MODELS FOR VARIABLE RECRUITMENT (continued)
The other model commonly used to relate recruitment strength with the size of the parental
spawning population is a model developed by Beverton and Holt (1957, Section 6), which is on the
Recommended Reading
list.
We will use the same notation as before.
E
S
k
E
0
is the total number of eggs, which then become larvae, then juveniles,
and then adults.
are the number of spawning adults, the parents of E.
is the average number of eggs laid per spawning adult.
is the total number of eggs laid.
E
0
= k·S
The Beverton and Holt stockrecruit model comes from the solution to the following simple
differential equation.
dE
dt
M
E

E
⋅
=
where
M
E
M
α
E
⋅
+
=
α
·E is the densitydependent
rate of natural mortality.
M is the densityindependent
rate of natural mortality.
Here the density dependent term is proportional to the cohort abundance, not to the parental
abundance (as in the Ricker model).
This form of mortality could arise from processes such as
competition for food or other scarce resources.
This differential equation is much more complicated to solve than the one that led to the Ricker
model.
As before, we start by separating
the variables.
dE
dt
M
α
E
⋅
+
( 29

E
⋅
=
dE
M
α
E
⋅
+
( 29
E
⋅
dt

=
But, now what do we do?
We can solve this equation by separating the left hand side, which is the product of two fractions,
into an equivalent sum of two fractions.
This is sometimes described as the method of
partial
fractions
.
The following example illustrates the general method.
We can integrate the sum on the right. The
integral of a sum is the sun of the integrals.
c
X
a

(
) X
b

(
)
⋅
A
X
a

B
X
b

+
=
We need to find values for A and B that satisfy the above equation.
To do so, multiply together the
two terms on the right hand side and then eliminate the denominator.
c
X
a

(
) X
b

(
)
⋅
A X
b

(
)
⋅
B X
a

(
)
⋅
+
X
a

(
) X
b

(
)
⋅
=
c
A X
b

(
)
⋅
B X
a

(
)
⋅
+
=
A X
⋅
B X
⋅
+
A b
⋅

B a
⋅

=
The left hand side of this equation contains no terms in X, which implies that
FW431/531
Copyright 2008 by David B. Sampson
Recruitment2  Page 82
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document0
A X
⋅
B X
⋅
+
=
and
c
A

b
⋅
B a
⋅

=
==>
B
A

=
and
c
A

b
⋅
A a
⋅
+
=
A a
b

(
)
⋅
=
==>
A
c
a
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 David B. Sampson

Click to edit the document details