Basic set­ting

In the world of “Puel­la Magi Mado­ka Mag­i­ca”, there are mag­i­cal girls fight­ing against witch­es. Human girls can be­come mag­i­cal girls (al­though it is de­bat­able whether mag­i­cal girls are still human, for the pur­pose of this sketch I will treat “hu­mans” and “mag­i­cal girls” as non-over­lap­ping cat­e­gories, to fa­cil­i­tate talk­ing about the re­spec­tive pop­u­la­tions; I could call the non-mag­i­cal-hu­mans “mug­gles”, but that starts with the same let­ter as “mag­i­cal girls”), they can kill witch­es and they can be killed by witch­es. Witch­es can spawn fa­mil­iars, those fa­mil­iars can also fight with mag­i­cal girls, and they can ma­ture into witch­es them­selves.

And mag­i­cal girls be­come witch­es too (and Tyler Dur­den is just a fig­ment of the nar­ra­tor’s imag­i­na­tion, and Doc­tor Mal­colm Crowe was dead all along, and Dum­b­le­dore dies, and Rose­bud was his sled, and Rim­mer com­plained about the soup being too cold while din­ing with the of­fi­cers). The show is slight­ly am­bigu­ous on how mag­i­cal girls be­come witch­es: we will have to re­turn to this point. On the other hand, it is clear how human girls be­come mag­i­cal girls: they are tricked into a con­tract by aliens with ul­te­ri­or mo­tives.

Any­way, we have four pop­u­la­tions: the hu­mans , the mag­i­cal girls , the witch­es and the fa­mil­iars , and their re­la­tions are as fol­lows:

I model these re­la­tions as an it­er­a­tion with dis­crete time steps (I’ll leave the mean­ing of the small­est time unit un­spec­i­fied, whether it means a day, a week, a month or a year won’t mat­ter)(I could use a sys­tem of dif­fer­en­tial equa­tions, but I pre­fer a model which can be run by a sim­ple spread­sheet ap­pli­ca­tion, to make it eas­i­er for fel­low otakus to check or alter my mod­els (see also Ap­pendix A)):

A word on the no­ta­tion: if “” and “” are two pop­u­la­tions, then “” de­notes a func­tion of the num­ber of mem­bers of the pop­u­la­tion turn­ing into mem­bers of the pop­u­la­tion , with as the pseu­do-pop­u­la­tion of the dead re­spec­tive­ly un­born.

This sys­tem is sub­ject to some dra­matur­gi­cal re­straints: the pro­duc­tion of fa­mil­iars can re­sult in an ex­po­nen­tial growth of witch­es which could quick­ly wipe out hu­man­i­ty; but since the show tells us that witch­es ex­ist­ed since the dawn of time, this can’t hap­pen quick­ly or with a high prob­a­bil­i­ty, that is, the pop­u­la­tion dy­nam­ics of our sys­tem should be most­ly sta­ble. On the other hand, for dra­matur­gi­cal rea­sons, it is de­sir­able that a mag­i­cal girl, on av­er­age, has to fight sev­er­al witch­es, has a real chance of dying in such an en­counter, and has a high prob­a­bil­i­ty of be­com­ing a witch her­self (oth­er­wise, being a mag­i­cal girl would lack sus­pense, drama and angst). The ques­tion aris­es whether these con­straints can be sat­is­fied si­mul­ta­ne­ous­ly. Es­pe­cial­ly, on the Puel­la Magi Wiki, pop­u­la­tion mod­els have been pro­posed which are ei­ther high­ly un­sta­ble or lack drama.

To tack­le this ques­tion, I pro­pose two sim­pli­fi­ca­tions of the basic model. First, since I’m search­ing a sta­ble so­lu­tion, I will treat hu­man­i­ty as an in­fi­nite pool where mag­i­cal girls can be re­cruit­ed. If the pop­u­la­tion of witch­es grows out of con­trol, the num­ber of hu­mans will de­cline, and with all hu­mans gone, it will be im­pos­si­ble to re­cruit new mag­i­cal girls. But with sta­ble (and com­par­a­tive­ly small) pop­u­la­tions of mag­i­cal be­ings, this should be ir­rel­e­vant.

