)
Catch of fish age a now - = x
Catch at age being the product of numbers, fishing mortality and age specific vulnerability to gear
Biomass being the sum of products of age specific numbers and mean weights
Recruitment as estimated by the Beverton and Holt SRR

40. Fishing mortality and age distributions
Estimating age specific mortality also yields key information for managers, e.g. ;
which parts of the stock are being fished hardest
in the identification of growth and recruitment overfishing
YFT
(SC-1 SAWP-1)
BET
(SC-1 SAWP-2)

41. F in MULTIFAN-CL
“Catch by age, time period, and fishery is determined by fishing mortality at age, time period and fishery applied to estimated abundance by age and region....”
i.e. C = F x B for each age, time period and fishery
Fishing mortality is a product of;
1. Fishery and time specific effort
2. A fishery specific catchability that can vary with time
3. A fishery and age specific selectivity that does not vary with time.
[Problem: In many fisheries, discarding is not well recorded and recorded retained catch is considered to equate to F]

42. F – BET SC
F adults; F juveniles
Initial F is high for older age classes, due to the predominance of the longline fishery. However the purse seine fishery on floating objects, and particularly drifting FADs since 1995, has led to high F on juvenile age classes also (Note: age classes are quarters)

43. F – BET SC
Impacts of
fishing on
total biomass
x gear

44. Comparing (current) F to
F – BET SC
Comparing (current) F to
F required to achieve
maximum sustainable
yield (MSY)

45. Calculating unfished biomass
MFCL models can be used to estimate biomass that would have occurred in the absence of fishing
i.e. fishing mortality (effort) can be ‘turned-off’
i.e. Z = M + 0  Z = M
Allows the assessment of biomass trajectories in the absence of fishing
Can be used to estimate the reduction in biomass as a result of fishing
 Impacts of fishing

46. F – BET SC
- Z (F + M)
- M only
Impacts of fishing

47. Summary – Natural Mortality (M)
M is a critical variable in describing population dynamics
Likely to vary with size/age of fish
M can be estimated using a variety of techniques.
Critical in producing ‘realistic’ models
Difficult to estimate (confounded with F, R)
As a result, the impacts of different rates of M are often examined in sensitivity analyses

48. Summary – Natural Mortality (M)
Age-structured models (like MFCL) can incorporate M in a variety of ways
Fixed estimate of M
Age-specific estimates of M
Can also estimate M
Incorporated in reference points, relative impacts of fishing, relative impacts of fishing methods etc
Solid biological data is required to provide at least seed estimates
Tagging studies are most likely to produce better estimates of Z, F and M (and other parameters)

49. Summary – Fishing Mortality (F)
Is the whole reason you are here!
F can be estimated within stock assessments and by other methods (e.g. tagging, effort series analyses etc)
In age structured stock assessment models, F is calculated for each time, age and fishery as a function of selectivity, catchability, and fishing effort
F estimation is critical in the calculation and interpretation of biological reference points - Fcurrent /Fmsy.
Estimating F-at-age is also important in the identification and type of overfishing (e.g. growth or recruitment overfishing).
F can be “switched off” within a model to estimate the impacts of fishing