Logistic growth (Pearl–Reed) forecasting: given populations P0 at time t0, P1 at time t1, and P2 at time t2 = 2 t1, what is the saturation (carrying capacity) Ps of the city?

Introduction:
Logistic (Pearl–Reed) population forecasting assumes growth slows as population approaches a saturation (carrying capacity) Ps. Using three known populations at equally spaced times, Ps can be estimated in closed form.

Given Data / Assumptions:

• Known: P0 at t0, P1 at t1, P2 at t2 with t2 = 2 t1 (equal time intervals).
• Model: Logistic growth toward Ps.

Concept / Approach:
Under the logistic law, 1/P varies linearly with time when expressed appropriately. Using three consecutive observations at equal time spacing, eliminating the intrinsic growth rate yields Ps in terms of P0, P1, P2. The standard result used in demographic engineering texts is Ps = (P1^2 - P0 * P2) / (2 * P1 - P0 - P2).

Step-by-Step Solution:
Start with the logistic form: dP/dt = r * P * (1 - P/Ps). Solve to get: P(t) = Ps / (1 + A * e^{-r t}). Write equations at t0, t1, t2 and eliminate A and r using equal spacing. Rearrange to obtain Ps = (P1^2 - P0 * P2) / (2 * P1 - P0 - P2).

Verification / Alternative check:
Plugging trial values (monotonically increasing P0 < P1 < P2) yields a finite Ps slightly above P2, consistent with saturation behavior.

Why Other Options Are Wrong:

• Other algebraic forms shown do not reduce to the logistic Ps when tested; signs/coefficients are inconsistent with the elimination step.
• “None of these” is incorrect because a standard closed-form expression exists and is listed.

Common Pitfalls:

• Applying the formula when observation intervals are unequal—this derivation assumes equal spacing.
• Confusing geometric growth with logistic growth.

Final Answer:
Ps = (P1^2 - P0 * P2) / (2 * P1 - P0 - P2).