Gompertz Model

  • Thread starter lockedup
  • Start date
  • #1
70
0

Homework Statement


The problem in the book:

a) Suppose a = b = 1 in the Gompertz differential equation. Since the DE is autonomous, use the phase portrait concept of Section 2.1 to sketch representative solution curves corresponding to the cases P0 > e and 0 < P0 < e.

b) Suppose a = 1, b = -1 in the Gompertz DE. Use a new phase portrait to sketch representative solution curves corresponding to the cases P0 > e-1 and 0 < P0 < e-1

c) Find an explicit solution of the Gompertz DE subject to P(0) = P0


Homework Equations


dP/dt = P(a-blnP)


The Attempt at a Solution


I used separation of variables to get:
dP/(P(a-blnP)) = dt

I let u = a - blnP and du = -bdP/P which leaves me with:
-b[tex]\int[/tex]du/u = [tex]\int[/tex]dt

I integrate to get:
-b ln (u) = t + C

ln (u-b) = t + C

eln (u-b) = et + C

u-b = Aet

(a - b ln P)-b = Aet

So how do I solve for P? :P Or, am I even close to having the right answer? LOL
 

Answers and Replies

  • #2
70
0
P = eAe-t/b?
 
Last edited:
  • #3
70
0
Or how about:

P = e(Ae-bt - a)/-b
 
Top