YIELDPERRECRUIT (continued)
The yieldperrecruit model applies to a cohort, but we saw in the Age Distributions lecture that the
properties of a cohort do not apply in general to a collection of cohorts, which is what we usually
encounter in practice.
Now we will explore some consequences for the yieldperrecruit model of
the simplifying assumption that the population is in equilibrium (constant survival, constant
recruitment, constant growth).
YieldperRecruit from Multiple Cohorts
The Beverton and Holt model for yieldperrecruit is based on equations for mortality and growth of
individuals within a single ageclass.
To derive the model we assumed that F and N(t
e
) were both
constant and we brought them from under the integral.
Y
t
e
t
λ
u
F N t
e
(
29
⋅
e
Z

u t
e

(
29
⋅
⋅
W u
(
)
⋅
⌠
⌡
d
=
This step is invalid unless F and N(t
e
)
are both constant through time.
==>
Y
F N t
e
(
29
⋅
t
e
t
λ
u
e
Z

u t
e

(
29
⋅
W u
(
)
⋅
⌠
⌡
d
⋅
=
If the fish population is in equilibrium (with constant annual recruitment, constant rates of natural
and fishing mortality, and constant growth parameters), then the yieldperrecruit model we derived
is also valid for the annual yieldperrecruit from the entire population.
There is a proof of this in
Beverton and Holt (1957) on pages 3738.
If the population is not in equilibrium, then the
yieldperrecruit model is only an approximation.
Recall the artificial age distributions that I constructed in a previous class.
Year ==>
Age
0
1
2
3
'95
100
'96
100
50
'97
100
52
26
'98
100
54
28
14
'99
100
56
30
15
If there are changes in annual
survival (or recruitment), the
age distribution changes from
year to year.
barb2se
barb2se
barb2se
barb2se
barb2se
barb2se
barb2se
barb2se
barb2se
S =
50%
52%
54%
56%
When there are changes in the rate of fishing mortality F or changes in the ageatentry t
e
, the
population will be out of equilibrium during a transition period, and the yieldperrecruit during the
transition will not
be the same as the equilibrium yieldperrecruit.
Below is an artificial population that illustrates the problem.
R'
y
denotes the number of fish entering
the fishery in year y, the recruitment.
Cohorts occur along each diagonal.
The equations that
describe survival for a cohort are reasonably simple because the recruitment term is a common
factor for each age class.
However, the pseudocohorts, which are the collection of cohorts
present in any given year, have complicated survival equations because there is no common factor
for recruitment.
FW431/531
Copyright 2008 by David B. Sampson
YieldPerRec4  Page 69
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Year
==>
Age
0
1
2
3
t
e
R'
0
R'
1
R'
2
R'
3
barb2se
barb2se
barb2se
t
e
1
+
R'
0
e
M

F
0

⋅
R'
1
e
M

F
1

⋅
R'
2
e
M

F
2

⋅
barb2se
barb2se
t
e
2
+
R'
0
e
2

M
⋅
F
0

F
1

⋅
R'
1
e
2

M
⋅
F
1

F
2

⋅
barb2se
t
e
3
+
R'
0
e
3

M
⋅
F
0

F
1

F
2

⋅
...
...
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 Fall '09
 Demography, Age class structure, Fish mortality, David B. Sampson, yieldperrecruit

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