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The logistic equation

Problem 6.348 · hard

A population follows the logistic model with growth rate \( \displaystyle r = 1 \) per year, carrying capacity 100, and initial population 20. Find \( \displaystyle P(t) \), \( \displaystyle P(4) \), and when the population reaches 50.

Equilibria: P = 0 and P = K = 100. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
Reviewed
\[ 4 \]
A.✓ Proved
\[ \frac{d}{d t} \frac{100}{1 + 4 e^{- t}} = \frac{100 - \frac{100}{1 + 4 e^{- t}}}{1 + 4 e^{- t}} \]
This P(t) satisfies the differential equation.✓ Proved
\[ 20 \]
and the initial condition.✓ Proved
\[ \frac{100}{\frac{4}{e^{4}} + 1} = \frac{100 e^{4}}{4 + e^{4}} \]
P(4).✓ Proved
\[ \ln{\left(4 \right)} \]
Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved

Answer \( P(t) = \frac{100}{1 + 4 e^{- t}},\ P(4) \approx 93.2,\ t = \ln{\left(4 \right)} \approx 1.386 \)

Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

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 constant A, and computes the requested values. The steps are logically sound and the algebraic checks confirm the results.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the logistic model parameters, derives the constant A, and computes the requested values. The steps are logically sound and the algebraic checks confirm the results.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the logistic model formula, calculates the constant A, and derives the specific values for P(4) and the time to reach half capacity. The logic is sound and the results are correct.
gpt-oss:20b: fail (error) 2026-10-10 — The solution never determines the constant A; step 2 merely states "4 = 4" instead of solving for A from the initial condition. Consequently the expression for P(t) is not justified, and the subsequent calculations are based on an unverified constant.

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 2026-10-10 with SymPy 1.14.0.