A little summary of Age-structured models for fisheries in particular yield-per-recruit. The slides were developed from part 2 of Chapter 2 in the fantastic book "Modeling and Quantitative Methods in Fisheries" by Malcolm Haddon. Authors: Daniele Baker and Derek Crane

1. Daniele Baker and Derek Crane
Developed from Chapter 2 (part 2) of
Modeling and Quantitative Methods in
Fisheries by Malcolm Haddon

2. Objectives
 Why develop age-structured models?
 Mortality rates (H vs. F)
 How to determine mortality or fishing
rate?
 Yield-per-recruit
Determining optimums
Model assumptions
Equations and definitions
Targets and conclusions

3. Logistic Model
 Brief stop…
��+1 = �� + ��� ൬1 −
��
�
൰− ��

4. Use of age-structured
 Why do you think it’s better to use age-
structured vs. whole-population models?
Growth rate, size, egg-production
http://afrf.org/primer3/ + http://www.fao.org/docrep/W5449E/w5449e06.htm (VERY USEFUL SITES)

5. 0
200
400
600
800
1000
0 4 8 12 16 20 24 28 32 36 40 44
PopulationSize
Time
Bt, Z=.25
0
200
400
600
800
1000
1200
1400
1600
0
200
400
600
800
1000
0 4 8 12 16 20 24 28 32 36 40 44
Biomass(kg)
PopulationSize
Time
Bt, Z=.25
Biomass
Age-structure example
 Length, weight,
fecundity
increase with
time
 Population
decreases with
time
 At some pt.
biomass peaks
0
2000
4000
6000
8000
10000
12000
0
50
100
150
200
250
300
0 4 8 12 16 20 24 28 32 36 40 44
Fecundity(#ofeggs)
Weight+Length
Age (yrs)
Length (in)
Weight(lbs)
Fecundity

6. Age-structure in Forestry
 “From a biological standpoint, trees and shrubs
should not be cut until they have at least grown to
the minimum size required for production
utilization… Trees and shrubs usually should not be
allowed to grow beyond the point of maximum
average annual growth, which is the age of
maximum productivity; foresters call this the
"rotation" age of the forest plantation.”
http://www.fao.org/docrep/T0122E/t0122e09.htm

7. Age-structured
 Why not apply the same
fishing mortality to all fish?
Short lived <1 year
Must pin point the time within the year in
order to catch more and allow for
reproduction

8. Age-structure btw. species
 Species vary in growth rate, fecundity,
age of maturity
 Makes some species very vulnerable
(sturgeon). WHY?
0
5
10
15
20
American
Shad
Bluefish Striped
bass
Winter
flounder
Shortnose
sturgeon
Age(years)
FishSpecies
First maturity
50% EPR
0
500
1000
1500
2000
2500
American
Shad
Bluefish Striped
bass
Winter
flounder
Shortnose
sturgeon
Fecundity(eggsinthousands)
FishSpecies
Data from Boreman and Friedland 2003

9. Annual vs. Instantaneous
 Compound interest- continuous vs. annual
 Which collects more interest ($)?
Positive interest
� = � ቀ1 +
�
�
ቀ
��

10. Annual vs. Instantaneous
 Which has greater annual mortality?
Negative
Exponential decay = draining bathtub
Larger decrease between .1 + .35 then .5 + .75
0
200
400
600
800
1000
0 4 8 12 16 20
PopulationSize
Time
Bt, Z=.1
Bt, Z=.25
Bt, Z=.5
Bt, Z = 1
� = −��ቀ1 − �ቀ��+1 = ���−�
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2
AnnualPercentMortality
Instantaneous Fishing Mortality F
H
F

11. Age-structured model
 Assumptions
○ Age-structure of fish population has attained equilibrium
with respect to mortality (recruitment is constant or one
cohort represents all)
○ r individuals at tr are recruited (tr = minimum age targeted)
○ Once recruited submitted to constant mortality
○ Fish older than tmax are no longer available
○ Minimal immigration/ emigration
○ Fishery reached equilibrium with fishing mortality
○ Natural mortality and growth characteristics are constant
with stock size
○ Use of selective-size actually separates out all fish > Tc
○ Have an accurate estimate of population size and good
records of total commercial catch

12. Age-structured model
 Equations
 Expected outcomes
Target fishing mortality (F)- determines constant
fishing rate harvest strategy
Target age at first capture (Tc)- determines gear
type
��+1 = ���−(�+��)
�� = �� − ��+1
�� = ��൫1 − �−ቀ�+� ቀ
൯

13. Conclusions
 Limitations
Don’t address sustainability of optimal F. Why?
Fo.1 instead of Fmax
 Overfishing
Growth-overfishing
Recruitment overfishing
 Other options. Which is best?
Egg-per-recruit
Dollar-per-recruit

14. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2
AnnualPercentMortality
Instantaneous Fishing Mortality F
H
F
Slight correction to this graph:
The red line plots the relationship of
Annual Mortality (as a FRACTION, not
a percent) to values of F, the
Instantaneous Mortality rate.
The dotted line is a 1:1 line (in other
words, on this line, the value of Y is
the same as that of X). What Haddon
is showing in this diagram is that at low
values of F, the corresponding annual
mortalities are about the same value –
a value of F = 0.1 produces an annual
mortality of 0.1 (i.e., 10% of the
population dies that year).
At higher levels of F, the red line
diverges from the 1:1 line – thus, at F
= 1, the annual mortality is around
0.63 (63%).
Etc.