poly: Seeking StarFlight Functional Forms

Sorry about the formatting on the message I just sent. I just switched from Eudora to Microsoft Outlook as my mail handler, because I bought a new computer, and it came with Outlook whether I wanted it or not. Outlook seems to have the property that the message as recieved does not have the same formatting as the message that was sent. When I sent it, all my comments were indented differently to the original text Robin wrote, but at some point this indentation was lost. I think Microsoft is trying to create an Email program that can only communicate correctly with other copies of itself, and then flood the market with that program, so as to drive everyone else out.

I will now try to mark my comments as opposed to Robin's original text. Only God and Bill Gates know what it will look like when you get it.

[Robin]
Hi. I'm working on a paper, tentatively titled "Burning the
Cosmic Commons: Evolutionary Strategies for Interstellar
Colonization," building on Carl's insight that we can derive
colonization behavior from the selection effect of what it
takes to stay on the frontier.
  
[Carl Feynman]
Beat me to it. Oh well, I'll just have to think of another publishable idea. Serves me right for procrastinating.

[Robin]
  
Part of the paper proves some general results independent
of specific functional forms, but I also want to include
computer simulations of specific cases. For such specific
cases I need to pick specific functional forms, which I'd like
your advice on choosing.

The basic functions in the model are these:
R(s) = the resources available after growing at an oasis for
       time s. R(0) = 1. R'(s) > 0
  
[Carl Feynman]
Let me suggest a sigmoid form for this. It's the usual form for growth proportional to the product of capital and virgin resources:

R(s) = r exp(alpha s) / (1 + exp(alpha s))

Here, r is the limiting amount of resources, and alpha is the growth rate. Lots and lots of things grow this way.
  
[Robin]
1/A(x*b(v,h)) = the fraction of probes that survive traveling
       distance x at velocity v, with hardness h. A(0) = 1.
       A'(x) > 0.
  
[Carl Feynman]
How about using Gompertz's law of mortality? The mortality rate as a function of age rises exponentially. The free parameters are the base rate and the doubling time. It's a generic feature of mortality in most species.

I was looking on the web for a closed-form equation for the resulting A, which I couldn't find, when I came across http://ariadne.if.uff.br/~tjpp/racco/racco.html, which as far as I know provides the first mechanistic explanation for why Gomertz's law is so general. I spent some time trying to come up with a model for it a few years ago; it's nice to know someone has done the job.
 
[Robin]
C(v,h) = the resource cost to make one probe with v,h.
 
[Carl Feynman]
There may well be economies of scale in the construction and launching of probes. Many of the good starflight proposals I have seen involve the construction of a reusable launcher.

[Robin]
Q(P) = the fraction of probes which land at open oases, given
       a fraction P of them are occupied.
  
[Carl Feynman]
Wouldn't this depend on where the probe was relative to the frontier?

--CarlF