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

TODO. Add the stability analysis. I'll do it later but my brain just melt from overheating and speculah overflow.

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)

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)