Logistic Growth Calculator
Growth with Limits
Biological populations rarely grow without bound. In the real world, factors such as limited resources and space constrain expansion. The logistic growth model captures this idea by incorporating a carrying capacity representing the maximum sustainable population. Initially, growth may appear exponential, but as the population approaches the carrying capacity, the growth rate slows, eventually leveling off.
The Logistic Equation
The logistic model expresses population over time using the formula
, which contains the initial population Pā, the carrying capacity K, and the intrinsic growth rate r. As time increases, the exponential term decays toward zero, leaving P(t) approaching K.
Using the Calculator
Enter the initial population size, carrying capacity, growth rate, and time. The script evaluates the logistic formula to estimate P(t). By adjusting the inputs, you can simulate scenarios ranging from bacteria in a petri dish to animals in a wildlife preserve. The calculator highlights how sensitive growth is to environmental limits and starting conditions.
Phases of Growth
The logistic curve has three distinct phases. First comes the lag phase, where growth is slow as the population acclimates. Next is the exponential phase, during which the population increases rapidly due to abundant resources. Finally, the plateau phase emerges as resources become scarce, leading to a steady state near the carrying capacity.
Example Scenarios
The table below demonstrates how different parameter sets shape the outcome after a fixed time horizon of ten units.
| Pā | K | r | P(10) | Context |
|---|---|---|---|---|
| 100 | 1,000 | 0.3 | 739 | Recovering wildlife population |
| 5,000 | 10,000 | 0.15 | 8,195 | Growing subscriber base |
| 50 | 500 | 0.6 | 494 | Rapid bacterial culture |
Contrast with Exponential Growth
Exponential growth, described by the simpler formula , lacks a carrying capacity. It predicts unrestricted increase, which is unrealistic for most natural settings. Comparing logistic and exponential models demonstrates how adding K dramatically changes long-term predictions.
Beyond Biology
Logistic growth also describes technology adoption, epidemiological curves, and market saturation. Marketers, epidemiologists, and conservationists use the same underlying equation to forecast how processes evolve in the presence of constraints.
Dive deeper into time-dependent models with the Exponential Growth & Decay Calculator, Population Projection Calculator, and the SIR Epidemic Model Calculator.