The gompertz model has been used to model population growth. \frac{dy}{dt

emancipezN

emancipezN

Answered question

2021-10-31

The gompertz model has been used to model population growth. 
 dy  dt =ryln(Ky) 
Where r=0.73 per year, k=33.800kg, y0k=0.27 
Use the Gompertz equation to find the predicted value of y(3).

Answer & Explanation

aprovard

aprovard

Skilled2021-11-01Added 94 answers

Step 1
Consider the following differential equation:
dydt=ryln(Ky) where r=0.73 per year, k=33,800kg, y0k=0.27
The objective is to find the predicted value of y(3) by using Gompertz model.
Step 2
Given dydt=ryln(Ky), it is variable separable equation as r and K are constant. This implies,
dyyln(Ky)=rdt
Integrate on both sides,
dyyln(Ky)=rdt
dyyln(Ky)=rt+c
Where c is an arbitrary constant.
Now, use u-substitution method as follows:
u=Ky
du=Ky2dy
dy=(y2K)du
Step 3
This implies,
1yln(u)(y2K)du=rt+c
yKln(u)du=rt+c
(Ku)Kln(u)du=rt+c
1uln(u)du=rt+c
Assume v=ln(u) which implies,
dv=1udu
du=udv
Substitute this in the above integration to get,
1vdv=rt+c
ln(v)=rt+c
ln(ln(u))=rt+c
ln(u
Nick Camelot

Nick Camelot

Skilled2023-06-14Added 164 answers

The Gompertz model is given by:
dydt=ryln(Ky) where r=0.73 per year, K=33,800kg, and y0K=0.27.
To find the predicted value of y(3), we'll solve the differential equation and evaluate it at t=3.
First, let's separate the variables and integrate:
1ydy=rln(Ky)dt
Integrating both sides:
1ydy=rln(Ky)dt
ln|y|=rtln(Ky)rt+C
Here, C is the constant of integration.
Next, we can simplify the equation by using the property of logarithms:
ln|y|=rtln(K)rtln(y)rt+C
We can rearrange the terms:
ln|y|+rtln(y)=rtln(K)rt+C
Now, let's exponentiate both sides using the property eln(x)=x:
eln|y|+rtln(y)=ertln(K)rt+C
Using the properties of exponents, we can rewrite the equation as:
eln|y|·ertln(y)=ertln(K)·ert·eC
Simplifying further:
|y|·yrt=ertln(K)·ert·eC
Since y represents a population, we can remove the absolute value:
y·yrt=ertln(K)·ert·eC
Now, we'll use the initial condition y0K=0.27 to determine the constant of integration C:
0.27=y0·y0r·0=y0·1=y0
Substituting this back into the equation:
y·yrt=ertln(K)·ert·eln(y0)
Simplifying:
y1+rt=ertln(K)·ert·y0
Now, let's solve for y:
y=(ertln(K)·ert·y0)11+rt
Finally, we can substitute the given values r=0.73, K=33,800kg, y0=0.27K, and t=3 to find the predicted value of y(3):
y(3)=(e0.73·3·ln(33,800)·e0.73·3·(0.27·33,800))11+0.73·3
Calculating this value will give us the predicted population at t=3.
madeleinejames20

madeleinejames20

Skilled2023-06-14Added 165 answers

Step 1:
To find the predicted value of y(3) using the Gompertz equation, we can plug in the given values into the equation:
dydt=ryln(Ky)
Given:
r=0.73 per year,
K=33,800 kg,
y0K=0.27
We want to find y(3), which represents the value of y at time t = 3.
To solve this, we first need to integrate the equation:
1ydy=rln(Ky)dt
ln|y|=rtrln(Ky)+C
Step 2:
Next, we can apply the initial condition at t = 0, where y = y₀:
ln|y|=0rln(Ky)+C
Simplifying the equation:
ln|y|+rln(Ky)=C
Now, we can substitute the given values:
ln(0.27K)+0.73ln(K0.27K)=C
Solving for C using a calculator or software, we find the value of C.
Step 3:
Finally, we can substitute t = 3 into the equation and solve for y(3):
ln|y(3)|=0.73×30.73ln(Ky(3))+C
ln|y(3)|+0.73ln(Ky(3))=0.73×3+C
ln|y(3)|+0.73ln(33,800y(3))=2.19+C
Again, using a calculator or software, we can find the value of ln|y(3)|+0.73ln(33,800y(3)).
Therefore, the predicted value of y(3) can be determined by solving the last equation.
Eliza Beth13

Eliza Beth13

Skilled2023-06-14Added 130 answers

To solve the given problem using the Gompertz model equation, we have the following differential equation:
dydt=r·y·ln(Ky) where r=0.73 per year, K=33,800kg, and y0K=0.27. We need to find the predicted value of y at t=3.
To solve this differential equation, we will separate variables and integrate both sides. Let's start by rearranging the equation:
dyy·ln(Ky)=r·dt
Now, we integrate both sides of the equation:
dyy·ln(Ky)=rdt
The integral on the left side can be solved using a substitution. Let's substitute u=ln(Ky), then dudy=1y. Rearranging gives dy=ydu. Substituting these values into the integral:
dyu=rdt
Simplifying the left side of the equation:
dyu=rdt
ln|u|=rt+C1 where C1 is the constant of integration.
Now, substituting u back in terms of y:
ln|ln(Ky)|=rt+C1
To determine the value of the constant C1, we can use the initial condition y0K=0.27 when t=0. Substituting these values:
ln|ln(Ky0)|=C1
ln|ln(33,800y0)|=C1
Now, we can rewrite the equation as:
ln|ln(Ky)|=rtln|ln(33,800y0)|
To find y at t=3, we substitute t=3 into the equation:
ln|ln(Ky)|=3rln|ln(33,800y0)|
Now, we can solve this equation to find the predicted value of y at t=3.

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