How to Calculate Exponential Growth Rate of Population
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Exponential Population Growth Rate Calculator
Determine the continuous growth rate between two time points.
Calculation Results:
How to Calculate Exponential Growth Rate of Population
Exponential growth occurs when the growth rate of a population is proportional to its current size. Unlike linear growth, which adds a fixed amount every period, exponential growth accelerates over time. This is commonly seen in microbiology, human population studies during specific eras, and compound interest models.
The Exponential Growth Formula
To find the continuous growth rate (r), we use the natural logarithm based on the standard exponential growth equation:
r = [ ln(Pt / P₀) ] / t
r: The exponential growth rate.
Pt: The population at the end of the time period.
P₀: The initial population at the start.
t: The time elapsed (measured in years, months, or days).
ln: The natural logarithm (base e).
Practical Example
Suppose a colony of bacteria starts with 500 individuals (P₀). After 6 hours (t), the population has grown to 2,000 individuals (Pt). What is the hourly growth rate?
Divide final population by initial: 2,000 / 500 = 4.
Take the natural log of 4: ln(4) ≈ 1.386.
Divide by time: 1.386 / 6 ≈ 0.231.
Result: The growth rate is approximately 23.1% per hour.
Why is this useful?
Calculating the exponential growth rate allows researchers to predict future population sizes and determine the "doubling time" (the time it takes for a population to double in size). The doubling time can be quickly estimated using the Rule of 70: divide 70 by the growth rate percentage.
function calculateGrowthRate() {
var p0 = parseFloat(document.getElementById('initialPopulation').value);
var pt = parseFloat(document.getElementById('finalPopulation').value);
var t = parseFloat(document.getElementById('timeElapsed').value);
var resultArea = document.getElementById('growthResultArea');
var resultOutput = document.getElementById('resultOutput');
if (isNaN(p0) || isNaN(pt) || isNaN(t) || p0 <= 0 || pt <= 0 || t <= 0) {
alert("Please enter valid positive numbers for all fields. Initial population and time must be greater than zero.");
return;
}
// Exponential growth rate formula: r = ln(Pt/P0) / t
var growthRatio = pt / p0;
var logRatio = Math.log(growthRatio);
var rate = logRatio / t;
// Percentage formatting
var percentageRate = (rate * 100).toFixed(4);
// Doubling time calculation: ln(2) / r
var doublingTime = (Math.log(2) / rate).toFixed(2);
resultArea.style.display = 'block';
var html = "Exponential Growth Rate (r): " + rate.toFixed(6) + "";
html += "Growth Rate as Percentage:" + percentageRate + "% per time unit.";
if (rate > 0) {
html += "Estimated Doubling Time: " + doublingTime + " units of time.";
html += "(This means the population will double every " + doublingTime + " time periods at this constant rate.)";
} else if (rate < 0) {
html = "Exponential Decay Rate: " + Math.abs(percentageRate) + "%";
html += "The population is shrinking at this rate.";
} else {
html = "The population has stayed the same. The growth rate is 0%.";
}
resultOutput.innerHTML = html;
}