694.74
0.990399
1/time
2.692308
0.359848
694.74
0.990399
1/time
2.692308
0.359848
The Carrying Capacity Estimator uses three consecutive population counts at equally spaced time intervals to estimate the carrying capacity (K) and intrinsic growth rate (r) of a population following logistic growth. This method provides a quick analytical estimate without requiring curve-fitting software.
Enter three population values at three equally spaced time points (for example, years 0, 1, and 2). The calculator uses the algebraic relationship between consecutive logistic growth values to solve for K and then estimates r from the growth between the first two time points.
From three consecutive population observations under logistic growth (N0, N1, N2 at equal time intervals), carrying capacity can be estimated as:
K = (2 x N0 x N1 x N2 - N1² x (N0 + N2)) / (N0 x N2 - N1²)
This formula is derived by solving the discrete logistic recurrence for K. Once K is estimated, the growth rate r is computed from:
lambda = N1 x (K - N0) / (N0 x (K - N1))
r = ln(lambda)
This method works best when the population is in the exponential-to-logistic transition phase and is sensitive to measurement noise.
Inputs
Results
With populations of 50, 120, and 250 at three time points, the estimated K is approximately 697 and r is about 0.84 per time unit.
Inputs
Results
When growth is visibly slowing (200 to 350 to 450), the estimated K of about 522 suggests the population is already at 86% of carrying capacity at time 2.
This algebraic method provides a rough estimate that is most accurate when the three data points span the transition from rapid growth to deceleration. If all three points are in the exponential phase (far from K) or all are near K, the estimate may be unreliable. For more accurate estimates, nonlinear regression fitting of the full logistic curve to many data points is recommended.
A negative K estimate indicates that the data do not fit the logistic growth model. This can occur if the population is not growing logistically, if there is too much measurement error, or if all three data points are in the exponential growth phase where the logistic curve is not yet decelerating. The calculator displays the absolute value in such cases.
Yes, this formula requires equally spaced time intervals between the three observations. If your data points are unevenly spaced, you would need to use numerical curve fitting methods instead. The time unit itself can be anything (days, months, years) as long as the spacing is consistent.
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