How Do You Solve and Apply the Logistic Growth Model?

[drdizzard]

Given logistics growth model dp/dt=kp(1-p/N)
p=population
t=time
k=unknown growth coefficient (constant)
N=unknown carrying capacity (constant)
1) Solve for p explicitly
2) Given collected population data for a given state approximate N and k for that state.


This is what I did:
dp/(p(1-p/N))=kdt

Using partial fractions I got (1/p + 1/(N-p))dp=kdt

Integrating both sides gave ln(p)+ln(N-p)=kt+c

pN-p^2=e^(kt+c) or pN-p^2=Ae^kt

I am pretty certain I did all of this right but not quite sure, can this be further manipulated to give an explicit solution with only p on one side of the equation

I'm not quite sure how to do the second part and would appreciate any suggestions that would lead in the right direction as to how to go about it.

 

[HallsofIvy]

Given logistics growth model dp/dt=kp(1-p/N)
p=population
t=time
k=unknown growth coefficient (constant)
N=unknown carrying capacity (constant)
1) Solve for p explicitly
2) Given collected population data for a given state approximate N and k for that state.


This is what I did:
dp/(p(1-p/N))=kdt

Using partial fractions I got (1/p + 1/(N-p))dp=kdt

Integrating both sides gave ln(p)+ln(N-p)=kt+c

No, you've integrated incorrectly. Let u= N-p in the second term and du= -dp.

pN-p^2=e^(kt+c) or pN-p^2=Ae^kt

I am pretty certain I did all of this right but not quite sure, can this be further manipulated to give an explicit solution with only p on one side of the equation

I'm not quite sure how to do the second part and would appreciate any suggestions that would lead in the right direction as to how to go about it.

 

[drdizzard]

thanks for checking my work, after working it through again I got:

p=(ANe^kt)/(1+Ae^kt)