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Let's modify the logistic differential equation of Example 1 as follows:

$ \frac {dP}{dt} = 0.08P ( 1 - \frac {P}{1000}) - 15 $

(a) Suppose $ P(t) $ represents a fish population at time $ t, $ where $ t $ is measured in weeks. Explain the meaning of the final term in the equation (-15).

(b) Draw a direction field for this differential equation.

(c) What are the equilibrium solutions?

(d) Use the direction field to sketch several solution curves. Describe what happens to the fish population for various initial populations.

(e) Solve this differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial populations 200 and 300. Graph the solutions and compare with your sketches in part (d).

a) 15 leave population every week

b)

c) $P=250,750$

d)

e) $$P(t)=\frac{250\left(3 e^{t / 25}-11\right)}{e^{t / 25}-11} \text { and } P(t)=\frac{750\left(e^{t / 25}+3\right)}{e^{t / 25}+9}$$

Differential Equations

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Zcz Z.

April 17, 2019

Zcz Z.

April 17, 2019

Missouri State University

Campbell University

Idaho State University

All right, so let's have a look at Question nineteen from Sexual or Chapter nine Raid's. Let's modify the logistic, the French for equation of example. One. This falls. So we are to consider this equation. DP over dt because Point I p Times one minus p over thousand minus fifty. The first question reads. So let's suppose it pfft represents a fish population of time. Tea with he's married Weeks explained the meaning ofthe the final term of the equation negative. So this leg of the fifteen here represents a decrease or fifteen fish for a unit of time, which is weeks that's constant throughout the evolution. So one way to interpret this is to think that, say, there's a fisherman who will care fifteen fish every week, no matter what, no matter how big or small. The population of fish in the lake, let's say is you always catch fifteen fish every week. That's why I have a constant night of fifteen. Turn here in the equation for the derivative after fish population proof, very time. So it's a simple simple is that kind of a Now I don't b s is to draw a direction field for this equation. Now, before we do that, it's a good idea to go ahead and factor the left hand side of this threats by the right hand side of disagree, it means she'll express it as a product of two terms. Ah, because that will make the whole question a lot easier. So how do we do there? Well, it could. He should start by distributing this product. Right? Because if you do that, we're gonna end up with a quadratic equation. All right? It's gonna look like it don't look like, uh, point Oh, a B minus point will be spread over a thousand and fifty, and so that's a quadratic equation. Can use the quadratic formula to factor it. So I'm going to go ahead on DH, skip that arithmetic and just tell you if you use the quadratic formula to find the roots, you will find the roots to one hundred and fifty, seven hundred and fifty. So if your Route two hundred fifty and seven hundred fifty factoring looks like this and you have a coefficient here at the beginning which is the coefficient ofthe B squared in your equation, which in our case is negative Final eight divided by a thousand. So that will give us names. You're a zero zero zero rate. That's our equation factor. Did just He's the quadratic formula to find these roots. And then at the coefficient in the front. So now that we have factored our equation, it becomes a whole lot simpler to draw. Ah, direction Field is Ah, let's add a new when they're here. So let's imagine a graph. Sorry. Imagine the Griff here, where this axis represents the time Gladys and weeks and this backs is represents the fish population. Right. So we have, you know, that at two hundred fifty and that at seven hundred and fifty this slow zero, that's what the equation we have previously thousands. So at both two hundred and fifty and seven hundred fifty, I can just draw as zero slow controlled this more accurately. All right? He has your slope. And also for seven. Fifty, you have zero slope. Something like that. Now, what about elsewhere? Well, noticed that this is a quadratic equation with a negative coefficient. Right? So what? The quadratic equations of negative coefficients looks like they looked like downward pointing prevalence. So between the two routes. You have something positive. And outside of both fruits, you have something negative. So, Francis here, When you go up from seven. Fifty, you're going outside of the roots. So you're going to start to have more and more negative slopes. So you're slow starts around here. Negative, but not very large. Say, and then it starts to become more and more native. The Freddy you go on any start negative, Not very large. And it becomes on negative. They're not very large. And you get more than one something like