The sec­ond sim­pli­fi­ca­tion is less ob­vi­ous: I’ll get rid of the pop­u­la­tion of fa­mil­iars. From the stand­point of mag­i­cal girls, witch­es and fa­mil­iars are dif­fer­ent: fight­ing them costs mag­i­cal en­er­gy which can be re­plen­ished using “grief seeds”, but only witch­es have grief seeds. A mag­i­cal girl ex­clu­sive­ly fight­ing fa­mil­iars will loose her mag­i­cal en­er­gy and turn into a witch. There­fore, the ratio of witch­es to fa­mil­iars might mat­ter for the rate of mag­i­cal girls turn­ing into witch­es. There are two rea­sons I’ll ig­nore fa­mil­iars nev­er­the­less: first, I ex­pect the ratio of witch­es to fa­mil­iars to be more or less con­stant. And sec­ond, I’ll model the tran­si­tion of mag­i­cal girls into witch­es as large­ly in­de­pen­dent of witch­es any­way (see below).

So in­stead of witch­es spawn­ing fa­mil­iars and the fa­mil­iars ma­tur­ing into witch­es, I sim­ply have witch­es spawn­ing more witch­es. Ac­cord­ing­ly, my model re­duces to this:

A sim­ple most­ly lin­ear model for the equi­lib­ri­um case

The model above has 5 func­tions we need to spec­i­fy. For four of these func­tions, I will pro­pose sim­ple lin­ear re­la­tions.

How do witch­es spawn fa­mil­iars, and how do fa­mil­iars ma­ture into witch­es? The show doesn’t ex­plain this in de­tail, so sev­er­al pos­si­bil­i­ties spring to mind. It might be that the fa­mil­iars need to spend some time being undis­turbed, so the pres­ence of many mag­i­cal girls could slow down the pro­duc­tion and ma­tur­ing of fa­mil­iars. Or per­haps a fa­mil­iar killing a mag­i­cal girl gets a boost and ma­tures quick­er or a witch needs to kill a mag­i­cal girl to pro­duce a fa­mil­iar. Al­though I can’t rule out these pos­si­bil­i­ties, I’ll ig­nore them for a sim­pler model: the num­ber of witch­es pro­duced via fa­mil­iars is sim­ply pro­por­tion­al (with some con­stant fac­tor ) to the num­ber of witch­es present:

How do mag­i­cal girls be­come witch­es? They can loose all their magic en­er­gy fight­ing witch­es and fa­mil­iars. The se­ries shows (in episode 10) an in­stance of this hap­pen­ing: Mado­ka de­feats Walpur­gis­nacht, but in the pro­cess loos­es so much en­er­gy that she in­stant­ly turns into a witch her­self. But I think that this is an ex­cep­tion­al case. I as­sume that usu­al­ly, a mag­i­cal girl can avoid this sit­u­a­tion by col­lect­ing grief seeds or avoid­ing fights with fa­mil­iars if her mag­i­cal en­er­gy runs low. In­stead, I as­sume that the main fac­tor turn­ing mag­i­cal girls into witch­es is them be­com­ing over­whelmed by grief and loos­ing their will to con­tin­ue fight­ing. There­fore, I model the tran­si­tion of mag­i­cal girls as being pro­por­tion­al to the num­ber of mag­i­cal girls:

