How Does the Difference in Growth Rates Affect Two Exponentially Growing Bacterial Cultures?

Exponential growth is a fundamental concept in biology, especially in microbiology and ecology, where it describes how populations can rapidly increase under ideal conditions. When two bacterial cultures grow exponentially but have different intrinsic growth rates, the difference between their population sizes over time is not straightforward. This article explains how the difference between two exponentially growing populations changes, what this means for microbial competition, and why it matters for understanding population dynamics.

What Is Exponential Growth?

Exponential growth occurs when a population increases at a rate proportional to its current size. This means that as the population gets larger, it grows faster. The classic equation for exponential growth is:

Nt=N0ertNt=N0ert

where:

• NtNt = population size at time $t$

• N0N0 = initial population size

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

• $r$ = intrinsic growth rate

• $t$ = time

In exponential growth, the per capita growth rate remains constant, and the population size forms a J-shaped curve when plotted over time.

Two Bacterial Cultures with Different Growth Rates

Suppose you have two bacterial cultures, A and B, both starting at the same initial population size (N0N0), but with different intrinsic growth rates (rArA and rBrB, where rA>rBrA>rB). Their population sizes at any time $t$ are:

NA(t)=N0erAtNA(t)=N0erAtNB(t)=N0erBtNB(t)=N0erBt

The difference between the two populations at time $t$ is:

ΔN(t)=NA(t)−NB(t)=N0(erAt−erBt)ΔN(t)=NA(t)−NB(t)=N0(erAt−erBt)

This difference is not linear; instead, it grows exponentially over time, because both terms are exponential functions, and the term with the higher growth rate will dominate as time increases.

How Does the Difference Change Over Time?

Let’s analyze how the difference between the two populations behaves:

• At $t=0$: Both populations are equal, so the difference is zero.

• As $t$ increases: The population with the higher growth rate (rArA) will grow much faster than the other. The difference between the two populations increases rapidly.

• Nature of the difference: The difference between two exponential functions with different rates is itself an exponential-like function. While the exact mathematical form is more complex than a simple exponential, for practical purposes, the difference grows exponentially, not linearly.

This means that the difference between the two populations does not increase or decrease linearly, but rather increases exponentially as time passes.

Why Is This Important?

Understanding how the difference between two exponentially growing populations changes is crucial for several reasons:

• Microbial Competition: In a mixed culture, the strain with the higher growth rate will quickly outcompete the other, leading to a rapid increase in the difference between their population sizes.

• Antibiotic Resistance: If one strain is resistant to an antibiotic and grows faster, it will dominate the population exponentially, not linearly.

• Ecological Modeling: Predicting how invasive species or beneficial microbes will spread in an environment requires understanding exponential growth dynamics.

Common Misconceptions

• Linear vs. Exponential Difference: Some might think that the difference between two growing populations increases linearly, but this is only true if both populations grow at the same rate (which is not the case here).

• Decreasing Difference: The difference does not decrease unless the slower-growing population somehow overtakes the faster one, which is not possible under constant exponential growth rates.

Graph Interpretation

If you plotted the difference ($\Delta N(t)$) over time, you would see a curve that starts at zero and rises rapidly, resembling an exponential curve. This is because the faster-growing population quickly outpaces the slower one.

Real-World Example

Imagine two bacterial strains in a nutrient-rich broth. One strain has a mutation that allows it to divide every 20 minutes, while the other divides every 30 minutes. After several hours, the difference in their population sizes will be enormous, and this gap will continue to widen exponentially.

Mathematical Insight

The difference between two exponential functions with different rates can be written as:

ΔN(t)=N0(erAt−erBt)ΔN(t)=N0(erAt−erBt)

For large $t$, the term with the higher growth rate (rArA) dominates, so:

ΔN(t)≈N0erAtΔN(t)≈N0erAt

This shows that the difference grows exponentially, following the faster-growing population.

Summary Table

The difference increases rapidly, following an exponential trend.

Conclusion

When two bacterial cultures grow exponentially with different intrinsic growth rates, the difference between their population sizes increases exponentially over time. This is because the faster-growing population quickly outpaces the slower one, and the gap between them widens at an ever-increasing rate.

Correct answer:
(3) Increase exponentially