{
    "name": "Exponential Population Growth Calculator",
    "url": "https://calculators.geekmidst.com/calculators/population-growth-exponential-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/population-growth-exponential-calculator.md",
    "category": "Science & engineering",
    "description": "Project a population with exponential growth, continuous (N₀e^(rt)) or per-period ((1 + r)^t), with the doubling or halving time and a growth curve.",
    "summary": "With unlimited resources a population grows by the same percentage each period, so it doubles at regular intervals. Choose continuous or per-period compounding.",
    "also_called": [
        "Exponential growth calculator",
        "Malthusian growth calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "n0",
            "label": "Initial population N₀",
            "type": "number",
            "default": "1000"
        },
        {
            "name": "rate",
            "label": "Growth rate per period",
            "type": "number",
            "default": "2",
            "unit": "%"
        },
        {
            "name": "t",
            "label": "Number of periods",
            "type": "number",
            "default": "10"
        },
        {
            "name": "model",
            "label": "Model",
            "type": "segmented",
            "default": "continuous",
            "options": {
                "continuous": "Continuous e^(rt)",
                "discrete": "Per period (1+r)^t"
            }
        }
    ],
    "outputs": [
        {
            "key": "nt",
            "label": "Population after t periods",
            "format": "number"
        },
        {
            "key": "change",
            "label": "Change",
            "format": "number"
        },
        {
            "key": "factor",
            "label": "Growth factor N ÷ N₀",
            "format": "multiple"
        },
        {
            "key": "doubling",
            "label": "Doubling or halving time",
            "format": "text"
        },
        {
            "key": "continuous_rate",
            "label": "Equivalent continuous rate",
            "format": "percent"
        }
    ],
    "formula": {
        "text": "Continuous: N = N₀ × e^(r t);  per period: N = N₀ × (1 + r)^t;  doubling time = ln 2 ÷ k, where k = r or ln(1 + r)",
        "variables": {
            "N₀": "initial population",
            "r": "growth rate per period (as a decimal)",
            "t": "number of periods",
            "k": "equivalent continuous growth rate"
        }
    },
    "worked_example": {
        "inputs": {
            "n0": "1000",
            "rate": "2",
            "t": "10",
            "model": "continuous"
        },
        "steps": [
            "r t = 0.02 × 10 = 0.2.",
            "N = 1,000 × e^0.2 = 1,000 × 1.22140 = 1,221.40.",
            "Doubling time = ln 2 ÷ 0.02 = 0.693147 ÷ 0.02 = 34.66 periods."
        ],
        "result": "A population of 1,000 growing continuously at 2% per period reaches about 1,221 after 10 periods and doubles every 34.7 periods."
    },
    "interpretation": "Exponential growth describes the early phase of a population with plenty of resources (bacteria in fresh medium, an invasive species). Over longer spans resources run short and growth slows, which the logistic model captures.",
    "assumptions": [
        "A constant growth rate (births minus deaths, plus net migration) over the whole period."
    ],
    "limitations": [
        "Real populations are limited by resources, so long projections overstate growth. Use the logistic model when a carrying capacity matters."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "What is the rule of 70?",
            "answer": "Doubling time ≈ 70 ÷ growth rate in %. It comes from ln 2 ≈ 0.693; at 2% it gives 35 periods, close to the exact 34.66."
        },
        {
            "question": "Continuous or per period?",
            "answer": "Use per period when growth happens in discrete steps (annual breeding seasons); continuous for overlapping generations such as bacteria or human populations."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Logistic Growth Calculator",
            "url": "https://calculators.geekmidst.com/calculators/logistic-growth-calculator"
        },
        {
            "name": "Half-Life Calculator",
            "url": "https://calculators.geekmidst.com/calculators/half-life-calculator"
        },
        {
            "name": "Exponent Calculator",
            "url": "https://calculators.geekmidst.com/calculators/exponent-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}