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The logistic equation

Problem 6.229 · hard

A population follows the logistic model with growth rate \( \displaystyle r = 1 \) per year, carrying capacity 500, and initial population 10. Find \( \displaystyle P(t) \), \( \displaystyle P(4) \), and when the population reaches 250.

Equilibria: P = 0 and P = K = 500. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
Reviewed
\[ 49 \]
A.✓ Proved
\[ \frac{d}{d t} \frac{500}{1 + 49 e^{- t}} = \frac{500 - \frac{500}{1 + 49 e^{- t}}}{1 + 49 e^{- t}} \]
This P(t) satisfies the differential equation.✓ Proved
\[ 10 \]
and the initial condition.✓ Proved
\[ \frac{500}{\frac{49}{e^{4}} + 1} = \frac{500 e^{4}}{49 + e^{4}} \]
P(4).✓ Proved
\[ \ln{\left(49 \right)} \]
Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved

Answer \( P(t) = \frac{500}{1 + 49 e^{- t}},\ P(4) \approx 263.5,\ t = \ln{\left(49 \right)} \approx 3.892 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the logistic model parameters, derives the specific function, and accurately computes the requested values and time. The logical flow is sound and the algebraic steps are verified.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the logistic model parameters, derives the specific function, and accurately computes the requested values and time. The logical flow is sound and the algebraic steps are verified.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the logistic growth formula, calculates the constant A, and derives the specific values for P(4) and the time to reach half the carrying capacity. The logic is sound and the results match the stated answer.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.