From Census Counts to a Growth Model
A population census gives a count at a particular time. Two counts can also suggest how quickly that population has been changing. To turn those observations into a prediction, we need a model that connects population size with time.
In this tutorial, we assume the population changes at a rate proportional to its current size, with the same proportionality constant throughout the time interval being modeled. This is the exponential-growth model. It is useful when a constant-rate approximation is reasonable, but census data alone cannot guarantee that the population will continue to follow it.
We will use \(P(t)\) for the modeled population \(t\) units of time after the initial census. The time unit might be years, months, or another unit, but it must be used consistently.
The condition \(P(0)=P_0\) anchors the model at the initial census. A later census at elapsed time \(t_1>0\) supplies a second piece of information, \(P(t_1)=P_1\), which we can use to estimate \(k\).
Deriving the Model Solution
The differential equation says that the rate of change \(P'(t)\) is \(k\) times the current population. Because the initial population is positive, the exponential solution stays positive, so we can divide by \(P(t)\) and use the derivative of the natural logarithm.
For a positive solution, \(P'(t)/P(t)=k\).
Since \(\frac{d}{dt}\ln P(t)=P'(t)/P(t)\), we have \(\frac{d}{dt}\ln P(t)=k\).
Using \(P(0)=P_0\), \(\ln P(t)-\ln P_0=kt\).
The resulting formula is \(P(t)=P_0e^{kt}\).
This derivation also shows why the initial condition matters: it determines the multiplicative constant \(P_0\). We can check the formula directly. Its derivative is \(P'(t)=kP_0e^{kt}=kP(t)\), and its value at \(t=0\) is \(P(0)=P_0e^0=P_0\). Thus the formula satisfies both the differential equation and the initial condition.
Estimating the Rate from a Later Census
Suppose the initial census is \(P_0\), and a later census reports \(P_1\) after \(t_1\) time units. Substituting those data into the model gives \(P_1=P_0e^{kt_1}\). Dividing by \(P_0\), taking the natural logarithm, and dividing by \(t_1\) isolates the rate constant.
This estimate requires positive population counts and a positive elapsed time. If \(P_1>P_0\), the estimated \(k\) is positive, consistent with growth over that interval. The units of \(k\) are the reciprocal of the time unit: if \(t_1\) is measured in years, \(k\) is measured per year. Once \(k\) is estimated, the model can be used to calculate a population at another elapsed time.
Worked Examples
Worked Example: Estimate a Rate and Project a Population
A town has an initial census count of \(10{,}000\). Two years later, its census count is \(12{,}100\). Fit the exponential model and project the population five years after the initial census.
Step 1: Identify the initial and later counts and the elapsed time.
Here \(P_0=10{,}000\), \(P_1=12{,}100\), and \(t_1=2\) years.
Step 2: Estimate \(k\).
Step 3: Write the fitted model.
Step 4: Project to \(t=5\) years.
Rounded to the nearest whole person, the model projects \(16{,}105\) people. Check the model against the later census: \(P(2)=10{,}000e^{2\ln(1.1)}=10{,}000(1.21)=12{,}100\), exactly the supplied count. This verifies the fitted model at both census times.
Worked Example: Use Calendar Census Years Correctly
A city has \(80{,}000\) people in its 2018 census and \(92{,}000\) in its 2023 census. Assuming a constant exponential growth rate, estimate the population in 2028.
Step 1: Measure elapsed time from the initial census.
The initial time is 2018, so the 2023 count occurs at \(t=5\) years, and 2028 occurs at \(t=10\) years. The data are \(P_0=80{,}000\) and \(P(5)=92{,}000\).
Step 2: Estimate the rate constant.
Step 3: Evaluate the model at \(t=10\).
The model projects \(105{,}800\) people in 2028. Check the fit to the 2023 census: \(P(5)=80{,}000e^{5k}=80{,}000e^{\ln(1.15)}=80{,}000(1.15)=92{,}000\). The time values are elapsed years, not the calendar labels themselves.
Worked Example: Check a Fitted Rate Against Another Census
A population is \(40{,}000\) at \(t=0\) and \(44{,}000\) at \(t=4\) years. A census at \(t=8\) reports \(50{,}000\). Fit the model using the first two counts, then compare its year-eight prediction with the third count.
Step 1: Estimate \(k\) from the initial and year-four counts.
Step 2: Write the fitted model and verify the year-four count.
Step 3: Predict at year eight.
The additional census reports \(50{,}000\), which is \(50{,}000-48{,}400=1{,}600\) more than the model prediction. Relative to the reported count, the difference is \(1{,}600/50{,}000=0.032\), or \(3.2\%\). The fitted model reproduces the first two counts, but it does not reproduce all three. This is evidence that the constant-rate model is an approximation for these data, not a perfect description across the full interval.
A Common Pitfall: Rounding the Rate Too Early
The rate estimate in the first example is approximately \(0.0953102\) per year. If we round it to \(0.1\) before making the projection, the model changes. Using this rounded rate at five years gives
This differs from the projection of \(16{,}105.1\) obtained using the unrounded rate. The rounded rate also fails the original census check: it gives \(P(2)=10{,}000e^{0.2}\approx12{,}214\), rather than \(12{,}100\). The correction is to keep the exact value \(k=\ln(1.21)/2\), or enough digits to preserve accuracy, through the calculations and round only the final population.
What a Census-Based Model Can Establish
With two positive census counts at distinct times, the constant-rate exponential model determines a rate \(k\) that fits both counts. That does not establish that the population actually followed exponential growth between the censuses, or that it will continue to do so afterward. The model assumes one constant rate across the times being considered.
Real populations can change as conditions change. Births, deaths, migration, policy, disease, and available resources can all affect the trend. Census counts may also carry measurement uncertainty. A forecast far beyond the observed interval therefore depends strongly on whether the constant-rate assumption remains reasonable.
Additional census counts are useful for checking the model. If one rate estimated from two counts also predicts other observed counts reasonably well, the model may be a useful approximation over that period. If it misses them substantially, the discrepancy may reflect changing growth conditions, measurement differences, or other features that the simple model does not include. A good fit to the data used to estimate \(k\) is expected; it is not, by itself, an independent test of future accuracy.
Check Your Understanding
Use the model \(P(t)=P_0e^{kt}\) and the census information stated in each question.
- A population is \(25{,}000\) initially and \(27{,}500\) after five years. Write the equation for the rate constant \(k\), including its units.
- For the data in question 1, write the fitted model and give the expression for the population after eight years.
- A census is taken in 2016 and another in 2020. If the first census is the initial time, what elapsed time should be used for the second census?
- Why does a model that matches two census counts not necessarily predict a later count accurately?
- What does a substantial difference between a model projection and an additional census suggest about the model's assumptions or fit?