How often do mag­i­cal girls and witch­es meet each other and en­gage in a fight? Dif­fer­ent as­sump­tions are pos­si­ble: it could be that a mag­i­cal girls search­es ex­act­ly one witch dur­ing one it­er­a­tion, and, pro­vid­ed that there are enough witch­es present, al­ways finds one. But I as­sume that a mag­i­cal girl en­gages in more fights if there are more witch­es present. The num­ber of fights should be pro­por­tion­al to . In each fight, ei­ther the mag­i­cal girl or the witch dies. Not ac­count­ing for ex­pe­ri­ence or mat­u­ra­tion, I as­sume that the prob­a­bil­i­ty of a mag­i­cal girl sur­viv­ing an en­counter with a witch is a fixed num­ber. I also as­sume that fights be­tween mag­i­cal girls and witch­es are the only way of them dying, and none of them dies of old age or other non-mag­i­cal caus­es. The num­ber of mag­i­cal girls dying is there­fore

And cor­re­spond­ing­ly the num­ber of witch­es dying is

The last re­main­ing func­tion we need to model is , the num­ber of human girls the aliens re­cruit dur­ing one it­er­a­tion step. This func­tion is slight­ly dif­fer­ent than the other four, since we can as­sume that the aliens choose this num­ber de­lib­er­ate­ly to achieve cer­tain goals. What would these goals be? I as­sume they want to achieve a sta­ble equi­lib­ri­um with the num­ber of witch­es equal to some de­sired tar­get num­ber . If they have set­tled for some , this re­strains the pos­si­bil­i­ties for the size of the pop­u­la­tion . Since we have

and we also know that for a sta­ble equi­lib­ri­um, , we have

And know­ing both and , we can at last com­pute for the equi­lib­ri­um case:

For il­lus­tra­tive pur­pos­es, I’ll pro­vide a model sce­nario with ex­plic­it nu­mer­i­cal val­ues: I as­sume that the aliens want to set­tle for an equi­lib­ri­um with 2,000 witch­es, so

Fur­ther­more, I as­sume that dur­ing each it­er­a­tion step, a mag­i­cal girl has a 4 % chance of be­com­ing a witch, and a 20 % chance of meet­ing a witch. Dur­ing a fight, she has a 10 % chance of dying. With these num­bers, a mag­i­cal girl lives long enough to ex­pe­ri­ence some angst and drama and suf­fer­ing, and she will most like­ly end up as a witch in the end: end­ing up as a witch is twice as like­ly as end­ing up being killed.

To achieve this, I set . I also set and . Now set­ting would con­strain , but I want to spec­i­fy in­stead as 1,000 (I want to in­ves­ti­gate the case where the mag­i­cal girls keep the num­ber of witch­es in check with­out drown­ing them, so I want twice as many witch­es than mag­i­cal girls). This, in turn, re­strains the pos­si­ble val­ues of to

For the set­ting I have cho­sen, this means a value of .

Now we have this sit­u­a­tion: there are 2,000 witch­es and 1,000 mag­i­cal girls. In each it­er­a­tion step, 40 girls be­come witch­es, and 20 girls are get­ting killed. Ac­cord­ing­ly, the aliens re­cruit 60 new girls. Fur­ther­more, 180 witch­es are killed, and 140 new witch­es are pro­duced via fa­mil­iars. A mag­i­cal girl, on av­er­age, lives for al­most 17 it­er­a­tions, while a witch lives for slight­ly more than 11 it­er­a­tions. On av­er­age, a mag­i­cal girl kills one and a half witch.

This av­er­age kill rate is rather low. It de­pends on how much witch­es come into being start­ing as mag­i­cal girls, and how many start­ed as fa­mil­iars. The more witch­es start as fa­mil­iars, the more witch­es a mag­i­cal girl will kill on av­er­age be­fore dying or be­com­ing a witch her­self, but the less like­ly it be­comes for her to loose a fight.

