Logistic growth and exponential growth are both models of how populations (or other quantities) increase, but they differ in how they respond to limited resources and space. Direct answer

• Exponential growth: growth rate is proportional to current size and remains constant per capita. The population grows unchecked, producing a J-shaped curve with no intrinsic upper limit.
• Logistic growth: growth rate slows as the population approaches a maximum sustainable size (carrying capacity). The curve is S-shaped (sigmoid), starting with rapid growth, then decelerating, and finally leveling off near carrying capacity.

Key distinctions

• Growth rate dependence
  • Exponential: per-capita growth rate is constant; dN/dt = rN.
  • Logistic: per-capita growth rate decreases as N approaches K; dN/dt = rN(1 - N/K).
• Carrying capacity
  • Exponential: no built-in limit; can unrealistically exceed resource constraints.
  • Logistic: includes a carrying capacity K that caps long-term growth.
• Curve shape
  • Exponential: J-shaped curve.
  • Logistic: S-shaped (sigmoid) curve with an inflection point where growth begins to slow.
• Real-world applicability
  • Exponential is a good approximation only in early growth phases with abundant resources.
  • Logistic better captures resource limitation, competition, and saturation in established populations or systems.

Common axes and parameters

• Exponential model typically uses parameters N0 (initial size) and r (intrinsic growth rate).
• Logistic model uses N0, r, and K (carrying capacity). The inflection point occurs near N = K/2, where growth is fastest, and then slows as N nears K.

Illustrative intuition

• In an environment with unlimited resources, the population can double or triple rapidly without restraint (exponential).
• In most natural systems, resources (food, space, nesting sites) limit growth. Early growth can be fast, but crowding leads to competition and mortality, causing the growth rate to drop and the population to level off at or near K (logistic).

If you’d like, I can provide a quick numerical example showing how N(t) evolves under each model, or derive the logistic equation step by step.