Question

Recall that the general form of a logistic equation for a population is given by P(t) = c/1 + ae−bt , such that the initial population at time t = 0 is P(0) = P0 . Show algebraically that

Answer 1

Consider that the logistic equation for a population:

P(t) = c/(1 + ae-bt)

 

Put t = 0

So,

P(0) = c/(1 + ae-b(0))

        = c/(1 + ae-b(0))

       = P0

 

Now,

{c – P(t)}/P(t) = {c – c/(1 + ae-bt)}/{c/(1 + ae-bt)}

                       = (c + cae-bt – c)/c

                       = cae-bt/c

                       = ae-bt

 

Similarly,

(c – P0)/P0 = {c – c/(1 + a)}/{c/(1 + a)}

                    = (c + ca – c)/c

                   = ca/c

                   = a

 

From the above calculations,

 

Hence,

c – P(t)/P(t) = (c – P0/P0)e-bt.

Hence,

c – P(t)/P(t) = (c – P0/P0)e-bt.