An im­por­tant dif­fer­ence be­tween dif­fer­ent model sce­nar­ios is whether or dom­i­nates witch pro­duc­tion. If the lat­ter dom­i­nates, most witch­es stem from fa­mil­iars, and we need ei­ther lots of mag­i­cal girls, or mag­i­cal girls al­most never loos­ing a fight to keep them at bay, both set­tings bound to re­duce drama and angst. On the other hand, if the for­mer term dom­i­nates, a mag­i­cal girl can’t fight many witch­es. As­sume a sce­nario with . Then the av­er­age mag­i­cal girl can, if the pop­u­la­tions are in equi­lib­ri­um, kill at most one witch, since mag­i­cal girls are the only source of witch­es in this sce­nario, and one mag­i­cal girl be­comes only one witch her­self. The more like­ly it is for a mag­i­cal girl to die in a fight with a witch, the less like­ly it be­comes that she will ever meet a witch in a fight at all.

We can par­tial­ly lessen this prob­lem if we as­sume that the num­ber of witch­es a mag­i­cal girl kills is dis­tribut­ed rather un­even­ly. Then we would have lots of in­ex­pe­ri­enced girls killing few or none witch­es at all, and a few ex­pe­ri­enced girls reg­u­lar­ly en­gaged in fights with witch­es.

Nev­er­the­less, at this point it seems doubt­ful whether it is pos­si­ble to have a sta­ble equi­lib­ri­um with drama and angst and lots of fights. For ex­am­ple: is the equi­lib­ri­um I just de­scribed sta­ble?

The suc­cess of rather dumb aliens

To an­swer this ques­tion, we need to know the be­hav­ior of our model out­side the equi­lib­ri­um case. So far, we have de­fined the func­tion only for the point . But what should the aliens do in sit­u­a­tions out­side the de­sired equi­lib­ri­um?

I think the most sim­plest and brain-dead an­swer to this ques­tion would be: sim­ply just do the same. I re-name the value as a con­stant and de­fine the func­tion for re­al­ly dumb aliens as

The com­plete model then looks like this:

For a spe­cif­ic sce­nario, I keep the val­ues for the other con­stants as above, that is

and set

Now what hap­pens if we set and ? It turns out that the pop­u­la­tion of the witch­es grows out of bounds: after 2,610 it­er­a­tion steps, the witch pop­u­la­tion ex­ceeds 2 mil­lion, while the pop­u­la­tion of mag­i­cal girls col­lapsed.

If we start with too few witch­es, that is, and , the sys­tem set­tles for a new equi­lib­ri­um:

We can also start with an ini­tial pop­u­la­tion of one mag­i­cal girl and zero witch­es. Al­though our aliens be­have in a re­al­ly dumb man­ner, the sys­tem set­tles for an equi­lib­ri­um, al­though not the de­sired equi­lib­ri­um:

It turns out that our sys­tem has two equi­lib­ri­um points. It is easy to see why: let be a fix­point of this model. For a fixed point, we have

Using these equa­tions, we can use the sec­ond equa­tion to ex­press in terms of , as we al­ready did above. Sub­sti­tut­ing this value in the first equa­tion, we get

This is a sec­ond-de­gree poly­no­mi­al with two so­lu­tions for (and there­fore two so­lu­tions for ). One of these so­lu­tions is our de­sired tar­get , the other is not. If we keep with the val­ues of the model sce­nario above, the tar­get so­lu­tion is 2,000 witch­es and 1,000 mag­i­cal girl, but it turns out that there is an­oth­er fix­point with 1,714.2857… witch­es and 1,050 mag­i­cal girls. Fur­ther­more, the tar­get fix­point is a re­pelling fix­point, while the sec­ond fix­point is at­trac­tive (the spec­tral ra­dius of the Ja­co­bian ma­trix is 1.00243… for the first fix­point, a value greater than 1, but 0.99747… for the sec­ond fix­point, see Ap­pendix B).

