What input format does the estimator expect?

One time, population pair per line. Separators can be comma, space, semicolon, or tab. Times must be strictly increasing and populations must be positive.

How many data points are required?

At least five points. More points that actually approach a plateau give a better-identified K.

Which model is fitted?

The Verhulst logistic N(t) = K / (1 + e^(−r(t−t₀))), solved by deterministic least squares on K, r, and t₀.

What does the plateau warning mean?

If the highest observed population is still below 0.6 times the fitted K, the series has not reached the carrying capacity, so K̂ is an extrapolation.

What statistics are returned besides K?

Intrinsic rate r̂, inflection time t̂₀, initial size N₀, R², RMSE, a point-by-point residual table, and a plot of the data plus the fitted curve and K line.

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Carrying Capacity Estimator (Fit K from Time-Series Data)

Least-squares logistic fit of K, r, t₀ from `time, population` observations, with fitted curve, residuals, and plateau warnings.

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2 examples

Gause's yeast culture → K̂ = 663.96, r̂ = 0.5448, R² = 0.9998

The classic Gause (1932) Saccharomyces growth series (10 points over 18 hours) recovers a carrying capacity K̂ = 663.96 with r̂ = 0.5448 per hour, inflection at t̂₀ = 7.81 h, and R² = 0.9998 — the yeast clearly reached their plateau, so K̂ is well identified.

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Synthetic series (K = 1000, r = 0.5) recovered exactly

Nine points sampled from a known logistic with K = 1000, r = 0.5, t₀ = 6 are fitted exactly: the estimator returns K̂ = 1000, r̂ = 0.5, t̂₀ = 6 with R² = 1 — a sanity check that the K-scan recovers the truth on clean data before you trust it on noisy field counts.

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ContentPaste or type the main input values.1

Time-series data (one `time, population` pair per line)textareaRequiredOne `time, population` pair per line (separator: comma, space, semicolon, or tab). At least 5 points with strictly increasing times and positive populations that approach a plateau.

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Carrying Capacity Estimator (Fit K from Time-Series Data)

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Examples

1. Gause’s 1932 yeast culture

2. Exact recovery of a known logistic

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