The Gompertz model has been used to model population growth \frac{dy}{dt}=ry\ln(\frac{K}{y}) where

leviattan0pi

leviattan0pi

Answered question

2021-11-14

The Gompertz model has been used to model population growth
dydt=ryln(Ky)
where r=0.71 per year, K=46,300 kg, y0K=0.38,y(0)=y0
Use the Gompertz model to find the predictet value of y(5).

Answer & Explanation

Tionant

Tionant

Beginner2021-11-15Added 17 answers

Given equation  dy  dt =ryln(Ky) is a separate-equation first order differential.
 dy  dx =xyln(Ky)
1yln(Ky) dy =π dx 
1yln(Ky) dy =π dx 
1t dt =π dx 
ln(t)=πx+C
ln(ln(Ky))=πx+C
ln(Ky)=eπxCKy=eπxC
now given y(0)=y0=0.38 K
K0.38K=eeceec=10.38
ky=eeπxec=(eec)eπx=(10.38)eπx
yk=(0.38)eπxy=K(0.38)eπx
y=46300(0.38)e0.71x
So y(5)=46300(0.38)e0.71x
y(5)=45030.88

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