Logistic Growth

Logistic Growth

Pure exponential growth is unrealistic — populations cannot grow forever. The logistic equation adds a carrying capacity $K$:

$\frac{dy}{dt} = r \cdot y \cdot \left(1 - \frac{y}{K}\right)$

• When $y \ll K$: growth is approximately exponential ($1 - y/K \approx 1$)
• When $y = K$: growth stops ($1 - y/K = 0$)
• When $y > K$: growth is negative (population declines toward $K$)

Exact Solution

$y(t) = \frac{K}{1 + \dfrac{K - y_0}{y_0}\, e^{-rt}}$

The solution forms an S-shaped curve (sigmoid), starting slow, accelerating, then leveling off at $K$.

Applications

• Population ecology: bacteria, fish stocks, human populations
• Epidemiology: spread of diseases before herd immunity
• Technology adoption: product diffusion models (Bass model)
• Neural networks: sigmoid activation function

Equilibria

The logistic equation has two equilibria:

• $y = 0$: unstable (any small population grows)
• $y = K$: stable (perturbations return to carrying capacity)

Your Task

Implement logistic(r, K, y0, t_end, n) using Euler's method.