Calculus Volume 2 · free preview
The Logistic Equation
The logistic equation models growth with a carrying capacity, its S-shaped solutions rising from an initial value toward the capacity K, separable by partial fractions and analyzed for equilibria and stability.
1. The model with capacity
The model with capacity
2. Solution by partial fractions
Solution by partial fractions
3. Analysis and applications
Analysis and applications
4. Beyond the pure logistic
Beyond the pure logistic
5. Carrying capacity effect
In the logistic equation dP/dt = kP(1 - P/K), what role does K play?
6. Logistic equilibria
The logistic equation dP/dt = kP(1 - P/K) with k positive has which equilibria, and which are stable?
7. Inflection population
At what population does the logistic curve change concavity, its growth fastest?
8. Early behavior
For an initial population P_0 much smaller than K, the logistic solution's early growth approximates what?
9. Separation decomposition
To integrate the separated logistic dP/(P(1 - P/K)) = k dt, which decomposition applies?
10. Logistic denominator
The logistic solution P(t) = K/(1 + A e^(-kt)) has A = (K - P_0)/P_0. For P_0 = K/3, what is A? Answer with a number.
11. Limit at infinity
As t approaches infinity, e^(-kt) approaches zero, so the logistic P(t) approaches what value? Answer with the single letter K.
12. Inflection time
The inflection occurs when P = K/2, at time t = ln(A)/k. As a single letter, what is A in this formula? Answer with an uppercase letter.
13. Stable equilibrium definition
An equilibrium P star of an autonomous equation is called stable when what holds?
14. Rate at capacity
At P = K the logistic rate kP(1 - P/K) equals zero. As a number, what is it? Answer with a number.
Practice this interactively
Free account · instant grading · spaced review that schedules itself.
Start this course for free