Expii

# 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.