Slight­ly less brain­dead aliens

The aliens are in no way obliged to be­have in such a blind-sight­ed way as de­pict­ed by the model above. In­stead, they can adopt the num­ber of girls they re­cruit ac­cord­ing to the cur­rent pop­u­la­tion sit­u­a­tion. A sim­ple model would work like this: the aliens de­sire a witch pop­u­la­tion of witch­es. At time , there are present. We apply this de­vi­a­tion of the de­sired sit­u­a­tion to the pop­u­la­tion of mag­i­cal girls and get a new, tem­po­rary tar­get for the pop­u­la­tion of mag­i­cal girls:

Now the aliens at­tempt to choose in such a way to as­sure that . We first de­fine an ide­al­ized ver­sion of , name­ly, , which de­scribes what the aliens would do if they could act with­out con­straints:

For the ac­tu­al func­tion , there are some con­straints. If we as­sume that the aliens never out­right kill mag­i­cal girls them­selves, then can never be neg­a­tive. Fur­ther­more, it seems nat­u­ral to as­sume that the re­sources of the aliens are lim­it­ed, such that there is a max­i­mal num­ber of mag­i­cal girls they can re­cruit with­in each it­er­a­tion step. At least, since the human pop­u­la­tion is fi­nite, and so is the num­ber of pubescent girls, the pool from which the aliens can re­cruit mag­i­cal girls is lim­it­ed. I there­fore as­sume that there is an upper limit for the num­ber of pos­si­ble re­cruits dur­ing one it­er­a­tion step. The ac­tu­al func­tion then be­comes

It should be noted that this model still has rather sim­ple-mind­ed and me­chan­i­cal aliens. Nev­er­the­less, it is sur­pris­ing­ly suc­cess­ful. For a con­crete ex­am­ple, I pick the same pa­ram­e­ters as above, with the only dif­fer­ence being the new def­i­ni­tion of the func­tion . For the re­main­ing un­spec­i­fied pa­ram­e­ter , I choose a value of 100 (in the equi­lib­ri­um case, the aliens re­cruit 60 girls dur­ing each it­er­a­tion step; re­cruit­ing 40 girls more dur­ing a time of cri­sis seems fea­si­ble). Com­pare the two mod­els:

Since in this sce­nario the mag­i­cal girls are rather ef­fi­cient witch killers and the pro­duc­tion of fa­mil­iars is slow, a pop­u­la­tion of 1,000 mag­i­cal girls and 9,115 witch­es still moves back to the de­sired equi­lib­ri­um:

Adding ran­dom noise

So far, both mod­els have been strict­ly de­ter­min­is­tic. It is to be ex­pect­ed that there are ran­dom vari­a­tions in the num­ber of mag­i­cal girls and witch­es, and a fea­si­ble alien strat­e­gy should be able to cope with such ran­dom de­vi­a­tions from a strict­ly de­ter­min­is­tic model.

A re­al­is­tic model would be to add some nor­mal dis­tribut­ed Gaus­sian noise to each func­tion. But I de­cid­ed against such a model, since, as men­tioned above, I want­ed to be able to com­pute mod­els with sim­ple spread­sheet func­tions to make it eas­i­er for oth­ers to com­pare and en­hance my re­sults. In­stead, I have cho­sen a rather sim­plis­tic method of adding some noise. Let be a pop­u­la­tion how it would had de­vel­oped from the state at time ac­cord­ing to one of the two the de­ter­min­is­tic mod­els de­vel­oped above, and let be a ran­dom dis­tri­bu­tion uni­form on the in­ter­val from 0 to 1 (as usu­al­ly pro­duced by soft­ware ran­dom num­ber gen­er­a­tors), and let be a pa­ram­e­ter to de­scribe the severe­ness of the ran­dom noise. Then the model pop­u­la­tion in the dis­turbed model is de­fined as

Let’s look again at our two mod­els from above with ex­treme­ly dumb and slight­ly less dumb aliens. For (which means that for each it­er­a­tion step, both pop­u­la­tions can ran­dom­ly go up or down about 5 %), one pos­si­ble out­come looks like this:

To make mat­ters worse for the model with dumb aliens, it takes on av­er­age about 1,200 it­er­a­tion steps until the model be­comes un­sta­ble and the witch pop­u­la­tion grows out of bounds. The “smart” aliens fare con­sid­er­ably bet­ter, their model stays sta­ble for long amounts of time.

