Expert: Paul Klarreich - 9/30/2009

Question
in study of population dynamics, one of the famous models for growing but bounded population is logistic equation

dP/dt= P(a-bP)

where a and b are positive constants,
how to solve this differential equation?

Answer
Questioner: fadzlan
Country: Malaysia
Category: Advanced Math
Private: No
Subject: engineering math 4
Question: in study of population dynamics, one of the famous models for growing but bounded population is logistic equation

dP/dt= P(a-bP)

where a and b are positive constants,
how to solve this differential equation?
.........................................
This looks like a 'separation of variables':

dP
-- = P(a-bP)
dt

 dP
--------- = dt
p(a - bP)

The right side will integrate to  t + C1

The left side is a fairly standard (except for symbolic constants) partial fractions thing.  (But see an alternative below.)

   1
---------, by a standard table, such as:
P(a - bP)

en.wikipedia.org/wiki/List_of_integrals_of_rational_functions which says:

does the partial fractions thing for us, and gives:

{     1
| --------- dx
} x(dx + c)

    1         dx + c
= - ---  ln | -------- |
    c            x

And, writing our integral as:
   -1
-----------
P(bP + a)

and matching  P <=> x,  d <=> b,  a <=> c,  we have:

    1         bP + a
= - ---  ln | -------- |
    a            P

    1            P
= + ---  ln | -------- |
    a         bP + a  


Put it together:

         1            P
t + C1 =  ---  ln | -------- |
         a         bP + a  

                  P
at + C2 =  ln | -------- |     <<< C2 is  aC1
                bP + a  
             P
C3 e^at =  --------
          bP + a
That is about as far as we can go, I think.  

But wait, there's more!  (Oh, yes, you don't have that stuff in Malaysia.  Lucky you!)

We can also do it this way:

   1
--------- =
P(a - bP)

  - 1
--------- =
P(bP - a)

  - 1
--------- =
bP^2 - aP

now complete the square:

  - 1
--------------------------------- =
b(P^2 - aP/b + a^2/4b^2) - a^2/4b

  - 1
------------------------- =
b(P - a/2b)^2 - a^2/4b

Now you can let  P - a/2b = U and write:

  - 1
--------------- =
bu^2 - a^2/4b

- 1        1
----  -------------- =
 b     u^2 - a^2/4b^2

- 1        1
----  -------------- =
 b     u^2 - C^2,  where  C = a/2b

which you can now look up.  [I think it involves hyperbolic functions.]


Are you glad you asked?