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Point) A population obeys the logistic model: It satisfies the equation dP P(11 - P) for P > 0. 1100(a) The population is increasing when(b) The population is de...

Question

Point) A population obeys the logistic model: It satisfies the equation dP P(11 - P) for P > 0. 1100(a) The population is increasing when(b) The population is decreasing when P >

point) A population obeys the logistic model: It satisfies the equation dP P(11 - P) for P > 0. 1100 (a) The population is increasing when (b) The population is decreasing when P >



Answers

Suppose that a population grows according to the logistic equation $p^{\prime}(t)=2 p(t)[7-2 p(t)] .$ Find the population at which the population growth rate is a maximum.

Suppose DP OVER DT describes the growth of a population P. Now we recall that the logistic differential equation is given by DP Over DT That's equal to k times p times m minus P. And in here M is the carrying capacity. So for D P over DT which is equal to 0.008 times p Times 700 -2. The carrying capacity is 700. Since that is our AMP in this equation for part B. Since our DP over DT if we expand it as equal to 0.56 p minus 0.8 P squared, which is an upside down parabola and we'll grow the fastest when it reaches maximum. Then since the vertex of this parable is the maximum we say that DP over DT grows fastest at its vertex which is given by P which is equal to negative B over two a in which B as the coefficient of P and is the coefficient of P squared. So based on what we have, this is equal to negative of 0.56. All over two times negative 0.008. And this is equal to 350. And for party, since you already have the value sp when DP over DT grows fastest, then the rate must be given by DP over DT That's equal to 0.56 times 3 50 -0.0008 Times The Square of 3, 15. And this is equal to 98

Suppose you have DP over DT as the growth rate of a population P. Now we recall that the logistic differential equation is given by DP OVER DT That's equal to k times p times a minus P in which M is the carrying capacity. So if that's the case, yes, DP over DT is equal to 0.002 times p Times 1200 -2 than the carrying capacity is 1200. Since this is the value of em in the formula. Now for part B Since DP Over DT if we expand it Is equal to 0.24 p -0.0002 piece squared is an upside down parabola and grows fastest at its maximum value. Then DP Over DT grows fastest at its vertex and that will be P which is equal to negative B over two A in which B is equal to the coefficient of P that's 0.24, that's divided by twice the coefficient of p squared, That's negative 0.0002. And so this is equal to 600. And since you already know that the rate of growth grows fastest at 600, then the rate must be defined by DP OVER DT and that's equal to 0.002 Times 600 times 1200 -600. That's equal to 72

All right, Question 56. We're gonna prove that, given our logistic function here, uh, that C minus p of tea divided by P over tea will C minus the initial piece. That's p of zero, um, tears. Initial B p. M zero times e to the negative. BT Right. So, um, let's go ahead. Start approaching this thing right now. Um, So I'm gonna simplify this right side. Um, So I'm gonna divide both ese terms here in here by P F T's. I get sea over P L t. Um, minus one equals, this is gonna be C over. Piece of zero over. Piece of zero, minus warm, some to begin. I'm dividing both of these five piece of zero times e to the t. All right, so from there, um, see, let's plug in our formula for P of tea, and so we ended. Kidding. Um, see, over C divided by one plus a e to negative bt, um minus one equals. Just leave this side for now. We'll play with that a little bit later. When we moved by by the reciprocal here, we just end up getting one plus a e to the negative B two, um, minus one. Is this this will cancel this move up to the top, Which equals, um, a me the negative Bt Okay, so that means that here's the negative. Here's the E to the negative dt here. Here's Annette. Need to negative Petey here. Which means that this a year this a year has to equal this this trying to show that this is a all right. So, um, first I want to do is, uh, figure out a formula for peace of zero eso piece of zero. Go back. The original is p of zero. So that's gonna be C over one plus a eats the negative b times zero. So anything to the zero is equal toe one. So piece of zero is actually equal to see over one plus a, which is also piece of zero. This All right, so now that I've got this expression here or a piece of zero, I'm gonna take that. I'm gonna plug it in here where I'm trying to show that what's in the princes here is a But I'm sure that that's a Then both sides will be put into each other. Okay, so let's go ahead with that. And so I've got see over. And then I'm gonna plug in my piece of zero formula. So let's see. Oversee, uh, c c turned out like he's, um, see over one plus a Then I've got the minus one, just like on the other side. When you simplify this you nimal by by the reciprocal of this. So that Cesaire gonna cancel the one plus a is gonna move up on top, Signed getting one plus a minus one, which is equal to a So that means that this side this right here this expression, people, eh? So this side is also a e negative b t. And I've shown with the left side was the right side. I proved an algebraic plea that these

And new problems 69 70 together. And in fact, I'm not exactly sure why you basically solve 70 if you solve 69. But anyway, it's kind of strange how they ask these questions. So they told us to look at this logistic logistic model um which is why equals L all over one plus eight times either the minus Katie where L A and K. Are parameters that would allow you to fit this to data. The first thing they ask is what at what rate is um why increasing at time zero? So we can take a derivative of this? Yeah. And then set equals zero. And we get that the rate of increase is eight times K times L. All over one plus a squared so that zero we have some some rate of of increase in this logistics model. Whatever we're modeling, it's increasing Atik or zero starts off is increasing. Now they ask us what described how the rate of growth fairies with team. Well, what you can see here is that um Well one way, one thing you can do is you can take take the second derivative and it's ugly but you can do it and simplifying it. What you can see is there is there is an inflection point. So what happens is if you look at this, it starts off with some slope kind of grows, but then as then decays and it's gonna approach as T gets very large, it's gonna approach approach l right, this goes to L. When this gets very he gets large, this gets very small. Um And so approach L. So it it decays the rate increase. The rate starts um what starts off not zero and it starts to increase but then it starts to decrease. So the rate decays as T. Increases for large enough t. So, you know, it it does something like this. Um So and then they ask us what what time is the population drawing most rapidly? Well that's, we need to figure out with the second derivative, so the second derivative would be we would find extremists of the rate. And so when this is zero at this inflection point, so that the rate of growth is the greatest at the inflection point, um which happens to be a natural log of a over K. So that is where the rate of growth is the greatest, I hear. And you can kind of see that the slope here be the greatest. Now in problems 70 they show that the inflection point of light is um curvy the growth curve. An example nine occurs at the time given by pharma left eight. Well, this is Formula Eight. Um Think let me just go back here and take a look pretty sure that that's Formula Eight. So yeah I'm not sure exactly. You know, basically to solve to solve that's um You know also gives us the this also gives us the time where the rate of growth is the greatest. So again I'm not sure exactly what why they wanted us to show that. But again basically just need to take two derivatives of this um and saturday equal to zero. And we can see that you know, we get a place where zero where T. Is teak was the natural log of a all overcame. So set this to zero and so fatigued. So that's the inflection point of this logistics growth curve. So depending on these parameters we can adjust where that collection point occurs.


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