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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.

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

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/
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Associated Topics:
High School Calculus

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