# Talk:Population dynamics/Archive 1

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## Correcting the model, again

Found some interesting result while working on the refined model, M(t) reach maximum at t= [lnC - ln(PD+PB)]/[F+BP-KP] with Mmax approximately equals c* (tmax - 1/(F+BP-KP)). Checked with the simulation and got 306 and 25337, which appears to be correct. Prima 18:10, 19 March 2011 (UTC)

I see. Good to see you're doing well; then you'll be able to find out why the equilibrium in this graph is 8e3 :)
It's a graph of the refined model, when K>F+B and the soul gem dims over time (I know, I just said I wouldn't do it, but whatever ...) --Homerun-chan 18:35, 19 March 2011 (UTC)

This graph makes absolutely no sense to me. lul Prima 18:39, 19 March 2011 (UTC)
Any chance you can provide a data dump of this on pastebin or somewhere. I can't image how W(t) managed to grow. Prima 18:49, 19 March 2011 (UTC)
Sure, what data do you want? I can't realistically print the values for the 3000 iterations, so tell me what values you want. In the meantime, here are what seems to be the values M(t) and W(t) converge to: M(3000) = 8090.3177; W(3000) = 189.8691
And here's the script: [1] --Homerun-chan 20:23, 19 March 2011 (UTC)
Also, don't forget that this is the model where soul gems dim over time. So the reason why W(t) could grow is simply that, at first, there were no witches, so no way for MSes to clean their soul gems, so after a while some of them turn into witches. That's all. --Homerun-chan 20:34, 19 March 2011 (UTC)

## Magical Girls to Witches over time

While it is not explicitly stated in the show that a Soul Gem gradually dims over time, I believe that it is implied to be the case. In episode six, we see proof of a constantly active link, of limited range, between a Soul Gem and the body it is linked to. I think it is reasonable to accept that this link would be magical in nature, and therefore provide a slight but constant drain on the Soul Gem's power. This would also allow Kyubei to start the cycle without needing to import witches from elsewhere, or use abilities not shown in canon to directly create a Witch. Jorlem 07:11, 20 March 2011 (UTC)

I believe this is a valid modification to the simple model. Perhaps we need to refactor the second portion of the page into Refinements to the simple model and allow these various refinements their own sections. Prima 07:30, 20 March 2011 (UTC)
In that case, I propose we make a third section ("Third Model", "Time-based Model" or something) with this new version of the model and explanations on the hypothesis (basically, whatJorlem said). I'd gladly make it now, but I'm kinda in a hurry. Plus, I think we should first find a rational conclusion on the graph I posted above, before moving it to the main page; posting a graph and saying "I don't understand it lol" isn't very professional --Homerun-chan 12:36, 20 March 2011 (UTC)
Below is how I plan to refactor this page. Thoughts? Prima 14:22, 20 March 2011 (UTC)

## Reasoning for removing P from the model

(besides that I am lazy) P is a constant factor that multiples only B, K, and D in every proposed model. Since the actual values of B, K, and D are arbitrary and indetermine to begin with, it does not in fact matter whether B, K, and D are modified by P or not for the purpose of analysis. If we define B',K', D' where B*P=B', etc, we achieve a functionally equivalent set of equation.

One caveat from this simplication means that the values used for B, K, and D may not be equivalent across the different proposed models. For the existing matlab simulations, I multiplied in the value of P into values used for B, K, D to avoid confusion. The outcome of the simulation should remain the same. Prima 09:22, 21 March 2011 (UTC)

Actually, I don't really see why you're so eager to remove P from the equations. Of course, "mathematically" speaking, they remains the same. The main advantage of keeping P as an external factor is that it is easier to read: the values for B, D and K can stay the same across the models, and in one glance you can see that "ah yeah in this model only 80% of girls are fighting", which you do not easily see when absorbing P into B, D and K. --Homerun-chan 16:14, 22 March 2011 (UTC)
It's one less variable to deal with, also in some ways simplify attempts to tackle closed form solutions and hand analysis of the refine models (in special case scenarios). Prima 16:45, 22 March 2011 (UTC)