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Modeling yeast populations Use the fact that the per capita growth rate of the yeast population in Table 1 is $0.55-0.0026 N$ to show that, in terms of the logistic equation $(4), r=0.55$ and $K \approx 210$

Proof on the video

Calculus 2 / BC

Chapter 7

Differential Equations

Section 1

Modeling with Differential Equations

Missouri State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:04

Modeling yeast populations…

01:07

01:20

The logistic differential …

02:55

The table gives the number…

01:30

Yeast population Shown is …

02:27

The number $Y$ of yeast …

08:18

a. Solve the logistic Equa…

0:00

Shown is the graph of the …

02:12

02:52

Logistic growth: $P(t)=\fr…

the per capita growth rate of just population is 0.55 minus 0.0 26 in. That is the variation of an over time we respect. The time is equal to zero point 55 minutes, 0.0 26 times in all that times in. We're going to show that in terms of the logistic equation R equals 0.55 k. It's approximately equal to 210. Okay, so the first thing we gotta do here is to remember what the logistic equation is. So I'm going to write it right here. The stick equation, this equation give us the variation of the population respected time. Okay, through the equations. If we're derivative and respected, t equals our times one minus x over que okay while minus an over K times in. So we have here two ways of new defying these equations this variation of end this it is art when an equals K and when an equal zero. So this is the logistic equation. And, uh, we know that the gist population is following or is fulfilling this equation. Creation of and respected time is equal to zero point 55 minus 0.0 26 times and times in this equation is the same aspiration of and respected time times one over in equals. Zero point 55 minus 0.0 26 times in. That is, um okay, the variation off and over time times went over and is equal to degrees growth rate off off the just the per capita growth rate of the population. So this is equal to zero point 55 minus 0.0 26. And so, if we look at the logistic equation and the equation that is satisfied by the genes population this one here we see that we have the same structure of thes two equations. We have the derivative respected time on the left and determine All right. We have some expression, will deploy I by end. So, uh, Thio be in accordance thio the logistic equation. We got to have that determine that multiplies in on the term of the right side on both equation is the same. So we got to have that our times one minus and over K must be equal to 0.55 minus 0.0 26 end and as we see the from the right side, we have a constant another costume that was adding or subtracting and the second constant constant. It's multiplying term. And if we rewrite thes expression on the left that came from the logistic equation to conform this expression here we have that are times Wonder is ar minus our times and over. Kate, that is our over k times and equals 0.55 minus 0.0 26 times in. And as we see here, we have a term here, attempt term here minus minus and some term or to playing in some term to play. And so we got to have that are must be equal to thes term here and that people the term are over came must be going to disturb here for the two suppression to have the same form then r equals 0.55. This is first thing we conclude and our vacay get to be equal to syrup on 00 26 studies okay is equal to are over so 0.0 26 that is K is equal to are we conclude it was equal to or it is equal to 0. 55 0 55 over 0.0 26. And this division here is about 211 point 54. So we can say that are is exactly equal to 0.55. End and K is about 211. Or if you want 210 she's a close value to this. We can say that if you want want to be a approximated to the unity, then we have 210. And that's, uh, but, uh, valid with answer. So we have found our in Key and K R 0.55 K about 210 or 211 in order to have a per capita growth rate off the juice population, um, mhm to be in accordance to the logistic equation. So we have the same form off the truth of the chiefs population, and the just the equation when we take this parameters are 0.55 k about 210

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