Logistic Population Growth (N_t = K/(1+((K−N₀)/N₀)·e^(−rt)))
Compute Verhulst logistic growth toward carrying capacity K, with instantaneous dN/dt and capacity percentage.
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Tool usage guide
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Key facts
Category
Health
Input types
number
Output type
json
Sample coverage
4
API ready
Yes
Overview
Compute Verhulst logistic growth N(t) toward carrying capacity K from initial size N₀, intrinsic rate r, and time t. The result also reports the share of K reached and the instantaneous rate dN/dt = r·N·(1−N/K), with a short note on whether growth is still accelerating, near its peak, or declining after overshoot.
When to use
When a population or culture is limited by a finite carrying capacity K rather than unbounded exponential growth.
When you need N(t), the percent of K reached, and whether dN/dt is still near its maximum around K/2.
When N₀ already exceeds K and you need the decline trajectory back toward carrying capacity.
How it works
1Enter initial population N₀, carrying capacity K, intrinsic growth rate r (must be greater than 0), and elapsed time t in the same time unit as r. Optionally set decimal places (0–8, default 2).
3It then computes the fraction of K and instantaneous dN/dt = r · N · (1 − N/K), and labels the phase (for example fastest growth when N is near K/2, or declining if N is above K).
Use cases
Predict cell counts in a nutrient-limited batch culture and see remaining headroom to K.
Project wildlife or fishery numbers under a known habitat carrying capacity.
Illustrate sigmoid growth, peak dN/dt near K/2, and overshoot decline in ecology or math-biology courses.
Examples
1. Culture still climbing toward K
Microbiology lab technician
Background
A starter of 100 cells is grown toward a nutrient-limited carrying capacity of 10,000, with r = 0.5 per hour.
Problem
Need size after 10 hours, how full the vessel is, and whether instantaneous growth is still near its peak.
How to use
Set Initial population (N₀) to 100, Carrying capacity (K) to 10000, Intrinsic growth rate r to 0.5, Elapsed time (t) to 10, and Decimal places to 2.
n0=100, k=10000, r=0.5, t=10, decimalPlaces=2
Outcome
N(10) = 5998.6 — 60% of carrying capacity. Instantaneous dN/dt ≈ 1200.14 per hour. Fastest growth — N near K/2.
2. Overshoot declining back toward K
Ecologist
Background
A population of 12,000 already exceeds a habitat carrying capacity of 10,000, with r = 0.5 per time unit.
Problem
Project size after 5 time units and confirm it is falling toward K.
How to use
Enter Initial population (N₀) 12000, Carrying capacity (K) 10000, Intrinsic growth rate r 0.5, and Elapsed time (t) 5.
FAQ
Which formula does this calculator use?
The Verhulst logistic equation N_t = K/(1+((K−N₀)/N₀)·e^(−rt)).
How is the instantaneous growth rate calculated?
dN/dt = r·N·(1−N/K). That rate peaks when N is near K/2.
Do r and t need the same time unit?
Yes. Use matching units for r and t (hours, days, years, and so on). The classic model also requires r > 0.
What if the starting population is already above K?
N(t) declines toward K and dN/dt is negative.
How does this differ from exponential growth?
When N is much smaller than K, logistic growth is nearly exponential. As N approaches K, growth slows and saturates at K.