Learn AIHL 5.16.3 Euler's method for coupled systems in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 5.16—Eulers method for 1st order DEs.
Learn AIHL 5.16.3 Euler's method for coupled systems in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 5.16—Eulers method for 1st order DEs.
The video discusses Euler's method for solving coupled systems of differential equations, specifically using the predator-prey model as an example. In this model, the population of rabbits (denoted as ) and foxes (denoted as ) are interdependent, with equations representing their growth rates:
The video demonstrates how to apply Euler's method with a step size of to predict the populations of rabbits and foxes after one year, starting with 1,000 rabbits and 100 foxes.
The results show that after one year, the rabbit population increases to approximately 2,837, while the fox population decreases to about 58, illustrating the dynamics of the predator-prey relationship.
00:00Hi guys, okay, so the
00:02next part of Euler's method
00:04is Euler's method for coupled
00:06systems. On a coupled system,
00:09it's essentially this with two,
00:11a couple of differential equations,
00:13dx, dt and dy, dt,
00:16where x and y are
00:17dependent, variables that are dependent
00:19on each other, and t
00:21is an independent variable. It
00:23is not dependent on anything,
00:25it's just time, time moves
00:26on, doesn't care about anything
00:28But x and y very
00:30much do care about each
00:31other. And in this situation,
00:34this is a common example
00:35that we use as the
00:37example I'm going to use
00:38is the predator prey model.
00:42And well, x is the
00:43population of rabbits. And y
00:45is the population of foxes.
00:47So obviously the foxes are
00:49the predators and the rabbits
00:50are the prey. And what
00:53happens is as the
00:56As the foxes eat the
00:59rabbits, so the foxes see
01:00the wrappers in our lovely
01:01lots of rabbits. Let's eat
01:02them. The population of rabbits
01:05decreases and the population of
01:08foxes increases because the foxes
01:10are getting lots of food.
01:12But as they eat lots
01:14and lots of rabbits and
01:16the population of rabbits decreases,
01:18there's not much food left
01:19because they've eaten all the
01:20rabbits. So the population of
01:22foxes starts to decrease
01:24because there's no food for
01:26them. And then as the
01:27population of foxes decreases, the
01:30population of rabbits, so it's
01:31the increase again because there's
01:32not that many foxes to
01:34kill them. And the whole
01:35thing just repeats. And the
01:41coupled system differential equations can
01:44model that very nicely. Okay,
01:47so this is our example.
01:49And you can see here
01:51within the differential equation
01:52This is the growth rate
01:55of rabbits. We subtract 0
01:57.01 y. So obviously, as
01:59y gets bigger, this dx
02:02dt decreases. And here, as
02:06x gets bigger, this dy
02:09dt increases. Because as I've
02:12said, as the rabbits, as
02:14x, which is the population
02:15of rabbits, increases, that's good
02:18for the foxes. The foxes
02:19are happy with that.
02:20the population stress to increase.
02:23Anyway, it says, right, we
02:26have an island, there's rabbits.
02:27Ecologists introduce a population of
02:30100 foxes to the island
02:31when the population grabs is
02:33a thousand. So there's a
02:34thousand rabbits and now 100
02:36foxes. Let X be the
02:38population of rabbits, why is
02:39the population of foxes teased
02:40the time in years? And
02:42these are the population growths
02:43of rabbits and foxes. Use
02:46orders method with a step
02:48sign
02:48of 0 .25 to find
02:50the populations of rabbits and
02:52foxes one year after the
02:54foxes were introduced. Okay. This
02:56is the order's method from
02:58the formal booklet. This is
02:59order's method for coupled systems.
03:01It's essentially the same thing,
03:04but we have an x
03:07and a y dependent on
03:09each other and this other
03:11independent variable t. So,
03:16What we want to get,
03:17what we want to do
03:18is find the next x
03:20and the next y. And
03:21we'll do the whole thing
03:22together. And it's a bit
03:23like, it's very like this,
03:25this example that we did
03:26in the previous video, but
03:28we're just going to do
03:28it twice essentially. But twice
03:31in the one go. So
03:33first, let's find these equations.
03:35The next x and the
03:37next y. So our next
03:40x are x and plus
03:42one is going to be
03:43our previous
03:44X, X of n plus
03:47H. Now H is the
03:48step length, which is 0
03:49.25. H times this, now
03:54this is f1, X, n,
03:56y, and t, and that
03:57looks quite complicated. That is
04:00just dx dt. And this
04:03is just dy dt. So
04:06dx dt is this here,
04:10but it's X, and I'm
04:11going to put n because
04:12it's the
04:12The x of n plus
04:141 is the previous n.
04:18And we're going to use
04:20it for the previous iteration.
04:22I feel like I'll explain
04:23that a bit more in
04:23a second. Times 2 minus
04:260 .01 yn. And our
04:31y of n plus 1
04:33is equal to yn plus
04:370 .25. That's the step
04:39length.
04:40n times 0 .0002 x
04:46n minus 0 .8 close
04:49bracket. Okay, and our T
04:52of n plus 1, our
04:53next T is just the
04:54T plus the H plus
04:55the step length. So we're
04:57gonna start with, let's start
04:59with the T. Our T
05:02is, our initial T is
05:04when T is 0. And
05:06then our next T is
05:080.
05:080 .25 because the t
05:10goes up in the step
05:11length, then 0 .5, 0
05:15.75, and 1. I'm going
05:18to stop at 1 because
05:19we're looking for the populations
05:20after 1 year. OK, fun.
05:25The next, sorry, the xn,
05:29this is our population of
05:31rabbits, at 0, it is
05:341 ,000 because that's
05:36the time as the start
05:38of this whole thing when
05:39the foxes were introduced, so
05:40we do a thousand rabbits.
