Logistic growth (with carrying capacity) differential equation (which version?)

Question

Hello,

I've come across two versions of the logistic growth differential equation with carrying capacity and they don't seem to be equivalent.

The version I've seen most often (including on Khan academy) is dP/dt = rP(1-P/k), where r is the growth rate, P is the population, and k is the maximum carrying capacity.

However, the Kaplan AP calculus book I'm using uses the equation dP/dt = rP(k-P)

This version removes the 'k' in the denominator and should (I would think) yield a different rate of growth, correct?

Are these two different equations or am I missing something? Thanks for your help!

Answer 1

You probably realized that the Kaplan equation should be dP/dt = (r/k)P(k-P), but you can just redefine r. You can also write the equation in terms of fraction P/k = x and it becomes dx/dt = r(1-x) with your original r after having divided both sides by k. Often, in differential equations courses, it is introduced as the more general equation dA/dt = εA -σA² emphasizing it's nonlinearity and that it has 2 constants without a set meaning as in the more applicable form. Stick with the form that makes sense to you. Take care.