💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Get the answer to your homework problem.

Try Numerade Free for 7 Days

Like

Report

Logistic growth A population is modeled by the differential equation$$\frac{d N}{d t}=1.2 N\left(1-\frac{N}{4200}\right)$$where $N(t)$ is the number of individuals at time $t$ (measured in days).(a) For what values of $N$ is the population increasing?(b) For what values of $N$ is the population decreasing?(c) What are the equilibrium solutions?

$$P=4200 \text { or } P=0$$

09:28

Sinisa S.

Calculus 2 / BC

Chapter 7

Differential Equations

Section 1

Modeling with Differential Equations

Campbell University

Harvey Mudd College

University of Nottingham

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

05:44

A population is modeled by…

02:52

Logistic growth: $P(t)=\fr…

02:32

Level $2 :$ Application/An…

02:35

In Exercises $23-26,$ the …

02:30

38:03

A population grows accordi…

01:59

Using a Logistic Different…

01:23

01:20

The logistic differential …

02:40

Suppose that a population …

all right. So today we're gonna try to find the values off end, where the population is growing, where it's decreasing and where it's constant. So first of all N is the number of the population. And this is the equation that governs the growth or to decrease or the how. If if the population remains constant, well, now that's thing. Dion over DT or D N Y D t represents the rate of change of the population with respected time now logically, if the rate of change is positive for greater than zero, that means we have a population growth. Now, if we look right here we have Dan by DT is less than zero. So that means that the population is decreasing. And then finally, if we have the end by DT is equal to zero. That means that there is no growth or decay now. The first thing we have to do is we have to look at the given equation. So this is the equation that they give us right here, and then we have to know set. We have to see what values offend. It's greater than zero or less than zero equal to zero So first of all, I'm going to start with the population growth, so I'm gonna make it greater than zero now, since it's greater than zero. First of all, I'm gonna do some algebraic manipulation to this inequality. The first step that I'm gonna do is I'm gonna divide by 1.2. So when I divide by one point to this goes away so again you're gonna divide on both sides now, zero divided by 1.2 is just 1.2 and then now when we go to the next, so we got rid of that 1.2. But now we also have an issue. Here we have only one variable here. We have a variable and a fraction. So we do what we usually do from normal numbers, we come up with a common divisor are a common denominator. In this case, this has a 4200 the divisor of N is just form. So I'm gonna multiply both the numerator and the denominator by 4200 again, this doesn't really change. And because 4200 divided by 4200 is just one. So this is like a sneaky one. All right, so you really changed nothing. So now you get 4200 and minus and squared, divided by 4200 and then you can get rid of that 4200 the denominator by cross small to play. And then here in the last step. What I did is I took an end out as a common factor. And now I'm gonna try to solve this normally, just using algebra. And if you recall to solve inequalities, what you do is, first of all, what makes this equal to zero this side right here we know if we plug in and equals to zero this side, all of it will become zero. So that's how he chose his first point. And the second point is, so how do I make so 4200 minus what number is gonna give you zero? Well, that's just 4200. So these are two points that make this and this, uh, this side of the equation the left hand side, this side equal to zero. But we don't want it to equal zero. We wanted to be creative and we're going to choose. So we So we see that these two numbers divide our number line into three parts. And the idea is, we're gonna choose some random points from these three different parts of our number line. I'm going to see what part makes this equation makes this left hand side equal to zero. Well, the first thing we can do is we're gonna realize that I really don't need to include this portion after zero. Why? Well, because and physically represents the number of the population. Remember, you can't have negative people, so we can easily just give rid of this part. All right, so now I'm gonna randomly choose a point. Let's start with this area right here. I'm gonna randomly choose a point. I'm gonna choose my easiest points for this area chosen number one. So that's set and any call to one. So now this becomes one times 4200 minus one. Well, is this greater than zero? Let's see. So this becomes one times. Let me just make this look a bit better. So this is one times are 4200 minus one is 4000 199 and this is obviously greater than zero. So now I just So I know that my solution is very here. This portion gives me a population, gives me population growth. So I have population growth when N is an element off or wood. You fixed that? So an end belongs to the set from zero all the way to 4200. All right, great. Now again, we're gonna help the same graph. But now we're gonna seem number line. But now we're going to choose a point. So we know. First of all, we're gonna neglect this part. So this part is neglected. This part is positive. So we're gonna try a number from right here after 4200. So let's try a number like 4201. So I plugged in 4201 So this is just 4201 200 animals. And now what's 4200 minus 4201? Well, that's kiss dated one, and that is definitely less than zero. So we shade this whole region right here, and then we say that four part for the for decay or decrease. We just need our end to be from that seal associated region. Our region a region is 4200 all the way to infinity. All right, great. Now finally, the last problem. We already solved that. When do we have When does it equal to zero? Well equals to zero and see. So this times this should equal zero. Well, if n is equal to zero I directly so your multiple zero times whatever number we get here. Still zero. So this is one solution and then finally So this window's 4200 minus and people zero you can do it logically was 208,200 minus one number gives you zero well, logically is going to be 4200 or you could solve it algebraic lee. And then you get it thes two points. You don't have population growth or population decay. So this is an equilibrium thes air. They call it the influence

Numerade Educator

Missouri State University

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

A population is modeled by the differential equation

$ \frac {dP}{d…

Logistic growth: $P(t)=\frac{C}{1+a e^{-k t}}$For populations that exhib…

Level $2 :$ Application/AnalysisAccording to the logistic growth equatio…

In Exercises $23-26,$ the logistic equation describes the growth of a popula…

A population grows according to the given logistic equation, where $ t $ is …

Using a Logistic Differential Equation In Exercises 55and 56, the logist…

The logistic differential equation Suppose that the per capita growth rate o…

Suppose that a population y grows according to the logistic model given by F…