Solving the Logistic Equation

The differential equation is (dP/dt)=a*P(1-b*P).  It has solutions of the form

DSolve[p'[t]==a*p[t](1-b*p[t]),p[t],t]

This can be written as p[t]=(1/b)(((e^(a t))/((e^(a t))+(1/b)(e^(Subscript[c, 1]))))).  Note that there are two parameters and an arbitrary constant in this form.  Also note that the end behavior is p[t]→0 as t→-∞ and p[t]→(1/b) as t→∞.

Now look at p[t]=α(1+Tanh[λ(t-(Subscript[t, 0]))]).  This has similar end behavior where 2α=(1/b).  We can rewrite this expression in exponential form as follows:

p[t]=α(1+((Exp[λ(t-(Subscript[t, 0]))]-Exp[-λ(t-(Subscript[t, 0]))])/(Exp[λ(t-(Subscript[t, 0]))]+Exp[-λ(t-(Subscript[t, 0]))]))).

Get a common denominator and combine the two terms to get:

p[t]=2α (((Exp[λ(t-(Subscript[t, 0]))])/(Exp[λ(t-(Subscript[t, 0]))]+Exp[-λ(t-(Subscript[t, 0]))]))).

Multiply the numerator and denominator by Exp[λ(t-(Subscript[t, 0]))] and then by Exp[2λ (Subscript[t, 0])] to get:

p[t]=2α(((Exp[2λ t])/(Exp[2λ t]+Exp[2λ (Subscript[t, 0])]))).

Compare this last expression with the above to see that b=(1/(2α)), a=2λ, and (Subscript[c, 1])=2λ (Subscript[t, 0])-Log[2α].  So one can see that p[t]=α(1+Tanh[λ(t-(Subscript[t, 0]))]) is another way to write the solution of the Logistic Equation.