Differential Equation for Logistic Growth - Expii

The logistic equation is \[\frac{dy}{dt} = ky\left(1 - \frac{y}{L}\right)\] where \(k,L\) are constants. It is sometimes written with different constants, or in a different way, such as \(y' = ry(L-y)\), where \(r = k/L\). In either case, the constant \(L\) is known as the carrying capacity limit, and the factor \(1 - \frac{y}{L}\) represents growth inhibition. All solutions to the logistic equation are of the form \[y(t) = \frac{L}{1 + be^{-kt}}\] for some constant \(b\) (depending on the initial conditions or other information). In particular, regardless of the value of \(b\), we see that \(y(t) \to L\) as \(t\to \infty\) (as long as \(L,k,r\) are positive), so \(L\) can also be thought of as the equilibrium value (as \(t\to\infty\)) in the logistic model.