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TOTAL EQUILIBRIUM YIELD
Recall that the yieldperrecruit model enabled us to examine the problem of growth overfishing
but, because we assumed that recruitment was constant, the model did not tell us whether a given
level of fishing might result in
recruitment overfishing
.
Now we will combine Beverton and Holt's
model for yieldperrecruit with their model for the stockrecruit relationship and build a model for
longterm equilibrium yield, which will let us examine the problem of recruitment overfishing.
Although this model was originally developed and described in Beverton and Holt (1957), it has
received relatively little attention from fisheries scientists.
Combining YieldperRecruit with a StockRecruit Model
For many fish species it is at least approximately true that the number of eggs laid per mature
female is directly proportional to the weight of the female.
( No. Eggs Laid Per Female ) = k · ( Female Body Weight )
If this relationship is valid, then the total number of eggs laid during the lifetime of a cohort of fish is
given by
E
k
t
R
∞
u
N
fem
u
( ) W
fem
u
( )
⋅
⌠
⌡
d
⋅
=
where
E
is the total number of eggs laid;
k
is the number of eggs laid per female body weight;
t
R
is the age at first reproduction;
N
fem
is the number of females at age;
W
fem
is the female weight at age.
If the amount of eggs as a proportion of bocy mass (k) is not constant with age
then parameter k cannot be brought from under the integral sign.
The formulation above assumes that egg production is a continuous process, but for many fish
species spawning is highly seasonal.
We will ignore this complication.
Also, we will assume that the sex ratio is constant and that both sexes have the same weight at age
so that we do not need separate integrals for males versus females.
Given these assumptions it
follows that
E
k'
t
R
∞
u
N u
( ) W u
( )
⋅
⌠
⌡
d
⋅
=
k'
t
R
∞
u
B u
( )
⌠
⌡
d
⋅
=
This equation uses
k' rather than k
because both sexes are included.
The cohort's egg production is proportional to its
spawning stock biomass
(SSB).
Our simplified
problem is to model the spawning stock biomass and then relate that to the subsequent production
of offspring and recruits to the adult population.
For simplicity let's assume that the age at recruitment t
r
is less than the age at first spawning t
R
,
and that the maximum exploitable age t
λ
is infinite.
We must consider two cases:
FW431/531
Copyright 2008 by David B. Sampson
EquilYield1  Page 107
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View Full Document M 
 M+F 
(a)
t
e
t
R
≤
——————————————————— Age, t
t
r
t
e
t
R
 M 
 M+F 
(b)
t
R
t
e
≤
——————————————————— Age, t
t
r
t
R
t
e
For t
e
≤
t
R
, the spawning stock biomass is
SSB
N t
R
( 29
t
R
∞
u
e
Z

u t
R

( 29
⋅
W u
( )
⋅
⌠
⌡
d
⋅
=
SSB
R e
M

t
e
t
r

( 29
⋅
⋅
e
Z

t
R
t
e

( 29
⋅
⋅
t
R
∞
u
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 Fall '09

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