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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). Read the explanation

(B). SEE GRAPHS

Differential Equations

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hi, everyone. So today I'm going to show you how to use a face trajectory to understand a predator prey relationship. So on the vertical access, I have a fox population and on the horizontal access, I have a rabbit population, and I have these various points marked throughout the face trajectory. So let's go ahead and analyze those. This left most point is T equals zero. This is the point in time at which we start to observe this. These two populations interact with each other, and at this time it looks like there are about 400 rabbits and 100 foxes. And as time progresses to Teeples one, it looks like maybe there's about 1200 rabbits and the fox population has come way down to 10. So pause the video real quick and think about why this makes sense. All right, you've paused the video and you've given all your thought. And you've come to the correct conclusion that this makes sense because if there are less predators in the area, the prey population is going to have a chance to thrive. There's not as much competition. So this continues. Toe equals two and the rabbit population has increased all the way to 2000. But the fox population has begun to come back up to, let's say 1 50 And this makes sense for the same reason. If there are more if there's more prey available, the predator population is going to begin to thrive. There's more food source allowing them to do well in numbers. So this continues up to t equals three This top most point and the fox population is ah 300 has reduced the rob population down to a measly about 600. Okay, so at this point, the fox population has begun to become an unsustainable size. The population has dwindled all the way back down to wear the fox population at the beginning started to fall off. So if we come back down to t equals zero, that's now T equals four. The rabbit population is back down to 400 the fox population has come all the way back down toe 100. There was no enough prey to sustain a 300 fox population size. If we did another lap around here, T equals one would be t equals five with the same data t equals two would become t equal Sex anti equals three would become t equals stuff him. And we could do this as many times as we want it. And we'd see the same cyclical relationship may be important to know, maybe not right now. But as you get further and differential equations somewhere between T equals three and T equals zero. That's like the equilibrium population where there are just enough rabbits to sustain a decent fox population. But of course, animals do what they do. They reproduce until there isn't enough resources available to them. They die off, reproduce again and the cycle just continues. But equilibrium becomes an important idea as you progress to differential equations may be no important for this particular topic, but good thing to have in the back of your mind. So we have all of our data extracted from the space trajectory, and we can use this to construct two separate plots of the rabbit and fox populations through time. So I've started the up down here, so we have in rad the fox population versus time and in green, the robin population versus time. So let's mark the points AT T equals zero. It was 100 for the boxes. How T equals one, uh, 10 t equals Chu was 1 50 anti equal. Stree was 300. And as we did another lap, we got the same data for fourth You something? Yeah. All right. So let's draw that. The fox population decreased, then started to thrive in the presence of all the rabbits, then overstayed their welcome decreased again. And the cycle continues for the rob population t equals zero was 400 t equals one was 1200 t equals two, was 2000 anti equals D with 600 and the cycle continues. So in the absence of the foxes, three Robert population increased. And then there were too many foxes. The rabbit population dwindled. Then, when the fox population overstayed their welcome, they were allowed to thrive again, and the cycle continues. So what we've done is we've analyzed this face trajectory, toe, understand the relationship between these two populations through time, and then check that information to construct two separate plots of the population