Background

    The exponential model    is used to study uninhibited population growth and solution is the exponential function  .  When the term   is added we obtain the logistic differential equation which is used to model inhibited population growth or bounded population growth.  The logistic differential equation is

            .  

One form of the solution is  

            .  

The terms have been carefully determined so that the initial condition is

            .

The limiting value  L  of  y(t)  is given by

            .

The graph is the so called "S-shaped" curve. The choice of parameters    creates the curve shown below.

Proof.

Symmetry

    The solution curve to the logistic differential equation

            .  

is given by

            .  

and it is symmetric about the point  .

Proof.

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(c) John H. Mathews 2004