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

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