Biostatistics: Pollen dispersal directionality

Question

What Information am I looking for?

Think about a tree that is sending pollen all over the place. Because of wind, most pollen grain will go toward one direction. Imagine, we split the 2D area around the tree where pollen grains fall into two half disks of equal size. We chose the disks so that the number of pollen grains falling into one half-disk is minimized and the quantity of pollen falling in the other half-disk is maximized.

The information I need is what proportion of pollen grain falls into each disk? Is it $\frac{0.5}{0.5}$ (in which case the wind would have no effect) or is it something like $\frac{0.8}{0.2}$?

Where to get the information from?

I was reading this paper about pollen dispersal directionality and was trying to extract the info I need.

On pages 4 and 5 they explain their analysis under the section statistical procedure. More specifically, in the first paragraph of the 5th page, they seem to describe the meaning of the parameters that are trying to estimate. One of them is the so-called directionality parameter $\delta$. I don't understand how to interpret this parameter $\delta$. This parameter is part of a logistic regression I think (although the authors do not characterize it as such) of "mating success" $y$ against variables $d$ ("distance") and $h$ ("height") and an angular variable $a = \cos(\alpha_0 - \alpha)$. ($\alpha_0$ is the "presumed prevailing direction of effective pollen dispersal," which apparently is not estimated from these data.) The corresponding parameters of the model are $\beta$, $\gamma$, and $\delta$, respectively, hence

$$\phi_j = \Pr(y_j = 1) = \frac{\exp\left(\beta d_j + \gamma h_j + \delta a_j\right)}{\sum_{k=1}^r \exp\left(\beta d_k + \gamma h_k + \delta a_k\right)}$$

where the index $j$ ranges over all $r$ "male(s) in the neighborhood."

Their Results

Using maximum likelihood, they found that the directionality parameter, $\hat\delta = 0.56$ (SE = $ 0.15$; bottom of page 6).

Question

How can I relate their finding to the information I need (even if it is just a rough approximation)?

I think I have to understand the so-called directionality paramter $\delta$ Can you help me to interpret the meaning of this parameter $\delta$?

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Reference:

Burczyk, Adams, & Shimizu, Mating patterns and pollen dispersal in a natural knobcone pine ... stand. Heredity 77 (1996) pp 251-260.

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Not receiving any answer on Biology.SE, I asked the same question here on Mathematics.SE but sadly, the question doesn't receive any attention!

Answer 1

(This isn't an answer, but hopefully it will help get it past the experimental design into just solving the equation.)

Where did you get that α0 was not determined from their data? On p. 10 (256), they state, "The prevailing direction of effective pollen dispersal within neighbourhoods (a0) that gave the best fit of the model was 91 degrees from north (nearly due east)."

The best-fit δ parameter value of 0.56 (which determines the size of the effect of the angle between the male tree and the wind) is also determined by their model (table 1, model 9) based on their sample set.

This math is too much for me to solve, but here is how I would go about getting through the experimental setup to just solving the math equation:

Since you're only looking for the effect of wind/directionality, you would set the tree height and distance between trees to be equal. So βd and γh would remain constant and cancel out when you solve for a.

Then you want to determine what angles would give you the highest and lowest pollen counts. Set a0 to wind direction and aj to the angle between a0 and the half disk. The highest pollen count would happen at aj = a0. The lowest pollen counts would happen at aj = (a0-180) or (a0-π or whatever unit of angle is appropriate. Stupid trig.).

With those settings, it should be possible to determine the values of ϕ where aj = a0 and where aj = a0-180.

Then you can get the proportion of pollen grain falls into each disk by finding the ratio of ϕj(a0-180) / ϕj(a0).

Assumptions: 1) There's a linear correlation between ϕ, "the mating success (i.e. fertility) of each outcross male in the neighbourhood" and the distribution of pollen grain dispersal.

2) That a0 ("The prevailing direction of effective pollen dispersal within neighbourhoods") is entirely due to wind. Here I could see terrain being a factor, which would make the effect of wind xδ, such that x<1. If this were the case, the calculated ratio of high pollen / low pollen would be higher than actual.