The Gompertz model has been used to model population growth.
(dy)/(dt) = ry ln((K)/(y)),
where r= 0.73 per year, k= 33,800 kg, (y_0)/(k) = 0.27
Use the Gompertz model to find the predicted value of y(3)
Round your answer to the nearest integer

1 Answers

Best Answer

Consider the following differential equation:
(dy)/(dt) = ry ln ((K)/(y)) where r= 0.73 per year, k= 33,800 kg, (y_0)/(k) = 0.27
The objective is to find the predicted value of y(3) by using Gompertz model.
Given (dy)/(dt) = ry ln ((K)/(y)), it is variable separable equation as r and K are constant. This implies,
(dy)/(y ln (K)/(y)) = rdt
Integrate on both sides,
int (dy)/(y ln (K)/(y)) = int rdt
int (dy)/(y ln (K)/(y)) = rt + c
where c is an arbitrary constant.
Now, use u-substitution method as follows:
u = (K)/(y)
du = = (-K)/(y^2) dy
dy = ((-y^2)/(K))du
This implies,
int (1)/(y ln (u)) ((-y^2)/(K))du = rt + c
int = (-y)/(K ln(u)) du = rt + c
int (-((K)/(u)))/(K ln(u))du = rt + c
int (-1)/(u ln (u))du = rt + c
Assume v = ln (u) which implies,
dv = (1)/(u)du
du = udv
Substitute this in the above integration to get,
int (-1)/(v)dv = rt + c
-ln(v) = rt + c
-ln(ln(u)) = rt + c (v = ln(u))
ln(u)=ce^(-rt)
u = e^(ce^(-rt))
(K)/(y) = e^(ce^(-rt))
y = (K)/(e^(ce^(-rt)))
Since r = 0.73, k = 33.800, (y_0)/(k) = 0.27
This implies y_0 = 9126 at t= 0, substitute the given value in the expression y=(K)/(e^(ce^(-rt))).
9126 = (33800)/(e^(ce^(0.73(0))))
9126 = (33800)/(e^(ce^(0)))
e^c = (33800)/(9126)
c = ln((100)/(27))
c=1.309
Now, we find the value of y(3), substitute all the known values in the expression y = (K)/(e^(ce^(-rt))).
y = (33800)/(e^(1.309)e^((-0.73)(3)))
=29193.96
~ 29194
Therefore, the value of y(3) by using Gompertz model is 29194.