# Sex Allocation in Solitary Bees and Wasps

### From www.norsemathology.org

## The Objective Functions to Minimize

To get the equation that Frank cites on page 317, we can minimize either of the following functions:

or

*F* is probably the more intuitive function: we can see that there's a trade-off point, λ, at which we switch from producing males to producing females. If we think of the two function *f*(*x*) and *g*(*x*) as "value" functions, with over the interval , and *h*(*x*) 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 λ = 0 or λ = 1, *F*(0) = *F*(1) = 0; because , we know that there is a maximum on the interior of the interval [0,1].

If we differentiate either expression, we will ultimately obtain the equation that Frank cites in his text:

which we can then solve for λ, given the functions *f, g, and h*.

*F*(λ) can be normalized without changing the maximizing value of λ: hence we could use instead

;

hence we can think of *F*(λ) as the product of the fraction of male value achieved by λ and the fraction of female value achieved from λ on out to the total resource allocation.

## The Minimization with Specific *f, g,* and *h*

Assume that *f*(*x*) = *x*^{r}, *g*(*x*) = *x*, and *h*(*x*) = κ*x*^{a}(1 - *x*)^{a}.

We expand

or

Given our choices of *f* and *g*, we want to evaluate

and

Now the equation that we're to solve for λ can be written λ^{r}*I*_{g} = λ*I*_{f} which we can reduce by dividing by λ^{r} (since λ = 0 is not a maximum); thus our equation becomes *I*_{g} - λ^{1 - r}*I*_{f} = 0, or

or

or

Here's a little script that shows some of those functions *P*.