05:42And our YN, our YN,
05:47is 100, is there was
05:49100 foxes. Okay, now we're
05:53gonna do this whole thing
05:55on the calculator. So definitely
05:57you want to get familiar
06:02with and able to use
06:03this spreadsheet, because this makes
06:04here.
06:04life quite easy. So this
06:07is actually going to be,
06:08let's call this t, this
06:10is xn, this is yn,
06:14and this just goes zero,
06:16zero point two five, zero
06:19point five, zero point seven
06:21five, and one. Our x
06:26of n, our initial x
06:28of n is a thousand,
06:30and our initial y of
06:31n is
06:33100. I mean this is
06:36the only tricky part of
06:37this whole question. You have
06:39to put what we have
06:42to write this in here.
06:45We have to write, let
06:47me just make sure we
06:48can see it. I have
06:49to write this, I can
06:54make that in smaller. I
06:55have to write this in
06:57here using like a sensor
07:01to Excel format. So I
07:03have to write equals, because
07:04I want this cell to
07:06equal, but I want it
07:07equal X of N, the
07:08previous X. So it's this
07:11X. So I can either
07:12click that, or I could
07:13just type B1, because this
07:15is cell B1. So it's
07:18B1 plus 0 .25 times
07:24B1 again. I'm just going
07:26to type B1 times
07:29Again, brackets two minus 0
07:32.01. And this time I
07:34want Yn, which is C1.
07:38So I'm going to do
07:39times C1. Close brackets, press
07:44Enter, and I get 1
07:46,250. Then this is, I
07:52put all of this in.
07:54So this is equal to,
07:56I'm going to
07:57go C1 plus 0 .25
08:02times C1 again times brackets
08:080 .002 times XN, which
08:16is B1 minus 0 .8
08:22closed brackets per center.
08:25So actually what's happened is
08:28after the first quarter of
08:30a year, the population of
08:34rabbits has actually increased in
08:35the population of foxes has
08:36decreased. So actually there's not
08:38enough, there's a thousand rabbits,
08:41but because there's a hundred
08:42foxes, there's just not enough
08:44rabbits for the foxes yet.
08:46So actually the population of
08:48foxes is decreasing. But as
08:51this decrease,
08:53and this increases, there's gonna
08:55come a point when the
08:57foxes are gonna go. All
08:58right guys, now there's enough
08:59rabbits, there's those are rabbits
09:00first. Let's start, let's start
09:03eating the rabbits and multiplying
09:05again. Okay, at this stage
09:08guys, you don't have to
09:09type that in over and
09:10over again, you can just
09:11drag this down and you
09:13can drag it down to
09:13where you wanna go, which
09:14is there. A nice function,
09:18because I know it's not
09:19as easy to do this
09:20on your calculator as it
09:21is
09:21me with my trackpad. If
09:24you press menu data and
09:27fill, then you just press
09:29down, down, down and press
09:31enter. And this just fills
09:33nicely automatically. So let's write
09:37out this first. It's 1,
09:402, 50, 1, 6, 0,
09:419. So 1, 2, 50,
09:441, 6, 0, 9.
09:49I'm just gonna round it
09:50guys 2 -1 -1 -9
09:51and 2 -8 -3 -6,
09:532 -1 -1 -9, 2
09:56-8 -3 -6. Actually, that's
10:022 -8 -3 -7 if
10:03we're gonna round that. That
10:04actually is important. And then
10:09I have 85, 73, 65,
10:1585, 73,
10:1765 and then 58 that's
10:23just rounded to the nearest
10:25Fox shall we say and
10:29now you may be thinking
10:31hang on the population of
10:32this is just increasing the
10:33population this is just decrease
10:34and it is they wanted
10:35us to stop after one
10:36year but let's just see
10:39what would have happened if
10:42we kept going so let's
10:44remove that down to here
10:45here and I move this
10:47down to here. So yeah,
10:50at this point you can
10:51see it takes a while
10:54but the so here at
11:02this point the population well
11:07the population of foxes starts
11:09to increase the population of
11:11rabbits still seems to be
11:13increasing here as well. Let's
11:16see if I go a
11:17little bit more. I want
11:22to see these rabbits start
11:23to decrease. Okay, so here
11:27now the foxes are start
11:28to the foxes start to
11:31increase a lot. And at
11:33this point, yeah, the rabbits
11:35are like they start to
11:36decrease. Anyway, okay, so
11:41That's basically it. You stop
11:44when when you says to
11:46stop and that's after one
11:46year. So it's here. So
11:48the question said find the
11:49population of rabbits and foxes
11:51after one year. So you
11:53you want to obviously clarify
11:55what's going here. Therefore after
11:58one year, after one year,
12:03population of, population of rabbits.
12:09is approximately because this is
12:13all approximately guys at 2837
12:17and Foxes is approximately 58.
12:29Okay, so that's that's it
12:31guys. That's the main main
12:32thing I wanted you to
12:34to get from this understand
12:37what's happening with pred
12:37predator prey models understand this
12:41this coupled system understand how
12:44to use the order's or
12:48there's equation or order's method
12:50equation in your from the
12:53formula booklet and apply it
12:54to this situation and then
12:56kind of most importantly and
12:58arguably the most difficult part
13:00of the of the question
13:01is to get this it's
13:04difficult I mean it's fine
13:05once once
13:06It's fine once you know
13:08how to do it to
13:09fill it in but in
13:10the middle of an exam
13:12high pressure situation With that
13:15really annoying calculator. Yeah, you
13:17can see that without practicing
13:18this a lot that could
13:20become problematic So I would
13:22recommend doing like about 10
13:24of these just to make
13:25sure you're really Yeah, no
13:28how to do it. Okay,
13:30that's it guys. I'll see
13:32you soon for the next
13:33video
13:34Thank you.