Two Ways to Describe Population Change
A population can grow by a certain number of people per year, but that number alone does not tell the whole story. An increase of 500 people per year is substantial for a population of 10,000, but proportionally smaller for a population of 100,000. To describe change clearly, distinguish the population’s absolute growth rate from its relative growth rate.
Let \(P(t)\) denote the population at time \(t\), with \(P\) differentiable at the time under discussion. The derivative \(P'(t)\) describes the population’s instantaneous change: a positive value means the population is increasing, a negative value means it is decreasing, and zero means its instantaneous rate of change is zero.
The derivative is an instantaneous rate, not a population count. For a short time interval \(\Delta t\), the change in population is approximately \(P'(t)\Delta t\). The relative rate describes that change in relation to the population currently present. Both quantities depend on the time at which they are evaluated; either may change as the population and its rate of change evolve.
Why the Relative Rate Is a Quotient
Over a short interval from \(t\) to \(t+h\), the population change is \(P(t+h)-P(t)\). Dividing by the initial population \(P(t)\) gives the change as a fraction of the current population. Dividing by the time interval \(h\) gives the fractional change per unit time. The instantaneous version follows by taking the limit as the interval shrinks to zero.
The second equality uses the definition of the derivative and the fact that \(P(t)\) is a fixed, nonzero number as \(h\) approaches zero. Multiplying the resulting fractional rate by 100 expresses it as a percentage rate. For example, a relative rate of \(0.04\) per year is a percentage growth rate of \(4\%\) per year.
This reasoning also explains why the current population must be positive for the relative rate to be defined. If \(P(t)=0\), division by \(P(t)\) is impossible. A model may still have a meaningful absolute derivative at such a time, but its relative growth rate, as defined here, does not exist.
Worked Example: Constant Absolute Growth
Worked Example: Constant Absolute Growth
Suppose a population is modeled by \(P(t)=12{,}000+800t\), where \(t\) is measured in years and \(P(t)\) is measured in people. Find its absolute and relative growth rates at \(t=3\).
First calculate the population at that time:
Differentiate the model to obtain the absolute rate:
The relative rate is the absolute rate divided by the current population:
As a percentage rate, this is \(100(1/18)\%=5.555\ldots\%\) per year, or approximately \(5.56\%\) per year to two decimal places. The model has a constant absolute rate, but the relative rate is not constant: as the population grows, the same gain of 800 people per year is a smaller fraction of the population.
As a check, the model’s change over an interval \(h\) is \(P(3+h)-P(3)=800h\). Dividing by \(h\) gives \(800\) people per year, agreeing with the derivative. Dividing that rate by the population of \(14{,}400\) gives the relative rate calculated above.
Worked Example: An Absolute Rate That Changes
Worked Example: An Absolute Rate That Changes
Suppose \(P(t)=5{,}000+200t+10t^2\), where \(t\) is in years and \(P(t)\) is in people. Find both rates at \(t=4\).
Evaluate the population:
Differentiate each term and evaluate the derivative:
Thus the relative rate is
Multiplying by 100 gives \(\frac{700}{149}\%\) per year, approximately \(4.70\%\) per year to two decimal places. To check the derivative directly, expand the change over an interval \(h\):
As \(h\) approaches zero, \(280+10h\) approaches \(280\), confirming the absolute rate. Dividing this rate by the population at \(t=4\) gives the relative rate above.
Worked Example: Equal Absolute Rates, Different Relative Rates
Worked Example: Equal Absolute Rates, Different Relative Rates
At a particular time, Population A has 10,000 people and is increasing at 500 people per year. Population B has 50,000 people and is also increasing at 500 people per year. Compare their relative and percentage growth rates.
For Population A:
For Population B:
Both populations have the same absolute growth rate, 500 people per year. Population A has the larger relative rate because that increase is a larger fraction of its current population. The calculations are consistent: \(0.05(10{,}000)=500\) and \(0.01(50{,}000)=500\) people per year.
Reading the Sign and Units
The sign of the absolute rate indicates the direction of population change. When \(P'(t)>0\), the population is growing at that instant; when \(P'(t)<0\), it is declining; and when \(P'(t)=0\), its instantaneous rate is zero. If \(P(t)>0\), dividing by the positive population preserves the sign, so the relative rate has the same sign as the absolute rate.
Units are an important check on interpretation. If population is measured in people and time in years, then \(P'(t)\) is measured in people per year. In the quotient \(P'(t)/P(t)\), “people” cancels, leaving per year. Multiplying by 100 changes the numerical expression into a percentage rate; it does not remove the time unit. A relative rate of \(0.03\) per year therefore means \(3\%\) per year, not simply \(3\%\).
| Quantity | Meaning | Units when time is in years |
|---|---|---|
| \(P(t)\) | Population at time \(t\) | People |
| \(P'(t)\) | Instantaneous absolute change | People per year |
| \(\frac{P'(t)}{P(t)}\) | Instantaneous change relative to population size | Per year |
| \(100\frac{P'(t)}{P(t)}\%\) | Relative change expressed as a percentage rate | Percent per year |
Common Mistake: Confusing People per Year with Percent per Year
Suppose a population of 5,000 people is increasing at 250 people per year. A common mistake is to report the absolute rate, 250, as a percentage rate, or to claim that the population is growing at \(250\%\) per year. That confuses a count-per-time quantity with a fraction-per-time quantity.
The correction is to divide by the current population before converting to a percentage:
The result has the right units and can be checked by reversing the calculation: \(0.05\text{ per year}\times5{,}000\text{ people}=250\text{ people per year}\). Always identify the quantity being requested and include its units.
Why Both Rates Matter
Absolute and relative rates answer different questions. The absolute rate asks how many people are being added or lost per unit of time. The relative rate asks how large that change is compared with the current population. The first is useful when the number of people affected is central; the second is useful when comparing populations of different sizes or describing proportional change.
For a short time interval \(\Delta t\), the derivative gives the approximation \(P(t+\Delta t)-P(t)\approx P'(t)\Delta t\). Dividing by \(P(t)\) gives the approximate fractional change over that short interval, and multiplying by 100 gives its percentage form. These are local approximations, not guarantees that the same rate will continue over a long interval. If the rate or population changes substantially, evaluate the rates at the relevant time rather than treating one instantaneous value as a constant over the whole period.
Check Your Understanding
Use the definitions and units in each question. Give percentage rates per unit of time where appropriate.
- A population is 24,000 people and its derivative is 720 people per year. What are its relative and percentage growth rates?
- Why can two populations with the same absolute growth rate have different relative growth rates?
- If \(P(t)>0\) and \(P'(t)<0\), what are the signs of the absolute and relative rates, and what do they mean?
- A model has \(P(t)=0\) at a particular time. Which rate in this tutorial’s definition is not defined there, and why?
- A population is 8,000 people and grows at 2.5% per year. What absolute growth rate does that percentage rate correspond to at that instant?