Even if we re­duce to the rather strict value of 70 (note that 60 new girls are need­ed in the equi­lib­ri­um case), the model usu­al­ly, in most runs, sur­vives sev­er­al thou­sand it­er­a­tion steps with­out the witch pop­u­la­tion ex­plod­ing. The fol­low­ing run shows how such a run al­most gets out of bounds around it­er­a­tion step 9,000:

The move­ment of the point through phase space looks like this:

For com­par­i­son, with we get a move­ment through phase space like this:

On the other hand, with as high as 100, the model usu­al­ly sur­vives even ran­dom noise of mag­ni­tude for a sub­stan­tial amount of time. The even­tu­al break­down might even be con­sid­ered to be con­sis­tent with the TV se­ries: after all, the sys­tem of mag­i­cal girls and witch­es is sup­posed to be in place since pre­his­toric times, but nev­er­the­less in most time­lines doesn’t sur­vive the ap­pear­ance of the witch Walpur­gis­nacht.

A pos­si­ble crit­i­cism against the pro­posed sce­nario could be that mag­i­cal girls fight against too few witch­es. We can rec­ti­fy this by in­creas­ing the num­bers of witch­es pro­duced by fa­mil­iars and also in­creas­ing the num­bers of witch­es killed in a fight: we can set , keep , slight­ly ad­just (there being half as much mag­i­cal girls means that each of them has to fight more often), set­ting , which means that each girl sur­vives 19 out of 20 fights, and can raise to . This means that in each it­er­a­tion step, there are 400 fights, 380 witch­es are killed and 340 new witch­es are pro­duced via fa­mil­iars. Fur­ther­more, each mag­i­cal girl, on av­er­age, kills more than 6 witch­es, al­though she still has a 2/3 chance of turn­ing into a witch her­self. With her life ex­pec­ta­tion un­changed, this new sce­nario means that her life will be full of fight­ing (even more so if we as­sume a heav­i­ly skewed dis­tri­bu­tion be­tween lots of girls killed early in their ca­reers and a few ex­pe­ri­enced vet­er­ans). Even with and , this doesn’t pose a prob­lem for the model in most runs, and witch­plo­sions rarely ever occur (I failed to ob­serve one with­in 200,000 it­er­a­tion steps).

Like­wise, if we want to drag out the mis­ery of the av­er­age mag­i­cal girl, we can set , , and , while re­tain­ing . In this sce­nario, a mag­i­cal girl kills 13 witch­es, still has a 2/3 chance of end­ing as a witch her­self, and in each fight has a 2,5 % chance of being killed, so in­di­vid­u­al fights are still some­what dan­ger­ous. Still, even with , this doesn’t pose a prob­lem for the abil­i­ty of the model to keep the witch pop­u­la­tion in check.

If, on the other hand, we want to make life mis­er­able for the mag­i­cal girls and ac­cept that they die like flies, we can also set the prob­a­bil­i­ty for each it­er­a­tion step that they be­come a witch to 10 % and like­wise set the prob­a­bil­i­ty that they get killed by a witch to 10 %. Fur­ther­more, a fight with a witch should be re­al­ly dan­ger­ous, with a mag­i­cal girl loos­ing every third fight. We can achieve this by the fol­low­ing set­ting:

Once again, this sce­nario is most­ly sta­ble:

