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Problem 12 Easy Difficulty

In Exercise 10 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows:
$ \frac {dA}{dt} = 2A(1 - 0.0001A) - 0.01AL $
$ \frac {dL}{dt} = -0.5L + 0.0001AL $
(a) In the absence of ladybugs, what does the model predict about the aphids?
(b) Find the equilibrium solutions?
(c) Find an expression for $ dL/dA. $
(d) Use a computer algebra system to draw a direction field for the differential equation in part (c). Then use the direction field to sketch a phase portrait. What do the phase trajectories have in common?
(e) Suppose that at time $ t = 0 $ there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change.
(f) Use part (e) to make rough sketches of the aphid and ladybug populations as functions of $ t. $ How are the graphs related to each other?

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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 9

Differential Equations

Section 6

Predator-Prey Systems

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Amer A.

November 22, 2020

The Lotka-Volterra system predator-prey system modeling the populations of aphids and ladybugs is given by the below equations

Hahhaha P.

November 21, 2020

The Lotka-Volterra system predator-prey system modeling the populations of aphids and ladybugs is given by the below equations ?? ?? = 3? ? 0.01?? ?? ?? = ?0.6? + 0.0001 ?? a. Compute the equilibrium solutions and explain their significance. [2 marks] b.

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Video Transcript

Hey guys. So today I'm gonna be talking about how we would go about analyzing lots of terror system which is just a very popular predator prey model system. So let us consider two populations X. and Y say we have dx over D. T. Equal to just some function. I'm just dependent on action. Y. And then Dy over DT which is another function let's say G. That's why. And these are both going to be dependent on X. And Y. Because um it's a predator price. So the populations are going to be interacting with each other. Okay, so for our first step, we want to find the equilibrium. So how we do this is just setting are directed events Equal to zero. So we have um zero equal to S. Of X. Y. People in G. Of X. Ply and we want these when these are simultaneously equal to zero. Um So we want a software plane let's say XDR why star this will be our constant value Since the door but it was 34- zero. And what this does is tell us what the system will support of the number of individuals in each population. Um Okay, so for a second step we want to try to get an expression for Dy over Diaz. Um And what this will tell us is how when population relates to the other, rather the change in one population relates to the other. And we can apply the train will define this. So we know that Dy DT is equal to D by the x. Friends detox every DT. And so we want to solve for this just by dividing the dy DT by dx over D. T. Um And then from here um we could try to um get what we call a directional a field direction field for this partial differential equation. The dy over dx. So we can use any type of computer algebra system personally, I would use matt lab um but you can use python or any other type of uh computer algebra system. So um what we're trying to do is get um vector field where we get vectors that point in the direction that they change. So the change in X change and why you'll get back to the field. So let's say for our system we have trajectories that go like this. So we kind of just have, he was elliptical looking directions, I would say. So you can see when experts to increase why it's these kind of study for a little bit and then it both increased at the same time. However, there is a certain point that X reaches where it starts to decrease and why such to decrease as well. So we kind of just want to get in that pattern of trying to are analyzed uh the patterns and the way that the arrows are pointing, try to understand how these populations, the change relates to the other. Uh directly. Um So let's say from here, we want to consider um from an initial population, the trajectory of um of starting from that point. So let's say we have like our X. Y. And then we have a point X. Zero Y. Zero. So how will we how we would try to get the trajectory from this? We would use our directional field let's say this corresponds to this point here and then just follow the direction that the arrows go. So we could just you have something like this, some type of he looked at school. Yeah. And then from here we can also side note we can use this uh And also so that in matt lab and but the trajectory for that starting from an initial point, you just define it. And then from here we want to look at is how each population changes it with respect to time. So for um talking about a predator prey system we most likely have something of the sort. So yeah. Um So let's say I suppose up this off leads and then for something like that. So um we can also use matt lab to provide these absolution for Y and X. With respect to time. But if we want to try to use the directional field we just kind of see the pattern of um how the changes relate to each other, what would happen to X and Y separately with respect to time. So um if we're thinking of a predator prey system, this makes sense that we would have oscillations because each one depends on the other and then there's only a certain amount that one could provide for the other. So for example if we're considering the prey um what they need pray um the predator needs to pray the feed on. So while they're feeding on the prey they're going to be increasing their population. However the price will be decreasing because the predators obviously getting rid of that population are these proportions. Um Whereas if there um you know there's only a limited amount our max value, so after a certain point, no cap off and then the prey will start increasing rather. So yeah this is kind of just um the general the general approach of how of what we could use to try to analyze predator uh system model. Yeah, that is all I have for today.

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