How to Calculate Doubling Time for an Exponentially Growing Population

Population doubling time is a fundamental concept in ecology, microbiology, and demography, describing how quickly a population can double in size under ideal, resource-rich conditions. This article explains what doubling time means, how it is calculated for populations growing exponentially, and why this measure is important for understanding and managing population dynamics.

What Is Doubling Time?

Doubling time is the period required for a population to double in size, assuming a constant growth rate. This concept is widely used in biology to describe the rapid growth of organisms such as bacteria, in finance to estimate investment growth, and in demography to project human population trends. Doubling time is especially relevant for populations undergoing exponential growth, where resources are unlimited and the environment is stable.

Exponential Growth and Doubling Time

Exponential growth occurs when a population increases at a rate proportional to its current size. The standard equation for exponential population growth is:

N(t)=N0ertN(t)=N0ert

where:

• $N(t)$ is the population size at time $t$,

• N0N0 is the initial population size,

• $r$ is the per capita growth rate (intrinsic rate of increase),

• $e$ is the base of the natural logarithm (~2.718),

• $t$ is time.

Doubling time is the value of $t$ when the population has doubled, i.e., when N(t)=2N0N(t)=2N0.

Deriving the Doubling Time Formula

To find the doubling time, set N(t)=2N0N(t)=2N0:

2N0=N0ert2N0=N0ert

Divide both sides by N0N0:

2=ert2=ert

Take the natural logarithm of both sides:

$$\ln (2)=rt$$

Solve for $t$:

t=ln⁡(2)rt=rln(2)

Thus, the doubling time (tdoubletdouble) is:

tdouble=ln⁡(2)rtdouble=rln(2)

Since $\ln (2)\approx 0.693$, the formula is often written as:

tdouble≈0.693rtdouble≈r0.693

Alternatively, if the growth rate $r$ is given as a percentage (e.g., 5% per year), the Rule of 70 is commonly used for approximation:

tdouble≈70r%tdouble≈r%70

where r%r% is the growth rate in percent.

Evaluating the Given Options

Let’s review the options provided for the doubling time of a population growing exponentially with growth rate $r$:

1. N0×2N0×2

   • Interpretation: This is simply twice the initial population, not a time period.

   • Incorrect.

2. ln⁡(2)rrln(2)

   • Interpretation: This is the correct formula for doubling time in exponential growth.

   • Correct.

3. $\lambda \ln (2)$

   • Interpretation: This is not a standard formula for doubling time. $\lambda$ is sometimes used in physics or chemistry for decay rates, but not here.

   • Incorrect.

4. $\ln (r)\times 2$

   • Interpretation: This is not a recognized formula for doubling time in population ecology.

   • Incorrect.

Why Is Doubling Time Important?

Doubling time is a useful metric for:

• Population management: Predicting how quickly a population will grow under optimal conditions.

• Conservation: Estimating the potential for invasive species to spread or for endangered species to recover.

• Public health: Understanding the spread of infectious diseases or the growth of tumor cells.

• Resource planning: Forecasting future resource needs based on population projections.

Real-World Examples

• Bacteria in a laboratory: Under ideal conditions, E. coli can double every 20–30 minutes, reflecting a very high growth rate.

• Human populations: With a growth rate of 2% per year, the doubling time is approximately 35 years (using the Rule of 70: $70/2=35$).

• Investments: A 7% annual return will double an investment in about 10 years ($70/7=10$).

Common Misconceptions

• Doubling time is not simply twice the initial population: It is a time period, not a population size.

• Doubling time depends only on the growth rate: The initial population size does not affect the doubling time in exponential growth.

• Doubling time is only meaningful for exponential growth: In logistic growth (where resources are limited), the concept of doubling time becomes less relevant as the population approaches its carrying capacity.

Summary Table

Conclusion

For a population growing exponentially with growth rate $r$, the doubling time is given by:

tdouble=ln⁡(2)rtdouble=rln(2)

This formula allows you to estimate how quickly a population will double in size under constant growth conditions.

Correct answer:
(2) ln⁡(2)rrln(2)