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