Logistic Curve y = L / (1 + e^(−k(x − x₀)))

S-shaped growth that saturates at carrying capacity $L$. Steepness is $k$ and midpoint is $x_0$. At $x = x_0$, $y = L/2$. As $x\to\infty$, $y\to L$; as $x\to -\infty$, $y\to 0$.

Slide capacity $L$, steepness $k$ and midpoint $x_0$. The lab plots the S-curve and reports the value at the midpoint $L/2$ and the maximum slope $kL/4$ (at $x = x_0$). Try $L = 1$, $k = 2$, $x_0 = 0$ for a sharp transition near zero.

Exponential growth $y = y_0 e^{kx}$ assumes infinite resources, which never holds long-term. Logistic adds the carrying capacity $L$ — the maximum population a habitat can sustain. COVID-19 case counts, market saturation of smartphones in India, and bacterial cultures in a Petri dish all follow logistic, not exponential, curves.

Logistic growth solves $\dfrac{dy}{dx} = ky(1 - y/L)$. At low $y$ it behaves like $dy/dx \approx ky$ (exponential); near $y = L$ growth stalls. This is the simplest nonlinear ODE most students meet. The lab shows the solution; sketching the ODE's slope field is a great follow-up.