The number of yeast cells in a laboratory culture increases...

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function $$ n=f(t)=\frac{a}{1+b e^{-0.7 t}} $$ where $t$ is measured in hours. At time $t=0$ the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of $a$ and $b$. According to this model, what happens to the yeast population in the long run?

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function
$$
n=f(t)=\frac{a}{1+b e^{-0.7 t}}
$$
where $t$ is measured in hours. At time $t=0$ the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of $a$ and $b$. According to this model, what happens to the yeast population in the long run?

Key Concepts

Logistic Growth Model

This concept describes population growth that starts out exponentially when resources are abundant but slows as limiting factors, such as resource depletion or competition, come into play, ultimately approaching a maximum sustainable value known as the carrying capacity. The logistic model is represented by a sigmoidal (S-shaped) curve and is widely used in biology and ecology to model growth dynamics.

Initial Conditions and Parameter Estimation

Initial conditions provide the starting values for a system, such as the initial population size and its rate of change, and are essential for accurately estimating the parameters of a model. In problems involving differential equations or nonlinear functions, these conditions allow one to determine constants that tailor the model to observed data or specific scenarios.

Asymptotic Behavior

Asymptotic behavior refers to the long-term trend of a function as the independent variable, in this case time, goes to infinity. In logistic growth models, this behavior is characterized by the population leveling off at a fixed value, which represents the carrying capacity, indicating that the population will stabilize in the long run.

Transcript

00:01 Okay, so for this problem we are given that we have a number of yee cells in a laboratory culture that are increasing rapidly.

00:11 And so this f of t is showing how rapidly or how the population is growing.

00:18 So this is our population.

00:20 And so we're given that f of 0 is equal to 20, f prime of 0 is equal to 10, and we're asked to find a and b given this information.

00:30 So let's start with f of 0 is equal to 20.

00:36 So that means f of 0, we can say that that is 20.

00:42 Then we need to do a over 1 plus b times e negative 0 .7 times 0.

00:55 That would be e to the 0, which is just 1.

00:59 So that is going to be a over 1 plus b.

01:04 And so this product right here is, let me highlight that, this right here is equal to our 20.

01:15 And i'm wanting to write a in terms of b or b in terms of a, just to help us get a better understanding for our next equation that we're going to do.

01:27 So then if we just multiply here, we'll get, well, we can cross multiply.

01:34 This is a one on the bottom.

01:36 So that's a equals 20 times 1 plus b.

01:41 So that's 20 plus 20 b.

01:44 And so this currently is as far as i can go with the first information given.

01:50 So now let's go on to the second one.

01:54 So f prime of t.

01:56 So we want to take the derivative of this first and foremost.

02:00 And so f prime of t.

02:04 And just so we can picture.

02:07 Better, i'm going to rewrite f of t without a numerator and denominator.

02:14 So this is going to be a times the quantity 1 plus b times e to the negative 0 .7t.

02:21 All that's to the negative first.

02:23 So now i can just use my power rule and it's a little easier for my eyes to see what we're doing with.

02:28 So that's going to be negative a times 1 plus b times e to the negative 0 .7t.

02:35 Subtract 1 to be negative 2 multiply by our derivative of the inside.

02:42 So that is negative 0 .7 b to e negative 0 .7t.

02:51 Okay.

02:52 And so this i'm going to rewrite it as 0 .7 times a times b to the e to the negative 0 .7t, all over 1 plus b times e to the negative 0 .7t.

03:08 This quantity squared.

03:10 It may look nasty, but since we are plugging in a zero for our t value, it will become a lot nicer to work with.

03:19 So we'll go ahead and plug in our zero here.

03:25 And so our e's, both of these, are going to be e to the zero.

03:30 So those are just going to cancel that to be 1.

03:32 So we're left with 0 .7 times a times b on top, divided by the quantity 1 plus b squared on the bottom.

03:42 And as we are given, our f prime of 0 is equal to 12...

Please give Ace some feedback