this. That's how parappa this work. And the same thing works here under two. Fifty. Or you're also going away from the root. So you start over negative slope. That's not very large. And that's so gross is you go through yours. No, it doesn't. So let's go look something like that. Now, here between two hundred fifteen seven hundred fifty, you're gonna have positive slopes, right? So say, starting here to fifty when you go a bit above, you gotta have a positive slope that's also not very large. Try to draw this less light, and then your slopes will begin to grow. Right. But then when you get closer to seven fifty day with decrees again, because they have to be zero when you get your seven fifty. So it works something like that, right? You have slopes like grow a little bit, then a decrease back down where they're always pasta. Here, write something like this at So it's not a very good drawing, but help pick. Explanation Mason. So this is some what what the but the direction field would look like, remembering that at seven fifteen to fifty, you have zero smoke. All right, so for birth sea Where are the equalization solutions? So equilibrium is exactly when the fish populations in equilibrium, which means it doesn't change. So that's what we've just figured out. It's seven fifty to fifty. That's when you get a constant line for the amount of fish. So if you start with either seven fifty or two fifty fish, then you'LL always have that state. Now it'LL never change those air. The equilibrium solution. Then party ass is. She used the direction few to sketch several solution curves, so let's have a look at what happens. So supposing start off a population of fish that's less than two fifty say that's supposed to start here now. The slopes are negative cigarettes that you go down, and in fact, as you go down the slopes get more and more negative. So you going to go down quicker and quicker and quicker, quick, and you're going to reach zero in a finer stock. Since you're your fish population is decreasing at a steadily increasingly negative wrecked it. So any fish population below two fifty is going to decrease rapidly and reach zero in a final time. Now what happens if you start off a fish population that's the above two fifty? Well, here the slopes are increasing, so you're going to increase. But as you get closer to seven fifty, your population starts. The smoke. Such decrease and say's you're increase becomes slower and slower and slower and slower. In fact, they will have seven fifty as a horizontal acid, so your fish population will steadily increase, but at a slower and slower rate and will attend at infinity towards seven fifty without actually ever reaching seven fifty. So that's what happens if it's top of the first population that's above two fifty, but smaller than seven fifty now, what happens if you start off a fish population that's greater than seven. Fifty? Well, saving say, start here or your slopes are negative. So it's going to you start to decrease, and it will keep increasing, but your slopes again in closing closing zero. So you're decrease will get slower and slower and slower again. You have a horizontal ass until at seven. Fifty, but you're never actually going to reach fifty. So if you start with a population of about fifty, it decreases, but at a slower and slower rate. And it tends towards some fifty and finish without ever actually screeching seven fifteen, at least in this small off course. So that's a heuristic sketch off what the solution curves looking like look like, according to our drawing off, the director of Shield. All right. So finally, Bar E asked us to solve this the French equation Explicitly. This one here, this he could have this. All right, so how do we solve this explicitly? Well, this is a separate ble equation, so we can write it thus just right. We can put our peace on one side of the equation off together, off the DP and we can put everything else of the other side together with the DDT, So it'LL look something like this. Over here, we have B and will be provided by me. Linus, you're fifty seven. Fifty. Right? And over there we have this proficient Franz DT, a hearse moving things from one side to the other. Now, the idea is that we want you to take the girl. Sorry. The idea is that we want to take the integral of those sites, just solve our friendship, create That's how separable equations work. But to take the integral off this left hand side, we have to right this fracture as a sub off two fractions is in the method of partial fractions. Let's quickly recall how that's done. They have this fraction here. Sorry. You have this fraction here. One over. Be right. It's your fifty. Sometimes the line a seven. You want to write that at some constant over feeling, Yusuf. If it worse so other constant over feeling a certain fish. No, hide that Well, let's simplify this equation by multiplying through Bye. This the nominated