Time to learn!

Today's post is alternatively titled "Sometimes it sucks when almost everyone you know doesn't understand what you do".

Anyways, I am bored so today I am going to teach some science - easy science that I think is pretty interesting and probably everyone should know the basics of.

In many of cases, it can be useful to estimate how quickly animal/bacteria populations grow at. Now, the only way a population can grow is for more animals to be born (from here on in the example will be of koala bears because they are oh so cute but also that unibrow is a little menacing so I don't mind talking about them dying). Of course, the population of koalas will decrease as they die. So the total growth (or decline) in population is however many are being born minus however many are dying:

Growth = Birth - Death

So far so good right! This math shit is easy. But there's more DUHN DUHN DUHN. Obviously, if you have 1000 koalas, they will be having alot more babies than if you only have 100 koalas. Likewise, there will be alot more koalas dying in a group of 1000 than a group of 100. This means that birth and death are both proportionate to the current number of koalas. This will require a change to our equation. Birth will be some number times the current number of koalas and Death will be some different number times the current number of koalas.
This will look something like this,

Growth = r*n - d*n.

Now, that looks kindave scary so let me explain. r is a growth rate term. It is determined by how many offspring the average koala has in a year. So say that the average koala has 1.2 kids a year. Then the total amount of koalas being born in one year would be r (which is 1.2) times n (which is the current population of koalas).

d is a death rate term. It is determined by the mortality rate for koalas. So say on average that 20% of koalas will die every year. Then d will become 0.2 and 0.2 times n would be the total amount of koalas that die in a year.

Do I still have everybody? I hope so because here is where it starts getting tricky. First of all, I am going to replace the word 'Growth' in our equation with the term 'dn/dt'. This may look stupid and complicated and I don't blame you because I didn't fully understand what it meant until second year calculus but it's actually quite simple. The "d's" in 'dn/dt' simply stand for 'change'. n stands for the number of koalas and t stands for time. So really 'dn/dt' is just a quick way of writing ' the change in koalas divided by the change in time'. Since we are working on a year to year basis (because that's what I just decided we were doing) the change in time will always be one year. So now 'dn/dt' can just be read as 'change in koalas per year' which is analogous to Growth but is much more official sounding because it has actual real units.

Okay, so now our formula looks like this:

dn/dt = r*n - d*n

and can be read as "The change in koalas per year equals the birth rate times the number of koalas minus the death rate times the number of koalas".

Now, time to add a couple more things and make it more fun. Let's say that the primary cause of death for all koalas is because some animal eats them (I don't know if that's true or not but let's roll with it and just for fun lets say that velociraptors eat them). This means that the death rate: d will depend on how many koalas a velociraptor needs to eat and the total number of raptors.

So lets say that d = c*p where c is how many koalas a raptor needs to eat to stay alive and p is the number of raptors. Now our equation looks like this,

dn/dt = r*n - c*p*n

"The change in koalas per year is equal to the growth rate times the number of koalas minus the number of koalas a velocirapter needs to eat times the total number of velociraptors times the total number of koalas".

Awesome, okay? Right around here we run into a bit of a problem though. The number of raptorss isn't going to remain constant (nature will always find a way). Maybe sometimes there are lots of raptors and maybe other times they've been extinct for 65 million years. It's hard to say. That being said it's really important to know just how many raptors there are because if there's alot that's gonna make the death rate really big and the koala population might all die out and nobody wants that. Likewise, if there's only one or two raptors, the death rate for the koalas will be really small and then there will be tons of koalas all over the place and the whole wide world will be a cuter place and people will stop having wars because there are just too many koalas that need cuddles.

But how in the hell can we find out how many raptors there are? Count them? Good idea but raptors are too clever for that. It's simple actually. We just make a second formula for the raptors. Let's start with the most basic one again,

Growth = birth - death

First of all, let's replace that Growth term with an official looking calculus 'dp/dt' (change in predators over change in time)

dp/dt = birth - death

Now, raptors are a little bit different than koalas. For one they aren't as soft and furry are dependent on koalas for food. If there aren't many koalas, raptors are gonna starve and die. So, let's take the death term from the koalas and use it as the birth term for the raptors with one little change. We're going to add a conversion rate term 'x' to it. The conversion rate term basically says " One koala bear is equal to this many velociraptors". So say if x = 0.05. You would need 20 koala bears before you could go to the bank and trade it in for a velociraptor (note to self: look at investing in raptor banks). More simply, a velociraptor would need to eat 20 koalas before it had the energy to make a new velociraptor. So let's add that term in

dp/dt = x*c*p*n - death

"The change in velociraptors per year is equal to the conversion rate times the number of koalas a raptor needs to eat times the number of raptors times the number of hares minus DEATH".

You see how by using that term if the number of koalas is really small, the birth of the raptors is also really small. Now, we have to deal with that death term. Luckily, nothing eats velocirapter (except for that one T-Rex but he wasn't hungry - just pissed off), so we don't need to worry about a predator. The death of velociraptors is just gonna be the natural mortality rate times the number of raptors, I'll call this 'd'

dp/dt = x*c*p*n - d*p.

Now, if we put our two equations together we have

dn/dt = r*n - c*n*p

and

dp/dt = x*c*n*p - d*p

and that right there is what is known as the Lotka-Volterra equation. It is a pair of autonomous first-order, non-linear differential equations. So if anyone ever gets on your case for not knowing anything you can say you know what an autonomous first-order, non-linear differential equation is and they will be all HOLY SHIT I'M SORRY I EVER THOUGHT YOU WEREN'T SUPER INTELLIGENT.


PS - This reminds me of something that makes me angry. People always exclaim "Oh! You must be really smart then" after I tell them I'm studying maths. I've only twice gotten the proper response out of someone. The first came from a friends parent who said " So why do you hate yourself" and the second came from a 17-year old whose mum is a chem prof and he said "Why math?".

I'm not really smart because I'm taking maths. I could be taking anything and still be just as smart as I am now. If I was taking something like Art History would they still exclaim that I'm smart? Probably not. They should be, man that shit is hard okay. Basically - moral of the story, if someone tells you they're in University odds are that they're fucking smart no matter what they're taking. It takes a shitload of time and effort and work and brains to get a degree in anything so don't assume just because the sciencey ones are harder for the general public to understand that it's harder to learn because it isn't. So the next time someone tells you that they're in university tell them that they must be really smart no matter what it is they tell you their taking... unless it's my ex-roomate.

Better moral of the story: If a skinny tall Korean with glasses who named himself after a hulking muscular greased up naked long-haired man tells you he's taking engineering, call him a fucking idiot and then punch him in the face.