WEBVTT
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Hey, it's glorious that when you married here.
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So we have part A. We're gonna find the
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carrying capacity, which we see as equal to DP
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over. DT is equal to 0.5 p minus Sierra
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0.0 five p square. Now we get the logistic
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model equation for finding the carrying capacity, which is
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equal to 0.5 p. Times one minus 0.1 p
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. We got a DP over DP, which is
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equal to 0.5 See Time's one minus p over 100
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and we get Kate to be equal to 0.5 or
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M is equal to 100 now. For part B
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, we see that the slopes are closest to zero
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near P as equal to zero and P is equal
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to 100. The largest positive soaps is on P
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is equal to 50 and the largest negative is when
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P is near or above 1 50 We see that
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it's increasing for positive values, or pee that are
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less than 100 are decreasing when the P value is
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greater than 100. Next, we're going to do
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some graphing So we see that piece of zeros,
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the initial population. So all the solutions they're gonna
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be going to P is equal to 100. So
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when ti, as time goes infinity, we see
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the equilibrium solution of the differential equation. It's gonna
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be 100. So you see, that piece of
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zero is bigger than 100. These are decreasing and
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piece of zero is smaller than 100 are increasing.
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You know that the slopes will be the biggest one
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p of equal to 50 and between zero and 100
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um, the negative of the slope will increase as
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p goes to above 100. And when t is
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equal to zero and he is equal to 100.
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Since these are the equilibrium solutions, a slope will
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be close to zero. And then for the inflection
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points, we see that it is when Pete it
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is a P is equal to 50 for one piece
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of zero is equal to 20 or 40. So
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this is the graph of 20 looks. This is
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a rough sketch. So we do this, it's
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gonna be 20 and it just goes like this.
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And then this is for 40 you see that it's
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getting more of like a humpback shape and then we're
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doing 60 and 80 so compared to 40 60 it's
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more has two more of the curve here. The
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beginning and then 60 and 80 pretty much look similar
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, but when we hit 1 20 we go all
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the way from up and it kind of goes down
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now for part D. We know we know that
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DP over d t has to be equal to zero
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. So we get DPS over DT. We get
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this part from part, eh? 0.5 p minus
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0.0 five. P square is equal to through and
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we get P is equal to thorough and P is
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equal to 100. Since we get 0.5 over a
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0.5 when P is equal to zero, it means
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that that there were no populations in the beginning,
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so at any time there's none. And when P
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is equal to 100 it shows that the it will
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reach equilibrium, which means that the number of deaths
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are equal to the number of birds. When the
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population becomes 100