If we multiply through by this, we get one equals. Now. Here. You already have a few minutes. You fifty. So you're only left even I have a seven fifty and he already have a people in seven fifty. So you only left for Beaver and it's your fifty. So Carly, figure a V where we can just plug in specific Clyde is for P and we get equations which we could himself. For instance, if we plug in because seven. Fifty, then seven. Fifty seven, fifty eight zero. So the coefficient of a here dies off then seven fifty minus two. Fifty is five hundred. So your occasional reed one equals eight and zero for speed sci fi ve And this means that be because one over five and similarly you can figure out by plucking and, uh, two fifty. Instead, the exact same thing will happen, except that this time we will have two. Fifty minus seven fifty. That's negative. Five. So when you work out a way, you would get Mega Kuzmin five hundred. So that's how the method of partial fractions work. X don't. So now let's go back to our integral. So we want you integrate at this integral ahead Here on the left hand side, we can write this as, uh, enter Grew off. So right, We have a over B minus two fifty. Where is negative? Another five hundred. Pull us over fifty first. And then he was one of five hundred Brian Speer, all of that. Now that these Constance one of the five hundred here and here, I can just factor out. So I'm gonna have, well, five hundred here first and then the one over even issue fifty. The integral of there is just ah, log. So we have just log off the absolute value off people's shit. Don't forget the night that sign we had here, right? Because of this night. The sign on the one of the five hundred and then But the indigo off one of the people. And seven fifty. That's his drug B minus seven. Fifty. All right, plus the constant. But this constant will show up spec plus a concert here. So the other integral was easy. This is just the integral off constant. So the integral off so and the girl. But this blue intergroup ofthe night of Syria's is eight e. R. This is zero eight baked B. It's a mother constant. Olek. Yes. So it can equate this. We have obtained here that one. Can I quit this with this right? That's what we get by taking into go off both sides. So let's do that. So we get no that. Let's see. On one side we had a longer he went seven fifty Dryness Log off, even Sue fifty. Right? If you look back to this side, let's ignore the one of her five hundred for now. And because we're going to move it over to the other side by multiplying through by five hundred. So he had Lord be one seven fifty minus log minus two fifty equals. So since we're multiplying through by Constant, uh, five hundred, we had a class. See that? Because it's five hundred c that will eat crow. Ah, on the other side, we had five hundred times and this year's ears is age t first. Thanks. Now it can just finish multiplying this. So here on this side we would have our people in a seven thirteen planets Park Su fifty and the rats move this five hundred c over to the other side. So we have ah, five hundred times this work cell tonight O forty Ah, first five hundred B minus five hundred six. Now this favor thie minus five hundred. Sea is just some other constant. All right? He's just a non viteri constant. And see, just a lavatory costed. So if you could just give this another name, I'd say e So we can write this. We can write this as, uh, that's it. Who are the seven fifteen minus far Sufism? Because I go for a pee breast some constantly. I doesn't really matter that we will define it by five hundred orders. I don't know if you recall a log rules. A difference off launch is the log off the cautioned seiken. Right, this is log off. So, Christine being right is to first because But the prosecution take this by using a long cruise. No, What can I do? I can just to get rid of the law, I can just expert in shape both sides. And so what happens if expert in shape both sides of peace that this equation well on the left hand side would just get rid of the Lord? Be Afghanistan, fifty over. You know, Sue first the right hand side we get now. Let recorded an exponential off. A soul is the product of thie exponential. So first you have it's exploding. Show Grimes, you have support. Transfer was near, raised your capital. Now this theory should capital E. We can just meet again. It's a constant since Jesus and lavatory cost. So let's rename this constant cape and let's right. Finally, the B minus seven fifty over being with Sue fifty because a constant K fine be regular. This is what the formula for our solution looks like this. No. Okay, so now the question asked us to use initial populations of two hundred and three hundred to graph what the solutions. So that's sort of two hundred. So let me rewrite this here. We're looking keeping in mind. So I had the one seven fifty over B minus your fifteen because a constant me to