Population Growth Calculator

Population Growth Guide

Exponential vs Logistic Growth

Exponential growth: dN/dt = rN. Population doubles at a constant rate, producing a J-shaped curve. Requires unlimited resources — occurs early in population establishment or in lab cultures. Logistic growth: dN/dt = rN × (K−N)/K. Growth rate slows as population approaches carrying capacity (K), producing an S-shaped (sigmoid) curve. The (K−N)/K term represents the 'unused capacity' — as N approaches K, growth rate approaches zero. Real populations typically follow logistic or more complex models.

Doubling Time and Rule of 70

Doubling time = ln(2) / r ≈ 0.693 / r. The Rule of 70 approximates this: doubling time ≈ 70 / (r as a percentage). For r = 0.02 (2% per year): doubling time ≈ 70/2 = 35 years. This applies to any exponential growth: bacteria, human populations, investment compound growth, carbon dioxide levels. The rule of 70 is the same calculation used in finance for compound interest doubling time — the mathematics of exponential growth is identical across disciplines.

Carrying Capacity and Limiting Factors

Carrying capacity (K) is the maximum population size an environment can support sustainably. Limiting factors that determine K: food and water availability, space and territory, disease and predation, waste accumulation. When N approaches K, competition intensifies, birth rates fall, death rates rise. The logistic model assumes K is constant — in reality, K can change with environmental conditions (drought reduces K; agricultural expansion increased human K significantly). Human K is debated: es

S-Curves in the Real World

The logistic S-curve appears throughout biology and beyond. Bacterial cultures: lag phase (slow start), exponential phase (J-curve), stationary phase (K reached), death phase (resources exhausted). Technology adoption: slow early adoption, rapid growth as critical mass reached, saturation as most potential users adopt. Epidemic spread: initial exponential growth, slowing as susceptible population reduces (this is the R₀ concept). Wildlife management: fisheries use logistic models to set sustaina