Logistic Growth Calculator Formula

Understand the math behind the logistic growth calculator. Each variable explained with a worked example.

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Formulas Used

Population

population = carrying / (1 + ((carrying - initial) / initial) * pow(e, -r * t))

Pct Capacity

pct_capacity = (carrying / (1 + ((carrying - initial) / initial) * pow(e, -r * t))) / carrying * 100

Inflection T

inflection_t = r > 0 ? log((carrying - initial) / initial) / r : 0

Variables

VariableDescriptionDefault
carryingCarrying Capacity (K)1000
initialInitial Population (P₀)10
rGrowth Rate (r)0.5
tTime (t)10

How It Works

Logistic Growth Model

Formula

P(t) = K / (1 + ((K - P₀) / P₀) × e^(-rt))

Where:

  • K = carrying capacity (maximum population)
  • P₀ = initial population
  • r = intrinsic growth rate
  • t = time

    • S-Curve

    Logistic growth starts exponentially, slows as it approaches the carrying capacity, and levels off. The inflection point (fastest growth) occurs at P = K/2.

    Worked Example

    Population starting at 10, carrying capacity 1000, growth rate 0.5, at time t = 10.

    carrying = 1000initial = 10r = 0.5t = 10
    1. 01P(10) = 1000 / (1 + 99 × e^(-5))
    2. 02= 1000 / (1 + 99 × 0.00674)
    3. 03= 1000 / (1 + 0.667)
    4. 04= 1000 / 1.667 ≈ 599.71

    Frequently Asked Questions

    What is logistic growth?

    Logistic growth is a model where a population grows exponentially at first, then slows as it approaches a maximum (carrying capacity). It produces an S-shaped curve.

    What is the carrying capacity?

    The carrying capacity (K) is the maximum population size that the environment can sustain indefinitely, limited by resources, space, and competition.

    Where is logistic growth used?

    Population biology, epidemiology (disease spread), market saturation models, technology adoption curves, and bacterial growth in limited environments.

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