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Age-structured models
Fish 458, Lecture 3
Why age-structured models?
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Advantages:
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Populations have age-structure!
More realistic - many basic population
processes (birth rate, death rate, growth,
movement) are age-specific.
Much of the data we collect are structured
by age.
Easily to build in actual processes and
highly flexible.
Why age-structured models?
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Disadvantages:
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Increased complexity.
The data needed to apply age-structured models
are often not available.
Some of the questions being addressed do not
require information on age-structure.
Age-lumped models often perform as well as agestructured models.
Still not that realistic (no predation, competition,
size and spatial structure)!
Some symbols
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Nt ,a
number of animals of age a at time t
ut
fraction harvested during time-step t
va
vulnerability of animals of age a to exploitation
x
oldest age considered
sa
survival of animals of age a from natural mortality
Et
egg production at time t
fa
egg production for animals of age a
g
offspring as a function of total egg production
Ct
catch during time-step t (biomass or numbers)
wa
mass of an animal of age a
State variables, Forcing
Functions and Parameters
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State variables:
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Numbers-at-age
Fraction harvested
Spawning biomass
Forcing function: catch
Parameters:
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Natural mortality, egg production-at-age, mass-atage, vulnerability-at-age, survival-at-age, oldest
age
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The Basic Age-Structured Model
 g ( Et 1 )
if a  0

Nt 1,a   Nt ,a 1 (1  ut va 1 ) sa 1
if 1  a  x
 N (1  u v ) s  N (1  u v ) s if a  x
t x 1 x 1
t,x
t x x
 t , x 1
Ct   ut va Nt ,a wa
a
Plus-group age
The Stock-Recruitment Relationship
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The function g determines the number of
offspring (age 0) as a function of the egg
production. Typical examples:
N t ,0   Et e   Et
Ricker
N t ,0   Et /(   Et )
Beverton-Holt
Note that this model has no stochastic
components, i.e. it is a deterministic model
(sometimes called an “age-structured
production model”).
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Some Assumptions of this Model
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The fishing occurs at the start of the year.
No immigration and emigration.
Fecundity, natural mortality, mass and
vulnerability don’t change over time.
Vulnerability and mass don’t change with
fishing pressure (i.e. no density-dependence
in these parameters).
Age x is chosen so that fecundity, natural
mortality, mass and vulnerability are the same
for all ages above age x.
Vulnerability, Selectivity and Availability
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Conventional definitions:
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Selectivity: The probability of catching an
individual of a given age scaled to the maximum
probability over all ages, given that all animals are
available to be caught.
Availability: The relative probability, as a function
of age, of being in the area in which catching
occurs.
Vulnerability: The combination of selectivity and
availability.
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The Basic Model Again-I
N y1 ,0
N y1 ,1
.
N y1 , x
E ( y1 )
N y1 1,0
N y1 1,1
.
N y1 1, x
E ( y11 )
.
.
.
.
.
N n 1,0
N n 1,1
N n 1, x
E (n  1)
N n ,0
N n ,1
N n, x
E (n)
The Basic Model Again-II
(The steps in setting up a model)
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1.
2.
3.
4.
5.
6.
Specify the initial (year y1) age-structure.
Set yc=y1.
Calculate the mortality (fishing and natural) during
year yc.
Project ahead and hence compute the numbers-atage for animals aged 1 and older at the start of year
yc+1.
Compute the egg production at the start of year
yc+1 and hence the number of 0-year-olds at the
start of year yc+1.
Increase yc by 1 and go to step 3.
Building Age-Structured Models
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Be careful of timing. In the previous model:
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These are not the only possible assumptions.
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Spawning: start of the year
Natural mortality: throughout the year
Exploitation: start of the year
Growth: instantaneous at the start of year
Southern hemisphere krill – no growth in winter!
The results may be sensitive to when
population dynamic processes occur
(especially if survival is low).
An Alternative Model
(northern cod-like)
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 Rt 1,2

Nt 1,a   Nt ,a 1 (1  ut vt ,a 1 )s
 N (1  u v )s  N (1  u v )s
t t ,x
t , x 1
t t , x 1
 t,x
Ct   ut vt ,a Nt ,a s wt ,a

a
if a  2
if 3  a  x
if a  x
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Assumptions of the alternative model
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The fishery occurs a fraction  after the start
of the year.
Vulnerability is age and time-dependent.
Natural survival is independent of age.
Only animals aged 2 and older are considered
in the model.
No stock-recruitment relationship, i.e. this is a
stochastic model.
What about a population in
equilibrium??
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Equilibrium implies:
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Constant recruitment: N t ,0  N 0  R
Time-invariant exploitation rate: ut  u
For the basic model therefore:

R

N a (u )   N a 1 (u )(1  u va 1 ) sa 1

(1  u vx 1 ) sx 1
 N x 1 (u )
1  (1  u vx ) sx

if a  0
if 1  a  x  1
if a  x
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Calculating the plus-group
Let: S a  (1  u va ) sa
Now: N x  N x 1 S x 1  N x 1 S x 1 S x  N x 1 S x 1 ( S x ) 2  ...
i.e.: N x  N x 1 S x 1 (1  S x  ( S x ) 2 )
1
Now :1  z  z  ... 
1 z
S x 1
So that : N x  N x 1
1  Sx
2
Building an age-structured
model-I
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There are two fisheries with different
vulnerabilities.
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One fishery operates from January-June and the
other from July-December.
Animals younger than 5 are discarded (dead) by
fishery 1.
Recruitment (age 0) is relate to egg
production according to a stochastic Ricker
stock-recruitment relationship.
Survival is independent of age.
The Equations
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N t 1,a
 Et 1 e   Et 1 e t

  N t ,a 1 (1  ut1v1a 1 )(1  ut2va21 ) s

1 1
2 2
1 1
2 2
 Nt , x (1  ut vx )(1  ut vx ) s  Nt , x 1 (1  ut vx 1 )(1  ut vx 1 ) s
if a  0
if 1  a  x
if a  x
Ct1, Land   wa ut1 v1a N t ,a s 0.25
a 5
Ct2, Land   wa ut2 va2 N t ,a s 0.75 [1  v1a ut1 ]
a
Note: This model implicitly ‘discards’ the catch of animals
younger than 5 by not including then in the landed catch.
Readings
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Burgeman et al. (1994); Chapter 4
Haddon (2001); Chapter 2
Au and Smith (1997). Can. J. Fish. Aquat. Sci. 54: 415-420.