# Agenda

Is this flight okay?

## Action Items

• Action items from our twelfth meeting:
• Grayson:
• Write up the Tale of the Decimated Female Cicada Killers!
• Katie:
• Create the mass/rwl non-linear model for the Cicadas, checking for species-specific models
• Continue to work on the histogram, using moving-window or kernel-smoothers.
• Register for the Nebraska Conference for Undergraduate Women in Mathematics -- you didn't actually agree to this, but I'm suggesting you do!;)
• We all agreed to start including our analyses on the "manuscript" we're writing (An article in response to Grant, including data), as we move toward a "final product".

## Something interesting

Some interesting linear regression insights, from Edward Tufte (and F. J. Anscombe -- The American Statistician, Vol. 27, No. 1. (Feb., 1973), pp. 17-21.):

 Anscombe's Quartet Anscombe's Quartet shows four extremely different data sets, all with the same best linear model (linear regression model).

• Further discussion of Peter Grant's article as we work towards An article in response to Grant
• Introduction
• Fisher's Equal Investment model: "In the absence of local mate competition the expected sex ratio of sexually dimorphic species is the ratio produced when maternal investment in the two sexes is equal( Fisher, 1930; Trivers & Willard, 1973; Myers, 1978 ). As female wasps are larger than males, males should be produced in greater numbers than females."
• "As female wasps are larger than males, males should be produced in greater numbers than females." [As Jon is fond of noting, however, the males may not live as long, as they are constantly fighting and causing trouble -- typical males!;). This should be factored into any census, of course.]
• "Therefore, to produce an excess of male offspring, a female could prey selectively on small prey. At variance with this expectation, however, wasps are believed to hunt selectively for large, and therefore usually female, prey and lay female eggs on them, because female prey outnumber male prey ( Lin, 1979b ). This would tend to yield a female-biased sex ratio."
• Three, non-exclusive, hypotheses of selective predation are tested.
• The first hypothesis is that wasps prey preferentially on one sex of their prey, the more valuable sex ( Trivers & Hare, 1976; Charnov, 1982 ), without specifying a priori whether this is male or female; Lin’s (1979b) argument implicitly assumes it is female.
• We can test this: we would want to include dates, however, because we may be seeing a preponderance of females simply because the males came out earlier.
• The second hypothesis is that wasps choose prey on the basis of the size of their prey (e.g. Kobayashi & Shimada, 2000 ).
• This is basically our hypothesis, so far, only we add in the component of transition to opportunistic as "unsuccessful hunt time" increases. Grayson has provided some initial examples of such a transition on his page.
• The third hypothesis is that wasps prey preferentially on larger, more conspicuous and hence easier to detect species of cicadas.
• Opportunistic predation, resulting in prey taken at random with respect to their traits, is the default hypothesis to account for data if they are not explained by selective predation.
• Determining whether predation is selective or opportunistic requires estimating prey captured in relation to prey availability. This is difficult to achieve by direct observation because the attacks usually take place at the tops of tall trees.
• This is not true. We don't need an accurate census to prove certain kinds of selectivity. For example, if field studies show that there is a super-abundance of small cicadas, but large wasps never take them, then we're sure that the cicada-killers are selective -- even though we don't know the actual abundances of the prey available.
• As a further extension, weights of individual wasps and of the cicadas they carried were used to test for a positive relationship between the two ( Lin & Michener, 1972 ), as this has so far not been done. [Andy's emphasis]
• We can test this: Strategy: paired t-test
• "...on average [cicada killers'] prey burden is almost twice as heavy as the wasps ( Coelho, 1997 )."
• We can test this: only we have rwl, rather than mass. So we need to model cicada mass based on rwl, and then do a conversion on all pairs.

Let's talk about Sex Allocation in Solitary Bees and Wasps (Vol. 146, No. 2. The American Naturalist. August 1995, and cited in Peter Grant's article). Frank has other papers available at his website, http://stevefrank.org/reprints-pdf/, including this paper which provides background on the one we're looking at.

"I develop a simple theory that predicts an increasing F/M allocation ratio with increasing F/M size ratio."

Why is it important?

• It gets to the question of why -- or whether -- wasps use a "one if by male, two if by female" strategy for allocating their eggs (or even a "four if by male, seven if by female" strategy!); or how they might behave under resource limitation (all males?).
• It helps us to understand "the psychology of the wasp".

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

${\displaystyle F(\lambda )=\int _{0}^{\lambda }f(y)h(y)dy\int _{\lambda }^{1}g(y)h(y)dy}$

or

${\displaystyle E(\lambda )=\ln(F(\lambda ))=\ln \left(\int _{0}^{\lambda }f(y)h(y)dy\right)+\ln \left(\int _{\lambda }^{1}g(y)h(y)dy\right)}$

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

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

${\displaystyle {\frac {f(\lambda )}{\int _{0}^{\lambda }f(y)h(y)dy}}={\frac {g(\lambda )}{\int _{\lambda }^{1}g(y)h(y)dy}}}$

which we can then solve for ${\displaystyle \lambda }$, given the functions f, g, and h.

${\displaystyle F(\lambda )}$ can be normalized without changing the maximizing value of ${\displaystyle \lambda }$: hence we could use instead

${\displaystyle F(\lambda )={\frac {\int _{0}^{\lambda }f(y)h(y)dy\int _{\lambda }^{1}g(y)h(y)dy}{\int _{0}^{1}f(y)h(y)dy\int _{0}^{1}g(y)h(y)dy}}=\left({\frac {\int _{0}^{\lambda }f(y)h(y)dy}{f_{T}}}\right)\left({\frac {\int _{\lambda }^{1}g(y)h(y)dy}{g_{T}}}\right)}$;

hence we can think of ${\displaystyle F(\lambda )}$ as the product of the fraction of male value achieved by ${\displaystyle \lambda }$ and the fraction of female value achieved from ${\displaystyle \lambda }$ on out to the total resource allocation.