The Objective Functions to Minimize
To get the equation that Frank cites on page 317, we can minimize either of the following functions:
F is probably the more intuitive function: we can see that there's a trade-off point, , at which we switch from producing males to producing females. If we think of the two function and as "value" functions, with over the interval , and as the probability density of the resources distribution (taken as a beta density by Frank), then we're saying that we want to maximize the product of the value of the males along with the value of the females. If we take or , ; because , we know that there is a maximum on the interior of the interval .
If we differentiate either expression, we will ultimately obtain the equation that Frank cites in his text:
which we can then solve for , given the functions f, g, and h.
can be normalized without changing the maximizing value of : hence we could use instead
hence we can think of as the product of the fraction of male value achieved by and the fraction of female value achieved from on out to the total resource allocation.
The Minimization with Specific f, g, and h
Assume that , , and .
Given our choices of f and g, we want to evaluate
Now the equation that we're to solve for can be written which we can reduce by dividing by (since is not a maximum); thus our equation becomes , or
Here's a little script that shows some of those functions P.