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Population P obeys the logistic model: It satisfies the equation dP 0.15P for P > 0. dt 2700(a) The population is increasing when(b) The population is decreasing...

Question

Population P obeys the logistic model: It satisfies the equation dP 0.15P for P > 0. dt 2700(a) The population is increasing when(b) The population is decreasing when P

population P obeys the logistic model: It satisfies the equation dP 0.15P for P > 0. dt 2700 (a) The population is increasing when (b) The population is decreasing when P



Answers

The logistic equation describes the growth of a population $P$, where $t$ is measured in years. In each case, find (a) the carrying capacity of the population, (b) the size of the population when it is growing the fastest, and (c) the rate at which the population is growing when it is growing the fastest. $\frac{d P}{d t}=0.0008 P(700-P)$

Have a population growth rate function, p primary tea, and our objective is to find the pft at which this is maximized. So let's re express this is F of P is equal to two times p times the quantity seven minus to he that's going to equal 14 p minus for squared. This negative four right here tells us we have a downward at the peace board, says proble. The negative forces down proble, which means the turn around the point where the slope equals zero is indeed the maximum points. So for this, if we find the first derivative of F and set it equal to zero, we'll have her maximum. So F prime is 14 minus two times for is eight times p. I will set that equal to zero. That means eight p equals 14 which means P equals 14/8, which means our answers. The population is 1.75 or you could say one in 3/4. That's a population at which our growth rate is maximized

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

Blue If Why Service fines distinguish in then we have. Do I worry? My Ah, they don't eat. It is your house. Kill me in my the operation. So religion says do did you like water? DTs fire Do can you give one my But putting you and I need you and my people to Yes What I my wife and I just where upon needy is so we have a back problem. But be where in back the population will be going faster when the wind By limiting this myth that if d why? When a day its maximum population, they find a maximum We know the technical points off your ability. When do these initiation off on the implementation off while respecting people's eo When they put the diva Squire, Why are tedious far? People do? Yes, I m minus and minus Now here After putting, we have this video people do. Mm. I wasn't you. The 1st 2 never happened. I left any sin populations. Yeah, First this will not happen. They don't happen. Progresses it for me. This m minus two. Well, no. We have or has two. When? Well, so why? So the police is doing? When it is how it's done in the past. Only the population is growing faster that well and reaches. Uh, yes, this is that, sir.

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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