Logistic Population Growth (N_t = K/(1+((K−N₀)/N₀)·e^(−rt)))

Logistic Population Growth (N_t = K/(1+((K−N₀)/N₀)·e^(−rt))) | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

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).
2The calculator evaluates N_t = K / (1 + ((K − N₀)/N₀) · e^(−r t)).
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.

n0=12000, k=10000, r=0.5, t=5, decimalPlaces=2

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Culture still climbing toward K

2. Overshoot declining back toward K

FAQ

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