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Problem 3 Medium Difficulty

The system of differential equations
$ \frac {dx}{dt} = 0.5x - 0.004x^2 - 0.001xy $
$ \frac {dy}{dt} = 0.4y - 0.001y^2 - 0.002xy $
is a model for the populations of two species.
(a) Does the model describe cooperation, or competition, or a predator-prey relationship?
(b) Find the equilibrium solutions and explain their significance.

Answer

(A). $$\begin{array}{l}
d x / d t=0.5 x-0.004 x^{2}-0.001 x y=0.5 x(1-x / 125)-0.001 x y \\
d y / d t=0.4 y-0.001 y^{2}-0.002 x y=0.4 y(1-y / 400)-0.002 x y
\end{array}$$
(B). There are four Equilibrium Solutions:
(1)$$x=0$ and $y=0$$
(2) $$x=0 \text { and } 400-y-2 x=0 \Rightarrow x=0 \text { and } y=400$$
(3)$$500-4 x-y=0 \text { and } y=0 \Rightarrow x=125 \text { and } y=0$$
(4) $500-4 x-y=0$ and $400-y-2 x=0 \Rightarrow y=500-4 x$ and $y=400-2 x \Rightarrow 500-4 x=400-2 x \Rightarrow$ $100=2 x \Rightarrow x=50$ and $y=300$

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Video Transcript

Hello. This is ah problem three. So are for their French equation. Is do you? Why? DT musical to 0.41 Mine is your poise, your one Why square when is your point? Shows your to next? Why? And we corrected as your point for Why times one minus Why I'm on a 0.8 x. Do you wrote it by 400. And this is called the logistic model. Okay. No Dereck City Could we written us 0.5 x times war minus X mines 0.25 Why do you want it by 125? And this temple rule is called the competition Because, um, when they interact, it decreases how the wife species, um is changing and is the same thing for the ec species. Okay, so, no, to find the equally room we could, uh, once before we could ah dio makes is equal to zero. So then is that the ec species is dead. Um and then we call it, um so we know that So next species IHS Dad, we still need a figure would Why is so we do Do you accept you? We're going to get 05 times You're one is your coin. When was your corn? 004 times your squared. Both of these Air zero, then minus 0.1 time. Zero times. Why the house stapler Zero. So this is good. We still have here. What is so no Do you want DT Has she could as your point for times. Why? And then we're gonna have one of minus Why my 0.8 times your dueted boy for 100. Then we know this, that this is zero. So remember, hostage reserve duty. So it was your also That is true. Then in order for this to be zero, um, then why has Chief Little 400? So that's it. So excessively zero. And why has secret of 400 nor for there to be equally room. Okay, so no way. Move on to part two. Um, so we kill it while you got zero. So the worst species is dead. So we had to figure out what X is. So we dio de axity history goto 0.5 x times one my sex minus 0.25 Time zero When we have to divide by 125 and this whole thing has to be zero. Uh, so, uh, this is you already? So, in this case, look, a few out that X has to be 125 for this whole thing to me. Is Europe okay? So, no, we were on to. So for relaxes a 25. There's just were for with do, why did tears well, or get sure a point for time Zero times. One line is euro minus 0.8 times 125 and then divided by 400. And we're going to see that this is gonna be zero. Uh, because this is gonna be we're here. It's gonna be 400. Um, Okay, So there's that, um, market. So no for problem. Sorry. Our part three. We could let both species We don't So species art. Yes. So let's verify that. So do you. Wind it t is equal. Is your court four times here? Many times. One times. Yeah. Okay. Uh, 1100.98 times. Seiver Hall or 400. Ah, And it's gonna be close euro journey. We do the same thing with your ex city. We're going to get zero point fine time. Zero times 110 Mine is your 1100.25 times. You delighted by 100 25 and this is also going to be zero. So we also reach equally room. Uh, the tricky one here. There's number four. The other way. Wouldn't push him or not. Zero. So first things first. So we could, um, from D y d t. All right. So it could just take a look at it. Uh, your entity. So let's take a look at this part right here. That part. Uh, we could factor out, Uh, you know, why wonder why and move this one. This one right here, to the right setting it equals zero. So if you were to do that, your gear zero point is euros. Your chew max B equals your point for when is your point su sure one. Um, And then that happens. Because you could factor out The Y says we said before ourselves, Europe, What is your shoes? You one away. It's gonna equals your appointment. Zeros your to it's why. And notice that, uh, this was right here much. So that's how you that. That's how we get this part right here. Okay. Ah, we move on to the second equation, so Okay, so let's go back again for we'll have the axity. Still originals better. Uh, so she's right down. Uh, the X 30 is in a Kodak's times. Your coin five minus your 50.4 x. Mine is your points. Euros. EUR one x y said it equal to zero. So we're gonna move this one to the right, and I have already factored out the X. Okay, so then no, um what could just rewrite it? So next time. Zero point fine. Minus your points. Roots 04 times X is gonna equals your for years. You're one. Why sewn owners in the rand exes to each other So we could say that you're born Joe jher. 11 is equal to 0.5 minus 0.4 Knicks. Okay, so, no, we have two equations. So we have this one here, and then we have the one that we just, uh, broke. Now, uh, which is this for? And actually, yeah. Okay, so the one that I'm talking about is actually going to be one. It's how in their equal to each other, but in the form no, we want it's this one. Okay, so you seen those two? Now we have a system of equations, so we're gonna rewrite them, like so. So I have 0.2 x z photo. Zero point for mine is your appointments. Your was your one. Why? And then we're in the right. Down 0.4 X is equal is euro 0.5 mine syrup or ju 01 way the room will supply the first equation by negative number and even oversupply. This you're gonna get you're a poll shows you're too x because 0.0 for X minus your poisoners air 0.2 x 00 0.2 x and then you will be 0.5 minutes point for which will be what Europe one. And then, uh, these will cancel out because it's always gonna be turning to positive. So you just salt. Um, And then he worked to solve this one. You're gonna get X is equal of 50. You saw this? Um all right, so then we still need to figure out what a wire so we could just plug it in and to year. So I'm gonna sell for Why first? So we do that. We're gonna get why is equal to 0.5 minus 0.4 And now we're gonna plug the accident. Right? So 50 and then it says we're solving for around, right? We're gonna deal. I'd buy this number here. Says it means your 0.1 was a simple fibers were going to get 300. So the fourth way, they could reach a cooler room iss with, um why equaling 300 and X equaling 50 and we're time.