The logistic differential equation models growth that begins almost exponentially but slows as resources become limited. It is useful when a population, quantity, or adoption process cannot increase forever. The solution forms an S-shaped curve that rises quickly at first, then levels off near a maximum value called the carrying capacity.
This makes the model important in biology, ecology, medicine, economics, and technology adoption.
The standard equation is dy/dt = r y(1 - y/K), where y is the changing quantity, r is the intrinsic growth rate, and K is the carrying capacity. When y is small compared with K, the factor 1 - y/K is close to 1, so growth is nearly exponential. When y gets close to K, the factor 1 - y/K becomes close to 0, so growth slows down.
The curve has an inflection point at y = K/2, where the growth rate is greatest and the graph changes from concave up to concave down.
Key Facts
- Standard logistic differential equation: dy/dt = r y(1 - y/K).
- K is the carrying capacity, the long-term maximum value the model approaches.
- For 0 < y < K, dy/dt > 0, so the quantity increases over time.
- The growth rate is greatest at y = K/2.
- The inflection point of the logistic curve occurs when y = K/2.
- A common solution form is y(t) = K/(1 + A e^(-rt)), where A = (K - y0)/y0.
Vocabulary
- Logistic differential equation
- A differential equation that models growth with a limiting carrying capacity.
- Carrying capacity
- The maximum population or quantity that the environment or system can support in the long run.
- Inflection point
- A point on a curve where concavity changes, from bending upward to bending downward or the reverse.
- Intrinsic growth rate
- The constant r that describes how quickly a quantity would grow when resources are not limiting.
- Equilibrium solution
- A constant solution where dy/dt = 0, so the quantity does not change over time.
Common Mistakes to Avoid
- Treating logistic growth as exponential growth, which is wrong because logistic growth slows as y approaches K.
- Forgetting the factor 1 - y/K, which removes the resource-limiting effect and changes the model into simple exponential growth.
- Saying the maximum growth rate occurs at y = K, which is wrong because dy/dt = 0 at y = K and the maximum occurs at y = K/2.
- Assuming the curve reaches K exactly in finite time, which is wrong because the standard logistic solution approaches K asymptotically.
Practice Questions
- 1 A population follows dy/dt = 0.4y(1 - y/1000). Find the carrying capacity and the population size where the growth rate is greatest.
- 2 For y(t) = 500/(1 + 9e^(-0.2t)), find the initial value y(0) and the carrying capacity.
- 3 Explain why a logistic growth curve is concave up at first but concave down after passing y = K/2.