Talk:Population dynamics

refined model analysis

Based off one of my courses, I made a quick analysis of the refined model in terms of convergence and equilibrium. The theorems used are [citation needed] though since they're directly taken from said course's notes (and I only have references for the whole course). Plus I'm not sure the vocabulary is correct in English (translated from french). Please correct it if necessary.

Here goes:

Let's rename the variables for easier notation. x1 will be M(t) and x2 will be W(t). We can generalize the system as follows:

Where a_ij is the contribution of population j to population i (so a12 is the effect witches have on magical girls, etc)

We can write this system using matrix notation:

Where A and C are matrices containing the coefficients. To be exact, let A be

And let C be

The matrix C is constant and hence does not influence the results on stability/equilibrium.

We can prove that (first citation needed), it A's determinant is different than zero, then the only equilibrium for both M(t) and W(t) is when M(0) = 0 and W(0) = 0. (see observations on the page). The determinant of the matrix A is

Let us now discuss the stability, and other equilibria when det A yelds zero (i.e. B=K or D=-B).

Equilibrium

For the further analyses, we ignore Kyubey's contribution (i.e. the matrix C). Basically, adding it would transform an equilibrium into a linear function, and so on.

If det A = 0 (i.e. if (B=K or D=-B), then the only equilibrium is at M(0)=0 and W(0)=0 (no witch, no MG => they can't exist by themselves). Otherwise, [second citation needed], all the equilibria are the states so that

This equation could only be satisfied if D=-B. This is inconsistent with the model (neither the amoung of girls dying nor the amount of girls becoming witches could be negative), so we'll discard this result. Hence, the only equilibrium would be M(0)=0, W(0)=0 when K=B

Stability

The system is always unstable (i.e. inserting a small amount of witches from the "ideal situation" [ W(t) = 0 ] will result in the Earth being taken over), unless K>B; in that particular case, the system is asymptotically stable (i.e. introducing a small number of witches will result in no witches after a certain amount of time). I can provide a partial mathematical demonstration of that, but it's quite long and it uses eigenvalues and complex numbers (and, as usual, it lacks citations for the theorems used). Do you want me to post it, or do we go by rational logic ("if B>K, the number of witches appearing is higher than the number of witches dying, so introducing a small amount of witches will result in exponentionnal growth. Otherwise, every girl kills more witches than what appears, so introducing witches from a state of equilibrium will result in the extinction of said witches over time")

A third option would of course be for someone to find a closed form of the system, so that the stability can be easily found by taking t going to infinity. --Homerun-chan 23:08, 19 March 2011 (UTC)

Unrelated

On a totally unrelated note, >mfw more than 350 tweets about this page in the past hour, and 20-50 more every minute...

Matrix calculus is not my strong point, so I will need some time to figure out your analysis. In other words, you're right about the tweets: http://mb.tweetbuzz.jp/entry/36044431 . Prima 13:13, 19 March 2011 (UTC)

I'll try and find the references, that'll make things easier. I'm pretty sure there are a lot of inconsistencies in my analysis too; I don't think ignoring Kyubey's contribution would be so easy... --Homerun-chan 13:16, 19 March 2011 (UTC)

Ah, the original tweet came from the a Sony developer, Keijiro Takahashi, whom lead the development of Ape Escape 3. Prima 14:09, 19 March 2011 (UTC)

I see. Too bad we can't put that on our resume ;_; --Homerun-chan 14:28, 19 March 2011 (UTC)

I see in the refined model, you changed the initial state of W(0) to 1. I think we just found our Walpurgis Night.

Simulations and graphs

I feel that you should add the assumed numbers right above the graphs, because those will affect greatly how the graphs turn out, and I think that putting the numbers up front will enable readers to agree with your starting numbers first.220.255.2.142 13:36, 19 March 2011 (UTC)

Thanks for the advice, adding it right now. Also, note that they are in the image's description field. --Homerun-chan 13:45, 19 March 2011 (UTC)

Twitter corrections

Along with the other tweets, I found this one that seemed interresting, but I'm not sure I understand it. Basically, they're saying the refined model only works when F+B+1-K>0?

aibery Original tweet	Refined modelで魔女が増加するのは F+B+1-K>0 の時だけのような気がするけど、たぶん私が間違ってるんだろう。discussionではmatrixまで用いられてて文系にはちとキツい。 Refined modelで魔女が増加するのは F+B+1-K>0 の時だけのような気がするけど、たぶん私が間違ってるんだろう。discussionではmatrixまで用いられてて文系にはちとキツい。

It might be nice to investigate it a little, or explain why it is the case if you know... --Homerun-chan 17:48, 19 March 2011 (UTC)

It is a stumbling block I located as well.

In the case when M(t) > W(t), ΔW=(F+B-K)W, that is, change in witch population depends solely on the current population, modified by the constant (F+B-K). If F+B-K is negative, then ΔW is negative, and the population spirals downwards to zero, and M becomes a linear function Ct. The population dynamics breaks down. The only way to avoid this is if F+B-K is nonegative, but I am not certain how we can justify it, unless we call Kyuubey as a rational agent again. Prima 17:57, 19 March 2011 (UTC)

Actually, it's just a logical consequence of our model. If we change the hypotheses so that soul gems dim over time, we should fall back to a working model. I wouldn't do it though, since there has been no proof of that in canon, and it means setting a model to show us what we want to show, instead of solving the inconsistency at its source. --Homerun-chan 18:20, 19 March 2011 (UTC)

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)