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Suppose that a population grows according to a logistic model with carrying capacity 6000 and $ k = 0.0015 $ per year.

(a) Write the logistic differential equation for these data.

(b) Draw a direction field (either by hand or with a computer algebra system). What does it tell you about the solution curves?

(c) Use the direction field to sketch the solution curves for initial populations of 1000, 2000, 4000, and 8000. What can you say about the concavity of these curves? What is the significance of the inflection points?

(d) Program a calculator or computer to use Euler's method with step size $ h = 1 $ to estimate the population after 50 years if the initial population is 1000.

(e) If the initial populations is 1000, write a formula for the population after $ t $ years . Use it to find the population after 50 years and compare with your estimate in part (d).

(f) Graph the solution in part (e) and compare with the solution curve you sketched in part (c).

a) $\frac{d P}{d t}=0.0015 P\left(1-\frac{P}{6000}\right)$

b) SEE GRAPH

c) Inflection occurs at a point If:

The graph is continuous at that point.

And the concavity of the graph reverses at that point.

d) By Euler method approximation, Population after 50 years is 1064

e) $P(t)=\frac{6000}{1+5 e^{-0.0015 t}} \quad P(50) \approx 1064$

f) SEE GRAPH

Differential Equations

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Missouri State University

Harvey Mudd College

Baylor University

Idaho State University

All right. So this problem asks us to suppose that a population grows according to a logistic model with carrying capacity six thousand and K equals zero point zero zero one five prayer. So part ay, they want you to write a logistic differential equation for this data. Well, this over here is our general form of this differential equation. Right. Well, we have that Kay's points here zero one five and M, which is the carrying capacity of six thousand. So this is how we can right this model here using the logistic differential equation. So there we go. Heartbeat now wants us to draw a direction field so you can do this by hand, or you can do it using the computer after system, I used the computer out of the system s o. Basically, all of these lines are telling you that if you had a point at any one of these points, that the way the vector is pointing is the direction that the curve would have to go. So you can see here that at six thousand, because we have basically a flatline here at six thousand. We see that we haven't equilibrium point here meaning that once the population hit six thousand, it's going to stay at six thousand. So if we start anywhere below six thousand, are functional increase until it hits up to six thousand. So taper off as it gets close to six thousand. On def, we start anywhere above six thousand, our population will decrease. I want skin taper off and converge to six thousand. We also see here that our population, which is is the vertical value here, does not depend on t meaning that if you pick any height, any population and he look all the way across well, all of those directions are pointing in the same direction. So if you move horizontally, that won't affect the direction here on the direction. All right, Part C asked us to draw curves on this direction field for initial populations of one thousand, two thousand, four thousand and eight thousand. So the one in red is with the initial population one thousand, and we can see here that the solution is Kong cave up for a little bit, which means that our population is increasing and increasing Richt and then around about ah, one thousand for tea about right here, we see that we have inflection point and then our solution looks Kong cave down after that. So our population is still increasing, but at a decreasing right, which makes sense because we need the population to taper off at six thousand at two thousand. It's concave up for a very short period of time and switches to con cape down once again to taper off as we get near six thousand. If we start with a population of four thousand, you can see that we're calm. Keep down everywhere, which means our population will increase at a decreasing right. If you look at eight thousand over here, you see that we're con cave up everywhere. So our population is decreasing because we need to taper off at six thousand but is decreasing slowly. It's it gets pretty steady and papers off at six thousand here. So these inflection points that we saw mainly with the initial population of one thousand, is when our population changes from increasing at an increasing rate to increasing at a decreasing right. All right now, part Dean asked us to use Oilers method so that you could program a calculator or a computer to use Oilers method with step size. Age equals one to estimate the population after fifty years if the initial population is one thousand. So, um, this formula over here is from section nine point two, I believe, and this is Oilers method in general. So if we have a differential equation, that's this Y part equals affects where they have X. We used tea in this problem and where they have why we used P. And so this is our initial condition right here. And this is how we're going to define our exes a CZ we keep going up. So for each value of X, we add the step size to it. In our case, that will be one on DH that gives us our new axe value. And so we get our new Why value Why? End is what we take the previous why value and add to it the step size So one in our case, times f of the previous point. So I have here some suit, a code for us. So where you want do this on the computer? You calculator. This is the general outline. So your first step? Well, we need to define our function, and that's going to be our logistic model that we did in part A. So you already know how to get this equation. Here are second step now. We need to input our initial start time and our initial population. So we wanted to start with the population of one thousand. So we have p zero equals one thousand here, and we're starting that at time. Zero. Next, we have to input our step size. So it's a Chico's one, and this an equals fifty here. It tells us that we're going to do this or into fifty steps, which makes sense because we need to get from your zero to your fifty to figure out with the population is after fifty years. All right, so apart for here is really important. And it's where we actually execute oil. Is that so? We're going to do a four loop and we're gonna do this fifty times, So from one to fifty. So the first part of this loop is Well, we need to figure out what f of teaser and P zero is, and I'm just going to call that a Well, then we can see our next population will approximate our next population with Oilers method in part be here. And so we're gonna take our initial population and add to it or step size times. Well, what we found in party here, then we're going to increase our time. So we're gonna say t one is the time we started off with, plus our step size, which is one in our case D here recently want to print the new time and the new population is so we can keep track of each step of the process here. And then Parts E and F here are redefining t zero and P zeros that we knew this loop again and again until we do it fifty times. And the other day, we need to end this program. So if you run this, you're going to see that after fifty steps, which means fifty years, the population well, this is going to spit out one thousand sixty four point zero three nine on and on and on. But we're talking about population here, so we want to talk about whole numbers. It doesn't make sense to talk about point zero three nine, but person so well, actually saying this gives us roughly one thousand sixty four as our answer here for using Oilers method. All right, so now Part E wants us to actually find a solution here if we have an initial population of one thousand. So that is what we're doing here. Thiss is our initial population. We're sorry. This is our solution to this differential equation and that comes from this section in the book. And so and once again, our carrying capacity is going to be six thousand. And this depends on this. A here. Well, a is going to be a carrying capacity minus our initial population, all divided by our initial population. So moving over here, we can compute that within the framework of this problem. So carrying capacity, this here is our M. And then here's AARP easier. That's one thousand and a ends of just being funny. So this here is our solution. It's a population function. So we get p of T equals six thousand over one plus five times E to the negative zero point zero zero one five t. And this part of the question also asked us to figure out what the population is. B after fifty years using this function so you can use your calculator here and you'LL see that once again we get roughly one thousand sixty four, which means that orders not it was pretty accurate. All right, so my late part f wants us to actually graph this function that we just found. And so this here on the left hand side, there, that is our graph this one over here of this function pft that we found in party. And so it also tapers off at six thousand. That's what this line is here. And you can see it has an inflection point roughly here where we go from Com cave up. So the population is increasing and increasing rate to Kaan cave down. So our function is still increasing, but at a decreasing right, because their populations why do taper off six thousand, which was our carrying capacity? Ah, and the graph I have here on the right is the graph that we drew using our direction field. So the red line on the graph on the right was also with the initial condition one thousand. And if we compare the red curve on the right to the red cover on the left, they look like they match up. So it all worked out