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The Logistic Model for Population Growth


Date: 03/29/98 at 02:42:58
From: Herve Duchemin
Subject: High School Calculus

I have a problem in my high school calculus class. It is known as the 
Logistic Model of Population Growth and it is:

   1/P dP/dt = B - KP

where B equals the birth rate, and K equals the death rate. Also, 
there is an initial condition that P(0) = P_0. I need to calculate 
P(t), which will predict the population at any time. I also need to 
find the limit of P(t) as t approaches infinity. 

Thanks for your help!


Date: 03/29/98 at 07:11:08
From: Doctor Anthony
Subject: Re: High School Calculus

We start with the following equation:

   dp/dt = p(b - kp)

Rearranging the equation so that p is on the left and t is on the 
right, we get:

      dp
   -------- = dt
   p(b - kp) 

Use partial fractions for the left hand side:

       1        A       B
   --------- = --- + ------
   p(b - kp)    p    b - kp

   1 = A(b - kp) + Bp

Then:

   p = 0 gives 1 = Ab and so A = 1/b

   p = b/k gives 1 = Bb/k and so B = k/b

The lefthand side can then be written:

     1        k
   [--- + ---------] dp = dt
     bp   b(b - kp)

   (1/b)[ln(p) - ln(b - kp)] = t + constant

   ln[p/(b - kp)] = bt + constant

   p/(b - kp) = Ae^(bt)            where A is a constant

   p = (b - kp)Ae^(bt)

   p = Abe^(bt) - Akpe^(bt)

   p(1 + Ake^(bt)) = Abe^(bt)

          Abe^(bt)
   p = --------------              (Equation 1)
        1 + Ake^(bt) 

At t=0, p = p(0), so we have:

   p(0)/(b - k*p(0)) = A           (Equation 2)

For any given problem, you can calculate A from (2) and then 
substitute for A in (1).

Dividing (1) top and bottom by e^(bt), we get:

            Ab
   p = ------------
       e^(-bt) + Ak 

and as t -> infinity, the term e^(-bt) -> zero, and we have:

                  Ab     b
   p(infinity) = ---- = ---
                  Ak     k
       
-Doctor Anthony,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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