How to Calculate per Capita Rate of Increase

Per Capita Rate of Increase Calculator

This calculator helps you determine the per capita rate of increase (r) of a population based on births, deaths, and initial population size over a specific time period.

function calculateRate() { var initialPopulationInput = document.getElementById("initialPopulation"); var birthsInput = document.getElementById("births"); var deathsInput = document.getElementById("deaths"); var resultDiv = document.getElementById("result"); var initialPopulation = parseFloat(initialPopulationInput.value); var births = parseFloat(birthsInput.value); var deaths = parseFloat(deathsInput.value); if (isNaN(initialPopulation) || isNaN(births) || isNaN(deaths)) { resultDiv.innerHTML = "Please enter valid numbers in all fields."; return; } if (initialPopulation <= 0) { resultDiv.innerHTML = "Initial population must be greater than zero."; return; } if (births < 0 || deaths < 0) { resultDiv.innerHTML = "Births and deaths cannot be negative."; return; } var changeInPopulation = births – deaths; var perCapitaRate = changeInPopulation / initialPopulation; resultDiv.innerHTML = "The per capita rate of increase (r) is: " + perCapitaRate.toFixed(4) + "" + "This means for every individual at the start of the period, the population changed by " + perCapitaRate.toFixed(4) + " individuals on average."; }

Understanding Per Capita Rate of Increase (r)

The per capita rate of increase, often denoted by 'r', is a fundamental concept in population ecology. It measures the rate at which a population's size changes per individual within the population, over a given period of time.

A positive 'r' indicates that the population is growing, a negative 'r' indicates that the population is shrinking, and an 'r' of zero suggests the population size is stable (births plus immigration equal deaths plus emigration, or simply births equal deaths if we ignore migration).

How is it Calculated?

The per capita rate of increase (r) can be calculated using the number of births (B) and deaths (D) within a population over a specific time interval, relative to the initial population size (N):

r = (B – D) / N

Where:

• B is the total number of births in the population during the time interval.
• D is the total number of deaths in the population during the time interval.
• N is the initial population size at the beginning of the time interval.

The term (B – D) represents the net change in population size due to births and deaths.</ Dividing this by the initial population size (N) gives the average contribution of each individual to the population change.

Interpretation

• r > 0: The per capita birth rate is higher than the per capita death rate, and the population is increasing.
• r < 0: The per capita death rate is higher than the per capita birth rate, and the population is decreasing.
• r = 0: The per capita birth and death rates are equal, and the population size is stable (zero population growth from births and deaths alone).

Example Calculation:

Imagine a population of 1000 deer at the start of the year. Over the year, 150 fawns are born, and 50 deer die.

• Initial Population (N) = 1000
• Births (B) = 150
• Deaths (D) = 50

r = (150 – 50) / 1000

r = 100 / 1000

r = 0.1

The per capita rate of increase is 0.1 per year. This means, on average, each deer contributed 0.1 individuals to the population growth over that year.

Alternative Calculation (using population change):

If you know the initial population size (N₀), the final population size (Nₜ), and the time period (t), you can also calculate r, especially if assuming exponential growth over a short period or if t=1:

If t=1 (one time unit): r = (Nₜ – N₀) / N₀

If the time period (t) is different from 1 and we assume a constant per capita rate, the relationship is more complex involving exponential growth (Nₜ = N₀ * e^rt), but the simple (B-D)/N over a period gives the average r for that period.