Daniele Baker and Derek Crane

Developed from Chapter 2 (part 2) of Modeling and Quantitative Methods in Fisheries by Malcolm Haddon

Objectives Why develop age-structured models? Mortality rates (H vs. F) How to determine mortality or fishing

rate? Yield-per-recruit

Determining optimumsModel assumptionsEquations and definitionsTargets and conclusions

Logistic Model Brief stop…𝐵𝑡+1 = 𝐵𝑡 + 𝑟𝐵𝑡൬1− 𝐵𝑡𝐾൰− 𝐶𝑡

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)

0

200

400

600

800

1000

0 4 8 12 16 20 24 28 32 36 40 44

Popu

latio

n Si

ze

Time

Bt, Z=.25

02004006008001000120014001600

0

200

400

600

800

1000

0 4 8 12 16 20 24 28 32 36 40 44

Biom

ass (

kg)

Popu

latio

n Si

ze

Time

Bt, Z=.25Biomass

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

Fecu

ndity

(# o

f egg

s)

Wei

ght +

Leng

th

Age (yrs)

Length (in)Weight (lbs)Fecundity

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

Age-structured Why not apply the same fishing mortality to all fish?

Short lived <1 yearMust pin point the time within the year in

order to catch more and allow for reproduction

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

(yea

rs)

Fish Species

First maturity

50% EPR

0

500

1000

1500

2000

2500

American Shad

Bluefish Striped bass

Winter flounder

Shortnose sturgeonFe

cund

ity(e

ggs i

n th

ousa

nds)

Fish Species

Data from Boreman and Friedland 2003

Annual vs. Instantaneous Compound interest- continuous vs. annual Which collects more interest ($)?

Positive interest 𝐴= 𝑃ቀ1+ 𝑟𝑛ቁ𝑛𝑡

Annual vs. Instantaneous Which has greater annual mortality?

NegativeExponential decay = draining bathtubLarger decrease between .1 + .35 then .5 + .75

0

200

400

600

800

1000

0 4 8 12 16 20

Popu

latio

n Si

ze

Time

Bt, Z=.1

Bt, Z=.25

Bt, Z=.5

Bt, Z = 1

𝐹= −𝐿𝑛ሺ1− 𝐻ሻ 𝑁𝑡+1 = 𝑁𝑡𝑒−𝑍

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1 1.5 2

Annu

al P

erce

nt M

orta

lity

Instantaneous Fishing Mortality F

H

F

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

Age-structured model Equations

Expected outcomesTarget fishing mortality (F)- determines constant

fishing rate harvest strategyTarget age at first capture (Tc)- determines gear

type

𝑁𝑡+1 = 𝑁𝑡𝑒−(𝑀+𝐹𝑖) 𝑁𝑍= 𝑁𝑡 − 𝑁𝑡+1 𝑁𝑍= 𝑁𝑡൫1− 𝑒−ሺ𝑀+𝐹 ሻ൯

Conclusions Limitations

Don’t address sustainability of optimal F. Why?Fo.1 instead of Fmax

OverfishingGrowth-overfishing Recruitment overfishing

Other options. Which is best?Egg-per-recruitDollar-per-recruit

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1 1.5 2

Annu

al P

erce

nt M

orta

lity

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.