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# A phase trajectory is shown for population of rabbits $(R)$ and foxe $(F).$(a) Describe how each populations changes as time goes by.(b) Use your description to make a rough sketch of the graphs of $R$ and $F$ as functions of time.

## (A) At $t=0,$ there are about 300 rabbits and 100 foxes. At $t=t_{1},$ the numberof foxes reaches a minimum of about 20 while the number of rabbits is about $1000 .$ At $t=t_{2},$ the number of rabbits reaches a maximum of about $2400,$ while the number of foxes rebounds to $100 .$ At $t=t_{3},$ the number of rabbits decreases to about 1000 and the number of foxes reaches a maximum of about $315 .$ As $t$ increases, the number of foxes decreases greatly to $100,$ and the number of rabbits decreases to 300 (the initialpopulations), and the cycle starts again. (B)

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Differential Equations

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##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

Hello. Ron does is probably fine. His party. We're talking about rabbits and crosses. So first we wanted, like a, uh, calling from that. So because they were her, So is going to do the rabbits Hawks is okay, and we're gonna start at zero. We're going todo 400 hungered 1200. Okay, so there. Yeah, and 1600 to does. Okay, show. And the waxes hold me started. 100. That 100. 300. That's just label 20 here usual later on. And that wasn't so birthday. Okay, so first, uh, when times a reserve where it's happening. So when times a zero, there are 300 rabbits and and 100 foxes. So then that would be What about so? Well, look at it. Right. So 300 rabbits. So So, like there and 100 foxes. Something with them is going to be t 0 to 0. Okay. And a T is equal to its a 21 We're gonna have 1000 ravage. So I look at it out here and 20 foxes. So I would say about their So this is going to re are to reason corner tee. Okay. Um and then let's say but another time. Jesus. Go to teach you. They're going to read 2400 province. So they'll be you're on here somewhere. And 100 hawks, So just put their own there. Okay, Um and then at another time, say, t sequel to because it was 20 to 2. Um, TZ goto t three. Let's say they're going to be 1000 rabbits. So we're going back. We hear, um, and 315 for So that would be ago. Here. Okay. Show one thing to know, Um, when at this point. So this point is going to be when they are, uh, 20 foxes. So that part here, 20 foxes. So that is the minimum source rate down minimum. Fox. Is that that corn and then, ah, the maximum amount of rabbits is in every at this point. Right here. Cheese. You to you, too. Show this is going to me. Breath mints. Okay, so now we need to connect to dust. So we start. Well, it is so anyone, but we'll just start here. You know what? Over soon. Thank you. Go. Oh, circles, but and he goes back to the first time. Okay, so, um So then what happens? Right? So we could write down last time. Increases the number Ah ha. CSIS increases to 100. When'd the rabbits the crease to 300. Okay, so I just picked on this was the three of the here. She's 33 so I just pig song. Four points and then from there, So apartment? Ah, yes. Could pick the points. Um, so for part B now, with what we have, they want us to draw some grants about how the rabbits change. Not well respected time. So wow, it's gonna be time Squint to the your XX iss and then or y axes air going to be the rabbits. So we'll start 300. That is starting pulling. And then we'll tow a dozen and then 2400. You could see isn't a drunk skill just so we could get a sense of how this is going to So was started. 300. And then we move to 2400. Reached that point. And then now we move back to 300 right? And they were just gonna keep going, right? So it's gonna keep possibly, uh, that's for rabbits. Okay. Now the boxes. Similar. So time XX is started zero and just kept going from there. And then her fox is going to be 20 100 300 And the hunter teen. Okay, Team Fox's OK, so in this case, change the coat. We started 100. All right was started here, and then we go down to 200 20 actually. Excuse me, and then we go up to 315 and this is just gonna keep going. So go on. Don't 20 years again and they go back now cares.

California State University - Long Beach

#### Topics

Differential Equations

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