ai
=
.
ares
=
.
astar
=
.
R
=
.
Wi
=
.
Cd
=
.
Noffspring
=
.
Variables are defined as in class, except that Noffspring is the number of offspring
produced by each females. As you will see, this is a constant, and drops out of the
solution.
This is Fisher's sex allocation problem. We what to show that the solution ares=1/2 is
a CSS.
Wi
=
Noffspring
R
H
1

ai
L
Cd
+
R
H
ai
L
Cs
H
1

ares
Lê
Cd
H
ares
Lê
Cs
Noffspring
H
1

ai
L
R
Cd
+
ai
H
1

ares
L
R
ares Cd
Firstder
=
D
@
Wi, ai
D
Noffspring

R
Cd
+
H
1

ares
L
R
ares Cd
Now we want to determine the effect of ares on the firstderivative in order to evaluate
convergence stability. Note that the first derivative is equal to zero at ares = 1/2.
Also note that the slope is negative at that point. Hence, the solution ares=1/2 is con
vergence stable. That means that if the population is away from ares=1/2, mutants
that are closer to 1/2 will be favored by selection, and the population will converge on
the equilibrium of one half.
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View Full DocumentNoffspring
=
5;
R
=
1;
Cd
=
1
ê
10;
Firstder
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 Fall '10
 Curtis
 Calculus, Derivative, Noffspring

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