# 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. > Canonical page: https://elysiatools.com/en/tools/carrying-capacity-estimator - **Category:** Health - **Keywords:** carrying capacity, logistic fit, population growth fit, verhulst model, sigmoid regression, k estimation, growth curve fitting, population ecology, yeast growth, ecology calculator, least squares, time series ## Overview The Carrying Capacity Estimator recovers carrying capacity K, intrinsic growth rate r, and inflection time t₀ by least-squares logistic (Verhulst) fit to time-series population counts. Paste at least five strictly increasing time, population pairs; the tool returns fitted parameters, R², RMSE, point residuals, a scatter-plus-sigmoid plot with the K line, and a warning if the series has not yet approached a plateau. ## Inputs - **Time-series data (one `time, population` pair per line)** (textarea): One `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. ## When to use - You have five or more time–population observations that look like they are leveling off. - You need to estimate K (and r, t₀) from lab or field counts such as yeast, microbes, or wildlife censuses. - You want a fitted S-curve, residual table, and an explicit check that the data actually reached a plateau. ## How it works - Enter one time, population pair per line, using comma, space, semicolon, or tab as the separator. - The estimator scans candidate K values, linearizes ln((K−N)/N) = −r·t + r·t₀, then refines K with least squares and ternary search. - It reports K̂, r̂, t̂₀, N₀, R², RMSE, a residual table, and an SVG of observed points, the fitted sigmoid, and a horizontal K line. - If the largest observed N is still below 0.6·K̂, it warns that K̂ is an extrapolation rather than a well-identified plateau. ## Use cases - Recover carrying capacity from classic microbial growth series such as Gause’s yeast counts. - Estimate K from sequential wildlife or fishery census data that appear to be saturating. - Sanity-check the fitter on a synthetic logistic series with known K, r, and t₀ before using noisy field counts. ## Frequently asked questions ### 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. ## Related tools - [Blood Alcohol Content (BAC) Estimator](https://elysiatools.com/en/tools/bac-calculator): Estimate your BAC from sex, weight, drinks and time using the Widmark formula. Returns g/dL and ‰, predicts sober time, and flags the impairment tier and legal driving limits. - [Fetal Development Timeline (by Gestational Week)](https://elysiatools.com/en/tools/fetal-development-timeline): See week-by-week fetal development from conception to birth (40 weeks). 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