the negative four. All right, so now let's suppose that we start with a zero okay, off two hundred. That's reason about these absolute values. Now, if I start over published of two acted and two hundred minus seven. Fifty is leg five. Fifty and two hundred minus two. Fifty is negative. Fifty. So both the top and the bottom are negative. And when I defined night by negative, I get a positive. So since this finger, I'm getting responsive. Anyway, I don't need the absolute violence sighing and I can just right without absolute fact is people in its own fifty performer give minus two fifty vehicles. Kay needed a night off for tea. Now let's figure out what the constant is by just plugging in two hundred. Right? So we plug in time equals zero time. It goes zero and a population equal to two hundred. So if Buzz we should plug that in tow, our equation we get here two hundred minus seven. Fifty over two hundred, my sou. Fifty four people. Okay, then here. I'LL have night of O four times zero zero and zero zero's Just what? So our case is thiss right? And if you work this math out, you will see that she get kay. It could happen. So they have k equals eleven. Then what can we do? So here we can write this as eleven isn't the whole forty. You know, Katie was eleven now. We haven't equation here with our peace on one side. And this in the other cell we can multiply through by B minus two fifty and then solve for Pete. This is just gonna work out to be a simple linear equation. Which MP? So you can solve this for B on. Do not be too much of your time. If the answer is B equal seven fifty miners twenty seven, fifty feet of the dragon for De decided by one of minus eleven. He didn't, for this is what you get as your answer. That's Blissett, Forman. And so if you graph this using our couch earlier and tried to that appear if you graph this is in your account, clear. A little practice at Syria. You started two hundred and then it would decrease rapidly to zero. So this point here is that so. It starts at two hundred and the crazies weapon reaches. Here you have two hundred and you can check numerically that it reaches zero at about thirty two. Thanks. The bridge is your but this is more or less what the graph of this will look like. You just use your calculator now, which is basically what we expected from Hard drone, right? A population of less than two fifty like two hundred who just decrease rapidly to zero and will be zeroing in on family life. So now that what let's do the math for what happens off population above two fifty, like three hundred? Well, we again start over formal of P five seven fifty over the two. Fifty. Because, Kay, I needed that Pegasus for the no. Uh, now we have to reason about the absolute Find his things carefully because no b zero, we're going to be easier to increase it. So if you put three hundred here, three hundred minus seven fifties night before happened, but three hundred minus your fifties fifty. So I get a night of over pasta, which is negative. So to really get the absolute value I need to flip the sign, right. So really gets something like, uh, negative. Meanwhile, seven. Fifty over him when sue fifty. Because, Kay, you know, forty. But to figure out what K is liken displayed in T equals zero and people's three hundred as before, this e to the whatever will become one. And I just have a formula for Kate. So if you do this, you will find this time Jake? Because Randall Yes, you'll find Keiko's night. Right? So given that cake was nine, I can write this. Oh, that's because mine needed the forty. All right? And then I have a linear equation here on one side. I have my piece on one side. I have my eating bananas over forty. I can multiply through by p minus you fifty and ourself this equation for B. And if you do this, you will get a phone for Be very similar to what we had before. But this time I get talent. Fifty plus three. Fifty. You did that. D all of that decided by one one Verse nine. Eat it. Oh, is your funnel for P to work out this simple Ah linear equation as before? So I should graph this using powerful here It's down here. What we have it would look something like this. Your population will start at three hundred at st three hundreds around here and grew It will change from cavity at some point and we'LL have seven fifty as a horizontal acid. So it will start at three hundred and it will have seven. Fifty as Horace until ass in here the uh, you'LL never actually reach seven fifty. If you work out the living off this expression of Steve goes to infinity, you will see that you indeed get seven. Fifty. Since this far here goes zero on this part. Here goes zero. So you only left of seven. Fifty over one to sell fish. So that's what the explicit equation for the solution looks like. And it behaves exactly as we had expected from our drawing off the directions for you. So that's it for this question. Thank you. Have a good thing.