What is the logistic equation satisfied by the population, p?

Question

The population of a certain planet is believed to be growing according to the logistic equation. The maximum population the planet can hold is 10¹⁰. In year zero the population is 50% of this maximum, and the rate of increase of the population is 10⁹ per year.

 

What is the logistic equation satisfied by the population, p.

 

p'=?

 

I have tried this so many times and have no idea what I'm doing wrong. Help would be appreciated! (university level math)

Answer 1

The Logistics Equation is the following:

  dP/dt = (r/K)*P*(K - P)0

 

   where K = 10¹⁰, r = 10⁹ and P(0) = K/2 (initial condition).

 

Keeping K as K for simplicity, with these values the particular equation is dP/dt = 0.1P(K-P)

 

dP/[P*(K-P] = 0.1 * dt => (1/P + 1/(K-P)) (dP/K = 0.1 *dt (Partial fractions)

 

                                   => (1/P + 1(K - P)) dP = 0.1K *dt

 

Integrate both sides: ln(P) - ln(K - P) = 0.1K t + C

 

With the initial condition P(0) = K/2: ln(K/2) - ln(K/2) = 0 = 0 + C => C = 0

 

∴ ln[P/(K - P)] = 0.1K * t => P = (K - P) *e^0,1Kt

 

                                       => P = Ke^0.1Kt -Pe^0.1Kt

 

                                      => P(1 + e^0.1Kt) = Ke^0.1Kt

 

                                     => P(1 + e^-0.1Kt) = K

 

                                     => P = K/(1 + e^-0.1Kt)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer 2

The logistic equation is a method to forecast population growth which ultimately reaches a plateau.

A discussion of that topic is too complex for this forum.

Please refer to the following site for a detailed explanation.

 

http://mathworld.wolfram.com/LogisticEquation.html