The logistic equation
Problem 6.310 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{5} \) per year, carrying capacity 5000, and initial population 1000. Find \( \displaystyle P(t) \), \( \displaystyle P(5) \), and when the population reaches 2500.
- Equilibria: P = 0 and P = K = 5000. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.Reviewed
- \[ 4 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{5000}{1 + 4 e^{- \frac{t}{5}}} = \frac{1000 - \frac{1000}{1 + 4 e^{- \frac{t}{5}}}}{1 + 4 e^{- \frac{t}{5}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 1000 \]and the initial condition.✓ Proved
- \[ \frac{5000}{1 + \frac{4}{e}} = \frac{5000 e}{e + 4} \]P(5).✓ Proved
- \[ 5 \ln{\left(4 \right)} = \ln{\left(1024 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{5000}{1 + 4 e^{- \frac{t}{5}}},\ P(5) \approx 2023.0,\ t = \ln{\left(1024 \right)} \approx 6.931 \)
✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the differential equation solved numerically (Taylor-series ODE solver) from P(0) |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly derives the logistic function, verifies it satisfies the differential equation and initial conditions, and accurately calculates P(5) and the time to reach half the carrying capacity.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly derives the logistic function, verifies it satisfies the differential equation and initial conditions, and accurately calculates P(5) and the time to reach half the carrying capacity.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the logistic growth formula, verifies the differential equation and initial conditions, and accurately calculates P(5) and the time to reach half the carrying capacity.gpt-oss:20b: pass 2026-10-08
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/logistic_equation, checked 2026-10-08 with SymPy 1.14.0.