Analytical Approach to the Simulations Already Undertaken

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It occurred to me to go about doing the analysis of the simulations, at least in the simplest cases.


Objective

to begin with a distribution (density ) of wasps (e.g. the truncated normal) and see how it evolves throughout future years.

Cicadas predated by wasps.jpg

Strategy

We assume that the cicadas are transformed into wasps as 25% of the mass of the cicada meals. We also assume the "one-if-by-male, two-if-by-female" strategy.

Given a distribution of cicadas. If wasps were free to pick up cicadas at random (and did so -- i.e., were opportunistic), then the (male) wasp distribution would be a scaled version of the cicada distribution. The wasps have a load limit, however, of anywhere from 2 to approximately 2.5 times their mass.

Hence, the average cicada that the wasps will collect will be

A little Fubini later, we have

The thing that I'm focused on in the above is the quantity

This is (almost a density) for the cicada of size y: it tells the relative number of cicadas of size y taken (but not the relative proportion, since it's not properly normalized). to normalize it, we simply rewrite it as

If those cicadas were turned directly into males (i.e. via a 1-if-by strategy), then what size would the wasps be? Why y/4, of course. So the relative proportion of each size cicada taken will give us a picture of the cicadas formed.

What's described above is a single iteration. We now do this over and over, which I do via the lisp code analytic_loop.lsp, to produce results such as these (for heft=2.2, and relative cicada populations of .1, .45, and .45 from smallest to largest):

Convergence of the wasp distribution from this cicada distribution.png

Cicada density

Convergence of the wasp distribution over generations.png

Ten generations go by....

Convergence of the wasp distribution from above.png

Final relatively stable equilibrium distribution.

Convergence of the wasp distribution.png

Wherever we start, we end up in the same place (there is a stable equilibrium distribution).