Con­clu­sion

There are re­straints which pre­vent cer­tain sce­nar­ios. It is im­pos­si­ble that al­most all witch­es are the re­sult of mag­i­cal girls be­com­ing witch­es and that si­mul­ta­ne­ous­ly witch­es pro­duce huge num­bers of fa­mil­iars which ma­ture into witch­es, or that al­most all mag­i­cal girls end up as witch­es while being a mag­i­cal girls means fight­ing lots of witch­es and that fight­ing a witch is very dan­ger­ous, all of this while at the same time this sys­tems re­mains sta­ble over long pe­ri­ods of times and was in place since the dawn of time (like the TV se­ries im­plies). But if we are will­ing to sac­ri­fice or relax at least some of these re­straints, fea­si­ble mod­els are pos­si­ble. It is pos­si­ble to have sta­ble sys­tems with sig­nif­i­cant ran­dom per­tur­ba­tions, it is pos­si­ble to have more witch­es than mag­i­cal girls, it is pos­si­ble for mag­i­cal girls to fight many witch­es on av­er­age, it is pos­si­ble for many girls to end as witch­es, or it is pos­si­ble for fights with witch­es to be dan­ger­ous. What­ev­er world we want to cre­ate, and what­ev­er fla­vor of angst and drama we pre­fer, it is pos­si­ble to con­struct sce­nar­ios which come close to these worlds, even with a rather sim­ple model that doesn’t ac­count for a more so­phis­ti­cat­ed be­hav­ior on the side of the aliens.

Ap­pendix A: Excel spread­sheet

As promised above, you can down­load an Excel spread­sheet to fid­dle with the dif­fer­ent pa­ram­e­ters of the two pro­posed mod­els for your­self.

The spread­sheet con­tains two ta­bles. The table la­beled “Dumb­Fix­point” is only rel­e­vant if you choose the model with dumb aliens. In this case, this table com­putes the two fix­points. Fur­ther­more, this table also com­putes the Ja­co­bian ma­trix for these fix­points, the cor­re­spond­ing two eigen­val­ues and the spec­tral ra­dius. A spec­tral ra­dius greater than 1 means that the fix­point is re­pelling, the greater the value, the faster points in the neigh­bor­hood of the fix­point are driv­en away. A spec­tral ra­dius small­er than 1 means that the fix­point is at­trac­tive, the small­er the value, the faster points in the neigh­bor­hood of the fix­points are con­verg­ing to­wards the fix­points. For the math­e­mat­i­cal de­tails, see also Ap­pendix B.

The table la­beled “Main” con­tains ev­ery­thing else. Pa­ram­e­ters you can set are shown in light green. I wired the cal­cu­la­tion in such a way that you don’t set the model con­stants di­rect­ly, but some (hope­ful­ly more mean­ing­ful) pre­cur­sors. If you set “Ran­dom Noise” to 0, you get a de­ter­min­is­tic model. You can choose be­tween the model with dumb and less dumb aliens by set­ting “Smar­tAl­iens” to “False” or “True”.

When you change a value, the spread­sheet im­me­di­ate­ly starts to cal­cu­late the new model for 10,000 it­er­a­tions. If your model has ran­dom noise and you want to start a new model run with dif­fer­ent ran­dom val­ues, sim­ply change an empty cell.

It may hap­pen that the pa­ram­e­ters you have cho­sen lead to an un­bound­ed growth of the witch pop­u­la­tion, lead­ing to pop­u­la­tion num­bers ex­ceed­ing the max­i­mum nu­mer­i­cal value Excel (or OpenOf­fice) can han­dle (in older ver­sions of OpenOf­fice, you can ac­tu­al­ly crash the ap­pli­ca­tion this way). To pre­vent this, the model stops when a pop­u­la­tion grows larg­er than a large thresh­old value, and restarts again with the ini­tial pop­u­la­tion val­ues. The value “Melt­down­Count” tells you how often this has hap­pened, which is a mea­sure of how un­sta­ble your model is. For ex­am­ple, if there is just one melt­down with­in the full 10,000 it­er­a­tions, your model is al­most sta­ble, and only rare and un­for­tu­nate com­bi­na­tions of ran­dom noise make it be­come un­sta­ble. On the other hand, if you en­counter sev­er­al hun­dred melt­downs, your model seems to be rather hope­less. If you en­counter no melt­down at all, this field is dis­played with a green back­ground, oth­er­wise, it is red.

