The logistic equation
Problem 6.308 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{2} \) per year, carrying capacity 5000, and initial population 250. 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
- \[ 19 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{5000}{1 + 19 e^{- \frac{t}{2}}} = \frac{2500 - \frac{2500}{1 + 19 e^{- \frac{t}{2}}}}{1 + 19 e^{- \frac{t}{2}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 250 \]and the initial condition.✓ Proved
- \[ \frac{5000}{1 + \frac{19}{e^{\frac{5}{2}}}} = \frac{5000 e^{\frac{5}{2}}}{e^{\frac{5}{2}} + 19} \]P(5).✓ Proved
- \[ 2 \ln{\left(19 \right)} = \ln{\left(361 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{5000}{1 + 19 e^{- \frac{t}{2}}},\ P(5) \approx 1953.4,\ t = \ln{\left(361 \right)} \approx 5.889 \)
✓ 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 identifies the logistic model parameters, derives the constant A, and computes the specific values for P(5) and the time to reach half the carrying capacity. The logic is sound and the steps are clearly justified.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the logistic model parameters, derives the constant A, and computes the specific values for P(5) and the time to reach half the carrying capacity. The logic is sound and the steps are clearly justified.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the logistic model parameters, calculates the constant A, and derives the correct expressions for P(t), P(5), and the time to reach half the carrying capacity. The algebraic steps are verified.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.