Reproduction Issue: Polynomial or Exponential?

[twoflower]

Hello,

let's have this scenario. We have a parameter n and a exponentially large (in n) universe of organisms. We are given a subset of this universe, which is only polynomially large. The only action which the organisms are able to do is that two compatible organisms join together and a new organism is created (the two original organisms remain there as well). The fact is that each organism is compatible with polynomially many other organisms from the universe and this polynom is the same for all the organisms.

The question is: if we let the given organisms reproduce as long as they are able to, may it happen that we end up with exponentially many organisms?

 

[Tac-Tics]

What do you mean by "exponentially large in n", "polynomially large", and "compatible"?

 

[Dragonfall]

What do you mean by "exponentially large in n", "polynomially large", and "compatible"?

http://en.wikipedia.org/wiki/Big-o_notation

 

[Dragonfall]

The question is: if we let the given organisms reproduce as long as they are able to, may it happen that we end up with exponentially many organisms?

You need to re-think your question. Does time flow independently of n?

 

[twoflower]

You need to re-think your question. Does time flow independently of n?

As I'm thinking about it again, now it seems clear to me that yes, we can end up with exponentially many organisms. I'm not sure what you mean for time to flow "independently of n", but we can imagine that time "happens" in discrete moments and only one thing can happen in each moment: two organisms join together and a new one emerge.

 

[Dragonfall]

"Time flows independently of n" means that the total population does not grow. This is as opposed to the total population growing wrt time. If the total population grows exponentially wrt time, then the subpopulation will never catch up.

 

[twoflower]

"Time flows independently of n" means that the total population does not grow. This is as opposed to the total population growing wrt time. If the total population grows exponentially wrt time, then the subpopulation will never catch up.

I see. The universe (total population) is fixed and of the same size the whole time. It's only the subset which is growing.

 

[Dragonfall]

If the total population is fixed, then you can't talk about "exponential growth". And it's obvious that in this case, even if 1 new specimen appears every 500 years, eventually the subpopulation will eclipse the total population.

 

[twoflower]

If the total population is fixed, then you can't talk about "exponential growth".

Why not? I can, indeed, end up with exponentially many organisms (exponentially many in n) - that was my concern.

And it's obvious that in this case, even if 1 new specimen appears every 500 years, eventually the subpopulation will eclipse the total population.

Is it really so obvious even if we take into account the limitation that each organism can reproduce only with polynomial (in n) amount of other organisms?

 

[CRGreathouse]

Is it really so obvious even if we take into account the limitation that each organism can reproduce only with polynomial (in n) amount of other organisms?

Since the original organisms remain, all you need is for tow to find each other... right?