There are also two di­a­grams with the pop­u­la­tions of the mag­i­cal girls and the witch­es, one for all 10,000 it­er­a­tion steps and one just for the first 1,000 it­er­a­tion steps. An­oth­er two di­a­grams show the move­ment of the model in the phase space: the x-axis is the num­ber of mag­i­cal girls, and the y-axis is the num­ber of witch­es. In a typ­i­cal growth-to­wards-an-at­trac­tive-fix­point sce­nario with ran­dom noise, the model will start in the lower left cor­ner, move to­wards the fix­point and then start to ran­dom­ly wan­der in the vicin­i­ty of this fix­point. Once again, the upper di­a­gram shows all 10,000 it­er­a­tion steps, while the lower di­a­gram is lim­it­ed to the first 1,000 it­er­a­tion steps.

Ap­pendix B: Fix­points in a two-di­men­sion­al phase space

The fol­low­ing sec­tion de­scribes a well-known piece of math, which might nev­er­the­less be of some in­ter­est for some­one try­ing to re­con­struct one the mod­els I de­vel­oped above. Be­fore we con­sid­er the two-di­men­sion­al case, let’s start with one di­men­sion: we have a func­tion and a point with , and we want to know whether this fix­point is at­trac­tive. A fix­point is at­trac­tive if, for every suf­fi­cient­ly small, ap­ply­ing re­peat­ed­ly to re­sults in val­ues com­ing clos­er and clos­er to . There­fore, we want to know if is ab­so­lute­ly small­er than , in which case the dis­tance to would be shrink­ing. This is the case if

But for small , the term on the left side of this equa­tion is al­most the same as the deriva­tion . So in the one-di­men­sion­al case, a fix­point of a func­tion is an at­trac­tive fix­point if

With this re­sult in mind, we turn to the two-di­men­sion­al case. Sup­pose we have two func­tions and , which both take two val­ues. Fur­ther­more, we have a point with and .

How do we com­pute the deriva­tion of such a func­tion pair? There are four dif­fer­ent ways: we have two par­tial func­tions, and we can com­pute the deriva­tion for two dif­fer­ent vari­ables. The Ja­co­bi ma­trix com­bines these four re­sults:

We are, of course, in­ter­est­ed in the fate of our point , so we have to com­pute

How do we know whether such a deriva­tion ma­trix is large or small, whether it stretch­es or shrinks things, whether an it­er­a­tion push­es neigh­bor­ing points far away or at­tracts them? Such a ma­trix can be char­ac­ter­ized cal­cu­lat­ing the eigen­vec­tors of the ma­trix, vec­tors which keep their di­rec­tion under the ap­pli­ca­tion of the ma­trix. Al­though the di­rec­tion of an eigen­vec­tor re­mains un­changed under the ap­pli­ca­tion of the ma­trix, its length might change about a fac­tor called an eigen­val­ue of the ma­trix. The eigen­val­ues de­scribe how much such a ma­trix shrinks or stretch­es dif­fer­ent di­rec­tions.

In the gen­er­al case, com­put­ing eigen­val­ues of a ma­trix is dif­fi­cult, but in the case of a 2×2 ma­trix, it is sim­ple enough. For a ma­trix of the form

the eigen­val­ues are

If the ab­so­lute val­ues of both eigen­val­ues are small­er than 1, then a point is drawn near­er. If one eigen­val­ue is greater 1, a neigh­bor­ing point is driv­en away in at least one di­rec­tion, and the fix­point is not at­trac­tive. We can ex­press this using the spec­tral ra­dius , which is de­fined as

A fix­point is at­trac­tive if the spec­tral ra­dius is small­er than 1.

In the dumb model above, we have the fol­low­ing func­tions:

Ac­cord­ing­ly, we have these par­tial derivates: